APPLYING NEWTON S LAWS

5-1

APPLYING NEWTON’S LAWS 5 Q5.1. Reason: For an object to be in equilibrium, the net force (i.e., sum of the forces) must be zero. Assume that the two forces mentioned in the question are the only ones acting on the object. The question boils down to asking if two forces can sum to zero if they aren’t in opposite directions. Mental visualization shows that the answer is no, but so does a careful analysis. Set up a coordinate system with the x-axis along one of the forces. If the other force is not along the negative x-axis then there will be a y (or z) component that cannot be canceled by the first one along the x-axis. Assess: In summary, two forces not in opposite directions cannot sum to zero. Neither can two forces with different magnitudes. However, three can.

Q5.2. Reason: Objects in static equilibrium do not accelerate and remain at rest. Objects in dynamic equilibrium do not accelerate and move with constant velocity. (a) The girder is moving at constant speed. We assume it’s being lifted straight up. If so, it’s in dynamic equilibrium. (b) Since the girder is slowing down it is accelerating, and therefore not in static or dynamic equilibrium. (c) Since the barbell is not accelerating and remains at rest it is in static equilibrium. (d) Once the plane has reached its cruising speed and altitude the plane is moving with constant speed and direction. It is in dynamic equilibrium. (e) A rock in free fall is accelerating due to gravity. It is not in equilibrium. (f ) The box is accelerating since the truck is accelerating and the box is not sliding relative to the truck. The box is not in equilibrium. Assess: For an object in equilibrium net 0.F =

Q5.3. Reason: Assume you are sitting in a chair, and that you are at rest. (Parts of your body may be moving, but if you model your body as a particle, then you generally aren’t moving much as you read.) The two forces that act on you are the gravitational force of the earth on you, directed down; and the normal force of the chair pushing up on you. These two forces are equal in magnitude and opposite in direction and so the sum (the net force) is zero. Assess: When you aren’t accelerating Newton’s second law says you experience a zero net force. So this analysis would apply equally to the case of you reading this while sitting in a chair on a smoothly moving (constant velocity) train.

Q5.4. Reason: Weight is due to the pull of gravity on an object. Mass is a measure of an object’s tendency to resist being accelerated. (a) The mass of an object is independent of location. Whether the object is on the earth or the moon or in space away from sources of gravity, its mass is the same. (b) The weight of an object depends on the acceleration of gravity at the location. The weight of an object on the moon is less than the weight of an object on the earth. (c ) Mass and weight are not the same thing in different units. Mass does not change depending on location, as in part (a), while weight does, as in part (b). Assess: Mass is an intrinsic property of an object. Weight depends on the acceleration due to gravity and so is not an intrinsic property of an object.

Q5.5. Reason: The reading on the moon will be the moon-weight, or the gravitational force of the moon on the astronaut. This would be about 1/6 of the astronaut’s earth-weight or the gravitational force of the earth on the astronaut (while standing on the scales on the earth).

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Assess: Some physicists and textbooks define weight to be “what the scale reads” in which case it will read the astronaut’s weight on the moon—by definition. But it won’t read the same weight as on the earth. While not all physicists agree on the best definition of “weight” our textbook uses a very standard and reasonable approach. The astronaut’s mass does not change by going to the moon.

Q5.6. Reason: The rope and two rocks are in free fall. Consider the following free-body diagrams.

(a) Considering mass M first, Newton’s second law gives

M MyT w Ma− − =

Since the block is in free fall, the acceleration of the rock is aMy = –g. The weight of the rock is wM = Mg. Solving the equation for T, we have

0 NM MyT w Ma Mg Mg− = + = − =

The free-body diagram for block m along with Newton’s second law gives

m myT w Ma− =

With aMy = –g and wM = mg. Solving for T gives

0 Nm myT w ma mg mg= + = − =

This also follows from the fact that the tension in a rope is the same all along the rope. (b) If the two masses are reversed, the results will be the same. The blocks and the rope are all in free fall. They all have the same acceleration, so there will be no tension in the rope. Assess: This answer makes sense. The blocks are accelerating at the same rate. For there to be a tension in the rope, the two rocks would have to be accelerating at different rates.

Q5.7. Reason: If we ignore air resistance as instructed, then the balls are in free fall. The net force in free fall is simply the force of gravity, which is the mass times g. Therefore, a ranking of the net forces is the same as a ranking of the masses, since g is constant for all four balls.

net 4 net 2 net 3 net 1F F F F> = >

Assess: The velocities are irrelevant to the net force. Note that although all four balls are in free fall (and have the same acceleration), they do not experience the same gravitational force.

Q5.8. Reason: You will pour the correct amount. The apparent weight of each object has increased due to the motion of the elevator, but each has increased by the same amount. The pan balance measures mass, not weight. Assess: On a spring scale, the increase in the apparent weight of the salt will lead you to pour too little.

Q5.9. Reason: The normal force (by definition) is directed perpendicular to the surface. (a) If the surface that exerts a force on an object is vertical, then the normal force would be horizontal. An example would be holding a picture on a wall by pushing on it horizontally. The wall would exert a normal force horizontally. (b) In a similar vein, if the surface that exerts a force on an object is horizontal and above the object, then the normal force would be down. One example would be holding a picture on a ceiling by pushing on it. The ceiling would exert a normal force vertically downward. Another example would be the Newton’s third law pair force in the case of you sitting on a chair; the chair exerts a normal force upward on you, so you exert a normal force downward on the chair.

Applying Newton’s Laws 5-3

Assess: We see that the normal force can be in any direction; it is always perpendicular to the surface pushing on the object in question.

Q5.10. Reason: During the upward part of the motion, the drag force and weight of the ball both act in the downward direction. During the downward part of the motion, the drag force acts upward while the weight of the ball acts downward. The net force accelerating the ball during the upward motion is greater than during downward motion, so the ball takes a shorter time on the upward part of the trip. Assess: The force of air drag always acts opposite the direction of motion of an object.

Q5.11. Reason: Use the simple model in Section 5.6 and assume that

214

D Avρ≈

For object 1: 20.20 m 0.30 m 0.060 m ;A = × = 2 2 2 2(6 m/s) 36 m /s ;v = = so 2 4 22.2 m /sAv = For object 2: 20.20 m 0.20 m 0.040 m ;A = × = 2 2 2 2(6 m/s) 36 m /s ;v = = so 2 4 21.4 m /sAv = For object 3: 20.30 m 0.30 m 0.090 m ;A = × = 2 2 2 2(4 m/s) 16 m /s ;v = = so 2 4 21.4 m /sAv = The density of air ρ is the same for all three objects, so it won’t affect the ranking. Therefore, 1 2 3.D D D> = Assess: Note that because v is squared, object 3’s greater cross-sectional area did not produce the largest drag force.

Q5.12. Reason: The skydiver is falling with a constant velocity just before she opens her parachute. At this point, the drag force on the diver is equal to her weight. When she opens her parachute, her effective area is increased, so this increases the drag force on the diver. This will cause a net force on the diver in the upward direction, which is greater than her weight and will decelerate her. Assess: This makes sense. The parachute slows the diver down further.

Q5.13. Reason: Raindrops would all have the same acceleration if they were in free fall, but if some started falling higher or earlier than others they could hit the ground at different speeds. However, raindrops are not in free fall; the air resistance is a significant factor. We can assume that the drops reach terminal speed before hitting the ground. Bigger drops experience a greater downward gravitational force than small drops because they have more mass. However, bigger drops also experience a greater upward air resistance drag force because their cross-sectional area is larger. But these two effects do not grow at the same rate; the mass (and hence the downward force) grows with r 3, while the cross-sectional area grows with r 2. Therefore, as the size of drops increases, the downward force grows faster than the upward force. So larger, more massive drops fall faster than smaller drops. Assess: Air resistance makes all the difference. If there weren’t any, all drops would have the same acceleration and they would be going very fast when they hit the ground. Rain drops generally do reach terminal speed, but not every drop has the same terminal speed. Larger drops have greater terminal speed.

Q5.14. Reason: As the plane’s thrust is decreased, the plane will start to decelerate. From Equation 5.14, as the plane’s velocity decreases so does the drag force on the plane. The plane’s velocity will decrease until the drag force equals the thrust force. At that point, the plane will stop decelerating as it reaches a new equilibrium. If the drag force continued decreasing, the thrust force would re-accelerate the plane to the point where it would stop accelerating, again reaching the same equilibrium. Therefore the plane will travel with a constant velocity once the new equilibrium is reached. Assess: Drag force decreases with decrease in velocity.

Q5.15. Reason: If you only consider objects dropped from rest and accelerating up to terminal speed you might think that is the maximum speed the object can go through the air. However, terminal speed is merely the condition when the gravitational force and the air resistance force have the same magnitude and sum to zero. That doesn’t necessarily mean it is the fastest possible speed. It would be quite possible to throw or fire an object straight down from a high cliff at greater than terminal speed. The higher speed would mean that the upward drag force is greater than the downward gravitational force, so the net force would be up, the acceleration would be up, and the object would slow down to terminal speed, at which time the forces would cancel and the downward velocity would be constant.

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Assess: The direction isn’t crucial either; and object can also go up at faster than terminal speed, but the important aspects of the issue are most easily shown in the case above. It is also possible to start from rest and speed up past terminal speed if there is another force which makes the net force non-zero and so acceleration can continue; an example of this would be a rocket-powered missile.

Q5.16. Reason: See the free-body diagrams below.

The object is in static equilibrium. Newton’s second law for the 5 kg box gives 0 N.y yT mg F ma− = = = Then .T mg= Because ,T mg= the spring scale reads the weight of the box, or 49 N. Assess: This makes sense, since the object has a mass of 5 kg.

Q5.17. Reason: The tension is 49 N. It reads the same as it would if the rope were attached to the ceiling. The role of the five kilogram mass on the left is to keep the system in equilibrium, but it doesn’t make the tension more than 49 N. Assess: Apply Newton’s second law to the mass on the right; the upward tension in the rope must equal the downward force of gravity. The scale reads the tension in the rope.

Q5.18. Reason: Consider the free-body diagrams below.

Since each object in the system is in static equilibrium, we can apply Equation 5.1. For m1, T1 = w1. For m2, T2 = w2. The pulley is massless, so it has no weight. The only forces on the pulley are the three tensions in the diagram, so T = T1 + T2. The tension in the rope is the sum of the tensions in the ropes directly connected to the objects, which is just the sum of their weights from the equations above. The tension is 98 N. Assess: This makes sense, since the scale is effectively supporting two 5 kg objects.

Q5.19. Reason: This question is very similar to Question 5.17. The tension is 49 N. It is the same as if the rope were attached to a wall on the left instead of the rope that goes over the left pulley. The role of the five kilogram mass on the left is to keep the system in equilibrium, but it doesn’t make the tension more than 49 N. Assess: Apply Newton’s second law to the mass on the right; the upward tension in the rope must equal the downward force of gravity. The pulley (in our simple model) merely changes the direction of the force.

Q5.20. Reason: The kinetic friction acts in a direction to oppose the relative motion, so on block 1 the kinetic friction is to the right and on block 2 it is to the left. Assess: We would expect them to be opposite since they are a Newton third law pair and the forces in a third law pair are always in opposite directions.

Q5.21. Reason: In this case there is not enough information to tell, because we don’t know which way the block would go if the friction were reduced. Think of extreme cases to see this. If block 1 were much, much


more massive than block 2 it would slide down the ramp if friction were reduced sufficiently; in that case (if the friction weren’t reduced) the static would have to be up the ramp to hold block 1 there. On the other hand, if block 2 were much, much more massive than block 1 then block 1 would slide up the ramp if friction were reduced sufficiently; in that case (if the friction weren’t reduced) the static friction would have to be down the ramp to hold block 1 there. Since we don’t know the masses we don’t know which extreme case is closer to our situation. So the answer is D. Assess: By examining limiting cases we get a good feel for the situation. It looks like block 1 is more massive than block 2, but we aren’t told, and there isn’t enough information to decide which way it would slide if friction were reduced.

Q5.22. Reason: The ball is in equilibrium. We will use Equation 5.1. See the free-body diagram below.

In the vertical direction we have

sin(50 ) sin(50 ) 0T w T mg° − = ° − =

Solving for T, we obtain 2(2.0 kg)(9.80 m/s ) 26 N

sin(50 ) sin(50 )mgT = = =

° °

The correct choice is D. Assess: Note that we did not need to use the horizontal components of the forces.

Q5.23. Reason: We will employ Newton’s third law in part (a) and Newton’s second law in part (b). (a) Newton’s third law says that if object A (you) exerts a force on object B (the ceiling), then object B (the ceiling) exerts a force with equal magnitude and opposite direction on object A (you). Therefore the ceiling exerts a force of 100 N on you. The correct choice is B. (b) This part is trickier; we must use the fact that while standing still, you are in equilibrium (i.e., the net force on you is zero). The individual forces on you are: the downward gravitational force of the earth on you (your weight), the downward force of the ceiling on you (which we just found to be 100 N), and the upward force of the floor on you (which we want to know). These must sum to zero. In other words the magnitude of the upward force of the floor must equal the sum of the magnitudes of the two downward forces, 690 N (your weight) and 100 N. The correct choice is D. Assess: Especially note that in part (b) the magnitude of the force of the floor on you is not the same as the magnitude of the earth’s gravitational force on you, as it would have been if you hadn’t been pushing on the ceiling.

Q5.24. Reason: We will use Equation 5.2 since neither the dog nor the floor is in equilibrium. (a)

From the free-body diagram above, we have .yn w ma− = Solving for the normal force,

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2 2(5.0 kg)(9.80 m/s ) (5.0 kg)( 1.20 m/s ) 43 Ny yn w ma mg ma= + = + = + − =

The correct choice is B. (b) The normal force on the dog is the force of the floor of the elevator on the dog. The force of the dog on the elevator floor is the reaction force to this. The correct choice is D. Assess: This result make sense, the normal force will be less than the weight of the dog, which is 49 N.

Q5.25. Reason: We must remember the east-west coordinate is independent of the north-south coordinate. The eastward component of velocity (4.5 m/s) will remain constant. (a) We treat the northward component of the motion as a constant acceleration problem. First we use =F ma to solve for a = F/m = (6.0 N) (3.0 kg) − 2.0 m/s2. Then we use = ,v a t∆ ∆ remembering that i 0.0 m/s.v = So the northward component of the velocity is

2f (2.0 m/s )(1.5s ) 3.0 m/sv a t= ∆ = =

The correct choice is C. (b) We have a northward component of 3.0 m/s and an eastward component of 4.5 m/s.

2 2 2 2north east( ) ( ) (3.0 m/s) (4.5 m/s) 5.4 m/sv v v= + = + =

The correct choice is B. Assess: This question is reminiscent of projectile motion problems with a constant velocity in one direction and a constant acceleration in a perpendicular direction. A puck with a mass of 3 kg is quite heavy, much more so than a hockey puck.

Q5.26. Reason: If the rocket’s thrust is constant, the force on the rocket will be constant. However, the rocket’s mass is decreasing as it burns fuel. The rocket’s acceleration should increase as it burns fuel due to Newton’s second law. The graph should reflect small accelerations in the beginning and larger accelerations as the rocket moves. The slope of the velocity-versus-time graph gives the acceleration. The correct choice is D, where the slope is small at the beginning and large near the end. Assess: Note that choice B represents constant acceleration, which is not correct. Choice C reflects a large acceleration at the beginning of flight and a small acceleration at the end, which also doesn’t match the motion.

Q5.27. Reason: This is still a Newton’s second law question; the only twist is that the object is not in equilibrium, i.e., the right side of the second law is not zero. The forces on Eric are the downward gravitational force of the earth on him w, and the upward normal force of the scale on him n (which we want to know). We note that a = –1.7 m/s2 and w = mg = (60 kg)(9.8 m/s2) = 5.88 N. This is a one-dimensional question in the vertical direction, so the following equations are all in the y-direction.

net

2(60 kg)( 1.7 m/s ) 588N 486N 500N

F man w ma

n ma w

=− =

= + = − + = ≈

The correct choice is C. Assess: Because the elevator is accelerating down, we expect the scale to read a bit less than Eric’s normal weight. This is the case. It is important that neither the question nor the answer specify whether the elevator is moving up or down. The elevator can be accelerating down in two ways: It can be moving up and slowing (such as the end of a trip from a low floor to a high floor), or it can be moving down and gaining speed (such as the beginning of a trip from a high floor to a low floor). The answer is the same in both cases.

Q5.28. Reason: For the two blocks to remain stationary, they must be in static equilibrium. We will use Equation 5.1. Refer to the figure below. We label the block on the left Block 1 and the block on the right Block 2.


For the 10 kg block,

21 1sin( ) (10 kg)(9.80 m/s )sin(23 ) 38 NT m g θ= = ° =

For the block on the right,

2 2sin( ) 38 NT m g θ= =

Solving for the mass of the second block,

2 22

38 N 6.1 kgsin( ) (9.80 m/s )sin(40 )

Tmg θ

= = =°

The correct choice is B. Assess: This makes sense, since the angle of incline of the second block is much greater.

Q5.29. Reason: We will assume a constant direction so that plus the “constant speed” means no acceleration. The sled is in equilibrium and the net force on it must be zero. In the horizontal direction there are two forces on the sled: the football player pushing on it, and kinetic friction acting in the opposite direction. These two must have the same magnitude. Equation 5.11 tells us that fk = µk n, but we don’t yet know n. Independently analyzing the vertical direction reveals that the magnitude of n is the same as the magnitude of w = mg = (60 kg)(9.8 m/s2) = 590 N. So the kinetic friction force is fk = µkn = (0.30)(590 N) = 180 N. And that must also be the magnitude of the football player’s pushing force. The correct answer is C. Assess: Choices A and B don’t seem very strenuous for a football player, but choice D seems like too much. Choice C is in the right range.

Q5.30. Reason: Friction will slow down and stop the sled once the players stop pushing. The only horizontal force on the sled while it is slowing down is the force of kinetic friction. See the diagram below.

In the vertical direction, Equation 5.1 gives .n w= The force of kinetic friction is given by Equation 5.11.

k k kf n mgµ µ= =

The net force in the horizontal direction is net k.F f=

We can find the acceleration of the sled using Newton’s second law.

2 2k kk (0.30)(9.80 m/s ) 2.94 m/sx

f mga gm m

µ µ− −= = = − = − = −

Additional significant figures have been retained in this intermediate calculation.

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We can find how far the sled slides before stopping using kinematic equations. We have the initial velocity of the sled is vi = 2.0 m/s. The final velocity of the sled is vf = 0.0 m/s. Using Equation 2.13 and solving for ∆x,

2 2i

2

( ) (2.0 m/s) 0.68 m2 2(2.94 m/s )x

vxa

∆ = = =

The correct choice is B. Assess: This result is reasonable. The sled would be expected to stop in a short distance.

Q5.31. Reason: For the Land Rover claim to be true, the vehicle must be able to at least sit on the hill motionless without slipping. So we’ll draw a free-body diagram with the vehicle stationary. We use tilted axes with the x-axis running up the slope.

First apply =netF ma in the y-direction.

θ− =cos 0n w

Then apply =netF ma in the x-direction.

θ− =s sin 0f w

With µs s=f n we rearrange the pair of equations into µ θ

θ=

=s sin

cosn wn w

Now the key is to divide the top equation by the bottom one. (This is mathematically legal, because since the two sides of the bottom equation are equal to each other, then we are really dividing both sides of the top equation by the same thing.) Remember that θ

θ θ=sincos tan .

µ θ=s tan Insertθ = °45 and we have µ = ° =s tan 45 1.0. The correct choice is D. Assess: The answer to this question is independent of the mass of the Land Rover! An equivalent way to express this is that w (and n) cancelled out. Also notice that by solving the equations with a variable θ and only inserting the value of 45° at the end, we are able to solve for the required minimum µs for any angle.

Q5.32. Reason: Friction will slow down and stop the truck once the truck starts to skid. The only horizontal force on the truck while it is skidding is the force of kinetic friction. See the diagram below.

In the vertical direction, Equation 5.1 gives n = w. The force of kinetic friction is given by Equation 5.11.

k k kf n mgµ µ= =

The net force in the horizontal direction is net k.F f=

We can find the acceleration of the truck using Newton’s second law.

k k 2 2k (0.20)(9.80 m/s ) 1.96 m/sx

f mga gm m

µ µ− −= = = − = − = −

Additional significant figures have been retained in this intermediate calculation for use later.


We can find how far the truck skids before stopping using the kinematic equations. We have the initial velocity of the truck is vi = 30 m/s. The final velocity of the truck is vf = 0.0 m/s. Using Equation 2.13 and solving for ∆x,

2 2i

2

( ) (30 m/s) 230 m2 2(1.96 m/s )x

vxa

∆ = = =

The correct choice is A. Assess: A speed of 30 m/s is almost 70 mph. Note that the truck takes nearly a quarter of a kilometer to skid to a stop. Problems

P5.1. Prepare: The massless ring is in static equilibrium, so all the forces acting on it must cancel to give a zero net force. The forces acting on the ring are shown on a free-body diagram below.

Solve: Written in component form, Newton’s first law is

net 1 2 3( ) 0 Nx x x x xF F T T T= Σ = + + = net 1 2 3( ) 0 Ny y y y yF F T T T= Σ = + + =

Evaluating the components of the force vectors from the free-body diagram:

1 1xT T= − T2x = 0 N 3 3 cos 30xT T= °

T1y = 0 N 2 2yT T= 3 3 sin 30yT T= − ° Using Newton’s first law:

1 3 cos 30 0 NT T− + ° = 2 3 sin 30 0 NT T− ° =

Rearranging:

1 3 cos 30 (100 N)(0.8666) 87 NT T= ° = = 2 3 sin 30 (100 N)(0.5) 50 NT T= ° = =

Assess: Since 3T

acts closer to the x-axis than to the y-axis, it makes sense that 1 2.T T>

P5.2. Prepare: The massless ring is in static equilibrium, so all the forces acting on it must cancel to give a zero net force. The forces acting on the ring are shown on a free-body diagram below. Note that the diagram defines the angle θ.

Solve: Because the ring is in equilibrium it must obey net 0 N.F =

This is a vector equation, so it has both x- and y-components:

net 3 1( ) cos 0NxF T Tθ= − = 3 1cosT Tθ⇒ =

net 2 3 3 2( ) sin 0N sinyF T T T Tθ θ= − = ⇒ =

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We have two equations in the two unknowns 3T and θ. Divide the y-equation by the x-equation:

3 2 1

3 1

sin 80Ntan 1.60 tan (1.60) 58cos 50N

T TT T

θ θ θθ

−= = = = ⇒ = = °

Now we can use the x-equation to find

13

50 N 94.3 Ncos cos 58.0

TTθ

= = =°

The tension in the third rope is 94 N directed 58° below the horizontal.

P5.3. Prepare: We assume the speaker is a particle in static equilibrium under the influence of three forces: gravity and the tensions in the two cables. So, all the forces acting on it must cancel to give a zero net force. The forces acting on the speaker are shown on a free-body diagram below. Because each cable makes an angle of 30° with the vertical, θ = 60°.

Solve: Newton’s first law for this situation is

net 1 2 1 2

net 1 2 1 2

( ) 0N cos cos 0 N( ) 0N sin sin 0 N

x x x x

y y y y y

F F T T T TF F T T w T T w

θ θθ θ

= Σ = + = ⇒ − + == Σ = + + = ⇒ + − =

The x-component equation means 1 2.T T= From the y-component equation

12 sinT wθ =2

1(20 kg)(9.8 m/s ) 196 N 113 N

2 sin 2 sin 2 sin 60 1.732w mgT

θ θ⇒ = = = = =

°

Assess: It’s to be expected that the two tensions are equal, since the speaker is suspended symmetrically from the two cables. This is 110 N to two significant figures.

P5.4. Prepare: The forces acting on the beam are shown on a free-body diagram below. You can model the beam as a particle and assume net 0F =

N to calculate the tensions in the suspension ropes.


Solve: The beam attached to the ropes will remain in static equilibrium only if both inequalities T1 < Tmax and T2 < Tmax hold, where Tmax is the maximum sustained tension. The equilibrium equations in vector and component form are

net 1 2

net 1 2

net 1 2

0 N( ) 0 N( ) 0 N

x x x x

y y y y

F T T wF T T wF T T w

= + + == + + == + + =

Using the free-body diagram yields:

1 1 2 2 1 1 2 2sin sin 0 N cos cos 0 NT T T T wθ θ θ θ− + = + − =

The mathematical model is reduced to a simple algebraic system of two equations with two unknowns, 1T and 2.T Substituting θ1 = 20°, θ2 = 30°, and w = mg = 9800 N, the simultaneous equations become

1 2 1 2sin 20 sin 30 0 N cos 20 cos 30 9800 NT T T T− ° + ° = ° + ° =

You can solve this system of equations by simple substitution. The result is 1 26400 N and 4400 N.T T= = The rope on the left side (rope 1) breaks since the tension in this rope is larger than 5600 N. Once the left rope is broken, the right rope will also break because now the whole weight of the beam will be applied to the rope on the right side. Assess: The above approach and result seem reasonable. Intuition indicates there is more tension in the left rope than in the right rope.

P5.5. Prepare: Draw a free-body diagram showing all three forces on the urn. The net force is zero because the urn is raised at a constant speed.

Solve: Use Newton’s second law.

2n = = 0 = = (25 kg)(9.8 m/s ) 25 N = 270 NetF T mg D T mg D− − ⇒ + +

Assess: With the drag force we intuitively expect the tension in the rope to be greater than the urn’s weight.

P5.6. Prepare: The femur is in static equilibrium. We can use Equation 5.1. Solve: See the free-body diagram below.

The direction of the force the femur exerts on the patella is indicated roughly on the previous diagram. The sum of the x-components of the forces must be zero. This gives

Q P sin(42 ) xT T F= ° +

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Solving for ,xF

Q P sin(42 ) 60 N (60 N)sin(42 ) 20 NxF T T= − ° = − ° =

The sum of the y-components of the forces must be zero also. This gives

P cos(42 ) (60 N)cos(42 ) 45 NyF T= ° = ° =

The magnitude of the force by the femur on the patella is then

2 2 2 2( ) ( ) (20 N) (45 N) 49 Nx yF F F= + = + =

Assess: This result is reasonable in magnitude, considering the magnitude of the forces exerted by the tendons and their directions.

P5.7. Prepare: The tension in the more vertical of the two angled ropes (the right one) will have a greater tension, so we apply Newton’s second law and set right = 1500NT and solve for .m leftT will be less than 1500 N and will not break. Solve:

right left= cos 45 cos 30 = 0xF T TΣ ° − °

right left= sin 45 sin 30 = 0yF T T mgΣ ° + ° −

There are various strategies to solve such a system of linear equations. One is to put the two leftT terms on the left side and then divide the two equations.

left rightsin 30 = sin 45T mg T° − °

left rightcos 30 = cos 45T T° °

Now dividing these two equations cancels leftT on the left (since we don’t need leftT ) and leaves tan30 .°

right

right

sin 45tan 30 =

cos 45mg T

T− °

°°

Solve for m and set right = 1500 N.T

right2

(tan 30 cos 45 sin 45 ) (1500N)(tan 30 cos 45 sin 45 )= = = 170 kg9.8 m/s

Tm

g° ° + ° ° ° + °

Assess: The answer seems reasonable, since if there were only one vertical rope it could hold 2(1500 N) / (9.8 m/s ) = 153 kg and here we have the left rope to help.

The original set of two linear equations with two unknowns could also be solved with matrices.

right

left

=cos 45 cos 30 0sin 45 sin 30

TT mg

° − °

° °

P5.8. Prepare: According to Newton’s second law F = ma, the force at any time is found simply by multiplying the acceleration by the mass of the object. Solve: We multiply each acceleration on the graph in Figure P5.8 by m = 0.5 kg and obtain the following force-versus-time graph.


P5.9. Prepare: According to Newton’s second law ,F ma= so the acceleration at any time is found simply by dividing the value of the force by the mass of the object. Solve: We divide each force on the graph in Figure P5.9 by m = 2.0 kg and obtain the following acceleration-versus-time graph.

P5.10. Prepare: According to Newton’s second law ,F ma= so the acceleration at any time is found simply by dividing the value of the force by the mass of the object. Solve: We divide each force on the graph in Figure P5.10 by m = 0.5 kg and obtain the following acceleration-versus-time graph.

P5.11. Prepare: Please refer to Figure P5.11. The free-body diagram shows three forces acting on an object whose mass is 2.0 kg. The force in the first quadrant has two components: 4 N along the x-axis and 3 N along the y-axis. We will first find the net force along the x- and the y-axes and then divide these forces by the object’s mass to obtain the x- and y-components of the object’s acceleration. Solve: Applying Newton’s second law to the diagram on the left:

5-14 Chapter 5

2net( ) 4 N 2 N 1.0 m/s2 kg

xx

Fam

−= = = net 2( ) 3 N 3 N 0 m/s

2 kgy

y

Fa

m−

= = =

Assess: The object’s motion is only along the x-axis.

P5.12. Prepare: Please refer to Figure P5.12. The free-body diagram shows five forces acting on an object whose mass is 2.0 kg. All the forces point along x- or y-axes. We will first find the net force along the x- and the y-axes and then divide these forces by the object’s mass to obtain the x- and y-components of the object’s acceleration. Solve: Applying Newton’s second law:

2net( ) 4 N 2 N 1.0 m/s2 kg

xx

Fam

−= = = net 2( ) 3 N 1 N 2 N 0 m/s

2 kgy

y

Fa

m− −

= = =

Assess: The object’s motion is only along the x-axis.

P5.13. Prepare: We assume that the box is a particle being pulled in a straight line. Since the ice is frictionless, the tension in the rope is the only horizontal force on the box and is shown below in the free-body diagram. Since we are looking at horizontal motion of the box, we are not interested in the vertical forces in this problem.

Solve: (a) Since the box is at rest, ax = 0 m/s2, the net force on the box must be zero or the tension in the rope must be zero. (b) For this situation again, ax = 0 m/s2, so net 0 N.F T= = (c) Here, the velocity of the box is irrelevant, since only a change in velocity requires a nonzero net force. Since ax = 5.0 m/s2,

2net (50 kg)(5.0 m/s ) 250 NxF T ma= = = =

Assess: For parts (a) and (b), the zero acceleration immediately implies that the rope is exerting no horizontal force on the box. For part (c), the 250 N force (the equivalent of about half the weight of a small person) seems reasonable to accelerate a box of this mass at 5.0 m/s2.

P5.14. Prepare: The force of friction between the crate and the horizontal floor surface is proportional to the crate’s mass. Specifically, fs max = µsn = µsmg = max. That is, the acceleration as the crate slows down is unchanged. We can now use kinematics equations to find the stopping distance. Solve: (a) The block will slide the same distance d. The acceleration is the same as before and the velocity is the same as before, so from Equation 2.13 the distance traveled d remains the same. (b) The block will slide a distance of 4d. Because the acceleration is unchanged, but the velocity is doubled, Equation 2.13 yields a stopping distance of 4d.

P5.15. Prepare: We assume that the seat belt supplies all the force necessary to decelerate the driver (that is, seatbelt net ),F F= and that the deceleration is constant over the time interval of 0.10 s. Set up a coordinate system

with the car traveling to the right along the x-axis. We use the kinematics equations from Chapter 2 to solve for the constant acceleration, and then netF ma= (with m = 70 kg) to solve for the force exerted by the seat belt. Solve: The definition of acceleration says

20.0 m/s 14 m/s 140 m/s0.10 s

xx

vat

∆ −= = = −

∆


where the negative sign indicates that the car (which is traveling to the right) is slowing down. 2

seatbelt net (70 kg)( 140 m/s ) 9800 NxF F ma= = = − = −

where the negative sign shows the force acting in the negative x-direction (the same direction as the acceleration). Assess: 9800 N is quite a bit of force, but so it is in a head-on collision at a significant speed. You can see from the equations above that if the crash had taken more time the force would not be so severe; save that thought for a future chapter.

P5.16. Prepare: We must first find the astronaut’s mass on earth and then multiply it with Mars’s acceleration due to gravity to find his weight on Mars. Solve: The mass of the astronaut is

earth2

earth

800 N 81.6 kg9.8 m/s

wmg

= = =

Therefore, the weight of the astronaut on Mars is 2

Mars Mars (81.6 kg)(3.76 m/s ) 310 Nw mg= = =

Assess: The smaller acceleration of gravity on Mars reveals that objects are less strongly attracted to Mars than to the earth, so the smaller weight on Mars makes sense. Also, note that astronaut’s mass stays unchanged.

P5.17. Solve: (a) The woman’s weight on the earth is 2

earth earth (55 kg)(9.8 m/s ) 540 Nw mg= = =

(b) Since mass is a measure of the amount of matter, the woman’s mass is the same on the moon as on the earth. Her weight on the moon is

2moon moon (55 kg)(1.62 m/s ) 89 Nw mg= = =

Assess: The smaller acceleration due to gravity on the moon reveals that objects are less strongly attracted to the moon than to the earth. Thus the woman’s smaller weight on the moon makes sense.

P5.18. Prepare: The true weight of an object and its apparent weight are connected by Equation 5.9: wapp = m(g + ay). After the box, with a passenger inside, leaves the rubber band (still moving upward), the box as well as the passenger are falling freely and their acceleration is equal to –g. Solve: (a) The true weight is mg, so the passenger’s true weight will be (75 kg)(9.8 m/s2) = 740 N. (b) The passenger’s apparent weight will be (75 kg)(9.8 m/s2 + (– 9.8 m/s2)) = 0.

P5.19. Prepare: The astronaut and the chair will be denoted by A and C, respectively, and they are separate systems. The launch pad is a part of the environment. In the following free-body diagrams for both the astronaut and the chair are shown at rest on the launch pad (top) and while accelerating (bottom).

5-16 Chapter 5

Solve: (a) Newton’s second law for the astronaut is

on A C on A A A A( ) 0 NyF n w m a∑ = − = = C on A A An w m g⇒ = =

By Newton’s third law, the astronaut’s force on the chair is 2

A on C C on A A (80 kg)(9.8 m/s ) 780 Nn n m g= = = =

(b) Newton’s second law for the astronaut is

on A C on A A A A( ) yF n w m a∑ = − = C on A A A A A A( )n w m a m g a⇒ = + = +

By Newton’s third law, the astronaut’s force on the chair is 2 2

A on C C on A A A( ) (80 kg)(9.8 m/s 10 m/s ) 1600 Nn n m g a= = + = + =

Assess: This is a reasonable value because the astronaut’s acceleration is more than g.

P5.20. Prepare: The passenger is subject to two vertical forces: the downward pull of gravity and the upward push of the elevator floor. We can use one-dimensional kinematics and Equation 5.8 for the three situations.

Solve: (a) The apparent weight from Equation 5.8 is

2app

01 1 (60kg)(9.8m/s ) 590Nyaw w w mg

g g

= + = + = = =


(b) The elevator speeds up from v0y = 0 m/s to its cruising speed at vy = 10 m/s. We need its acceleration before we can find the apparent weight:

210m/s 0m/s 2.5m/s4.0sy

vat

∆ −= = =

∆

The passenger’s apparent weight is

2

app 2

2.5 m/s1 (590 N) 19.8 m/s

yaw w

g

= + = +

= (590 N)(1.26) = 740 N

(c) The passenger is no longer accelerating since the elevator has reached its cruising speed. Thus, app 590Nw w= = as in part (a). Assess: The passenger’s apparent weight is his normal weight in parts (a) and (c), since there is no acceleration. In part (b), the elevator must not only support his weight but must also accelerate him upward, so it’s reasonable that the floor will have to push up harder on him, increasing his apparent weight.

P5.21. Prepare: We’ll assume Zach is a particle moving under the effect of two forces acting in a single vertical line: gravity and the supporting force of the elevator. These forces are shown in Figure 5.12 in a free-body diagram. We will use Equation 5.8 to find the apparent weight.

Solve: (a) Before the elevator starts braking, Zach is not accelerating. His apparent weight (see Equation 5.8) is

22

app0 m/s1 1 (80 kg)(9.8 m/s ) 784 N 780 Naw w w mg

g g

= + = + = = =

(b) Using the definition of acceleration,

f i 2

f i

22

app 2

0 ( 10) m/s 3.33 m/s3.0 s

3.33 m/s1 (80 kg)(9.8 m/s ) 1 (784 N)(1 0.340) 1100 N9.8 m/s

v v vat t t

aw wg

∆ − − −= = = =

∆ −

⇒ = + = + = + =

Assess: While the elevator is braking, it not only must support Zach’s weight but must also push upward on him to decelerate him, so the apparent weight is greater than his normal weight.

P5.22. Prepare: The passenger is acted on by only two vertical forces: the downward pull of gravity and the upward force of the elevator floor. Referring to Figure P5.22, the graph has three segments corresponding to different conditions: (1) increasing velocity, meaning an upward acceleration, (2) a period of constant upward velocity, and (3)

5-18 Chapter 5

decreasing velocity, indicating a period of deceleration (negative acceleration). Given the assumptions of our model, we can calculate the acceleration for each segment of the graph and then apply Equation 5.8. Solve: The acceleration for the first segment is

2f i 2

app 2f i

2

8 m/s 0 m/s 4 m/s4 m/s 1 ( ) 12 s 0 s 9.8 m/s

4(75 kg)(9.8 m/s ) 1 1040 N9.8

yy

av va w w mgt t g

− −= = = ⇒ = + = + − −

= + =

For the second segment, ay = 0 m/s2 and the apparent weight is

22

app0 m/s1 (75 kg)(9.8 m/s ) 740 Nw w mg

g

= + = = =

For the third segment,

3 2 2

3 2

22

app 2

0 m/s 8 m/s 2 m/s10 s 6 s

2 m/s1 (75 kg)(9.8 m/s )(1 0.2) 590 N9.8 m/s

yv vat t

w w

− −= = = −

− −

−⇒ = + = − =

Assess: As expected, the apparent weight is greater than normal when the elevator is accelerating upward and lower than normal when the acceleration is downward. When there is no acceleration the weight is normal. In all three cases the magnitudes are reasonable, given the mass of the passenger and the accelerations of the elevator.

P5.23. Prepare: In each case the frog is in equilibrium net( 0).F =

Solve: (a) The two forces on the frog act in the vertical direction: the weight (gravitational force of the earth down on the frog), and the normal force of the log up on the frog. The two must have equal magnitude; since w = mg (0.60 kg) (9.8 m/s2) = 5.9 N, then the magnitude of the normal force is also 5.9 N. (b) Draw a free-body diagram for the frog. Use tilted axes with the x-axis running up the log.

Apply netF ma= in the y-direction.

2

cos 0 cos cos (0.60 kg)(9.8 m/s )cos 30 5.1 N

n wn w mgθ

θ θ− =

= = = ° =

Assess: The answer is less in part (b) than in part (a), as we would expect. The static friction force is also helping hold up the frog in part (b). Notice that we solved the problem algebraically before putting numbers in. This not only allows us to solve a similar problem for a different frog or log, but it enables us to check our answer in this case for reasonableness. Take the limit as 0;θ → the slope approaches zero and the conditions revert back to part (a) as cos 1.θ → Then take the limit as 90θ → ° and the normal force decreases to zero as the log becomes vertical and there is no normal force on the frog.


P5.24. Prepare: We apply Newton’s second law to solve for the value of the normal force. Solve: (a)

(b) Use tilted axes with the x-axis running down the incline. Apply Fnet = ma in the y-direction.

2

cos 0 cos cos (23 kg)(9.9 m/s )cos 38 180 N

n wn w mg

θθ θ

− =− = = ° =

Assess: The answer is less than the child’s weight of 225 N, as we would expect, since only part of the weight is in the y-direction. The value seems to be in the right ballpark. Notice that we solved the problem algebraically before putting numbers in. This not only allows us to solve a similar problem for a different child or incline, but it enables us to check our answer in this case for reasonableness. Take the limit as θ → 0; the slope approaches zero and n tends toward the child’s weight as cos θ → 1. Then take the limit as θ → 90° and the normal force decreases to zero as the incline becomes vertical and there is no normal force on the child.

P5.25. Prepare: We assume that the safe is a particle moving only in the x-direction. Since it is sliding during the entire problem, the force of kinetic friction opposes the motion by pointing to the left. In the following diagram we give a pictorial representation and a free-body diagram for the safe. The safe is in dynamic equilibrium, since it’s not accelerating.

Solve: We apply Newton’s first law in the vertical and horizontal directions:

net B C k k B C2

net

( ) 0 N 350 N 385 N 735 N

( ) 0 N (300 kg)(9.8 m/s ) 2940 Nx x

y y

F F F F f f F F

F F n w n w mg

= Σ = + − = ⇒ = + = + =

= Σ = − = ⇒ = = = =

Then, for kinetic friction

kk k k

735 N 0.252940 N

ff nn

µ µ= ⇒ = = =

Assess: The value of µk = 0.25 is hard to evaluate without knowing the material the floor is made of, but it seems reasonable.

P5.26. Prepare: The truck is in equilibrium. Below we identify the forces acting on the truck and construct a free-body diagram. Solve: The truck is not accelerating, so it is in equilibrium, and we can apply Newton’s first law. The normal force has no component in the x-direction, so we can ignore it here. For the other two forces

5-20 Chapter 5

2 4net s s( ) 0 N sin (4000kg)(9.8m/s )(sin 15 ) 1.0 10 Nx x x xF F f w f w mg θ= Σ = − = ⇒ = = = ° = ×

Assess: The truck’s weight (mg) is roughly 40,000 N. A friction force that is 25%≈ of the truck’s weight seems reasonable.

P5.27. Prepare: The car is undergoing skidding, so it is decelerating and the force of kinetic friction acts to the left. We give below an overview of the pictorial representation, a motion diagram, a free-body diagram, and a list of values. We will first apply Newton’s second law to find the deceleration and then use kinematics to obtain the length of the skid marks.

Solve: We begin with Newton’s second law. Although the motion is one-dimensional, we need to consider forces in both the x- and y-directions. However, we know that ay = 0 m/s2. We have

netnet k 2( )( ) 0m/s yx

x y

FF f n w n mga am m m m m

− − −= = = = = =

We used k k( )xf f= − because the free-body diagram tells us that kf

points to the left. The force of kinetic friction

relates kf

to n with the equation k k .f nµ= The y-equation is solved to give .n mg= Thus, the kinetic friction force is k k .f mgµ= Substituting this into the x-equation yields

k 2 2k (0.6)(9.8 m/s ) 5.88 m/sx

mga gm

µ µ−= = − = − = −

The acceleration is negative because the acceleration vector points to the left as the car slows. Now we have a constant-acceleration kinematics problem. ∆t isn’t known, so use

22 2 2 2f i 2

(40 m/s)0 m /s 2 140 m2( 5.88 m/s )xv v a x x= = + ∆ ⇒ ∆ = − =

−

Assess: The skid marks are 140 m long. This is ≈ 430 feet, reasonable for a car traveling at ≈80 mph. It is worth noting that an algebraic solution led to the m canceling out.

P5.28. Prepare: The mule is acted on by two opposing forces in a single line: the farmer’s pull and the friction. The mule will be subject to static friction until (and if!) it begins to move; after that it will be subject to


kinetic friction. We give below an overview of the pictorial representation, a free-body diagram, and a list of values. We will calculate the force of maximum static friction and compare it with the maximum applied force.

Solve: Since the mule does not accelerate in the vertical direction, the free-body diagram shows that

.n w mg= = The maximum friction force is 2

smax s (0.8)(120 kg)(9.8 m/s ) 940 Nf mgµ= = =

The maximum static friction force is greater than the farmer’s maximum pull of 800 N; thus, the farmer will not be able to budge the mule. Assess: The farmer should have known better.

P5.29. Prepare: We show below the free-body diagrams of the crate when the conveyer belt runs at constant speed (part a) and the belt is speeding up (part b).

Solve: (a) When the belt runs at constant speed, the crate has an acceleration 20 m/sa =

and is in dynamic equilibrium. Thus net 0.F =

It is tempting to think that the belt exerts a friction force on the crate. But if it did, there would be a net force because there are no other possible horizontal forces to balance a friction force. Because there is no net force, there cannot be a friction force. The only forces are the upward normal force and the crate’s weight. (A friction force would have been needed to get the crate moving initially, but no horizontal force is needed to keep it moving once it is moving with the same constant speed as the belt.) (b) If the belt accelerates gently, the crate speeds up without slipping on the belt. Because it is accelerating, the crate must have a net horizontal force. So now there is a friction force, and the force points in the direction of the crate’s motion. Is it static friction or kinetic friction? Although the crate is moving, there is no motion of the crate relative to the belt. Thus, it is a static friction force that accelerates the crate so that it moves without slipping on the belt. (c) The static friction force has a maximum possible value ( fs)max = µsn. The maximum possible acceleration of the crate is

max smax

( )sf nam m

µ= =

If the belt accelerates more rapidly than this, the crate will not be able to keep up and will slip. It is clear from the free-body diagram that .n w mg= = Thus,

amax = µsg = (0.50)(9.8 m/s2) = 4.9 m/s2

(d) The acceleration of the crate will be a = µkg = (0.30)(9.8 m/s2) = 2.9 m/s2.

P5.30. Prepare: We will need to apply Newton’s second law in both the vertical and horizontal directions. We want to use the coefficient of static friction since we want the box to stay stationary.

5-22 Chapter 5

Solve:

= = 0 =yF n F mg n F mgΣ − − ⇒ +

s s s= 125N = 125N = 125N ( ) = 0xF f n F mgµ µΣ − − − +

Solve for .F

2

s

125 N 125 N= = (30 kg)(9.8 m/s ) = 63 N0.35

F mgµ

− −

Assess: 63 N is about half of the force on the rope; this seems reasonable given s.µ

P5.31. Prepare: We can find the drag force using Equation 5.14. Solve: Using Equation 5.15, ρ≈ 1 2

4D Av with ρ = 31.22kg/m . The area of the car is A = (1.6 m)(1.4 m) = 2.24 m2, where an additional significant figure has been kept in this intermediate calculation.

(a) ≈ ≈3 2 21 (1.22 kg/m )(2.24m )(10m/s) 68N4

D

(b) ≈ ≈3 2 21 (1.22 kg/m )(2.24m )(30m/s) 610N4

D

Assess: Note that the drag increases with the square of the speed, so that at 30 m/s, the drag force is nine times what it is at 10 m/s.

P5.32. Prepare: The bowling ball falls straight down toward the earth’s surface. The bowling ball is subject to a net force that is the resultant of the weight and drag force vectors acting vertically in the downward and upward directions, respectively. Once the net force acting on the ball becomes zero, the terminal velocity is reached and remains constant for the rest of the motion. An overview of a pictorial representation and a free-body diagram are shown later.

Solve: The mathematical equation defining the dynamical equilibrium situation for the falling ball is

net 0 NF w D= + =

Since only the vertical direction matters, one can write

net0 N 0 NyF F D w∑ = ⇒ = − =


When this condition is satisfied, the speed of the ball becomes the constant terminal speed term.v v= The magnitudes of the weight and drag forces acting on the ball are

( )

2

2 2 2 3 2 2term term

(9.80 m/s )1 0.25 ( ) (0.25 )(1.22 kg/m )(0.11 m) (85 m/s) 83.7 N4

w mg m

D Av R vρ ρ π π

= =

≈ = = =

The condition for dynamic equilibrium becomes

22

83.7 N(9.80 m/s ) 83.7 N 0 N 8.5 kg9.8 m/s

m m− = ⇒ = =

Assess: The value of the mass of the bowling ball obtained above seems reasonable. Such a ball is heavy so that it has a significant impact on the pins, but not “excessively” heavy so that it can be lifted and rolled by an average human player.

P5.33. Prepare: We assume that the skydiver is shaped like a box. The following shows a pictorial representation of the skydiver and a free-body diagram at terminal speed. The skydiver falls straight down toward the earth’s surface, that is, the direction of fall is vertical. Since the skydiver falls feet first, the surface perpendicular to the drag has the cross-sectional area A = 20 cm × 40 cm. The physical conditions needed to use Equation 5.15 for the drag force to be satisfied. The terminal speed corresponds to the situation when the net force acting on the skydiver becomes zero.

Solve: The expression for the magnitude of the drag with v in m/s is

3 22 21 0.25(1.22 kg/m )(0.20 m 0.40 m) N 0.0244 N4

D Av v vρ≈ = × =

The skydiver’s weight is 2(75 kg)(9.8 m/s ) 735 N.w mg= = = The mathematical form of the condition defining dynamical equilibrium for the skydiver and the terminal speed is

2net term term

735 0 N 0.0244 N 735 N 0 N 170 m/s0.0244

F w D v v= + = ⇒ − = ⇒ = ≈

Assess: The result of the above simplified physical modeling approach and subsequent calculation, even if approximate, shows that the terminal velocity is very high. This result implies that the skydiver will be very badly hurt at landing if the parachute does not open in time.

5-24 Chapter 5

P5.34. Prepare: The car and the truck will be denoted by the symbols C and T, respectively. The ground will be denoted by the symbol G. A visual overview shows a pictorial representation, a list of known and unknown values, and a free-body diagram for both the car and the truck. Since the car and the truck move together in the positive x-direction, they have the same acceleration.

Solve: (a) The x-component of Newton’s second law for the car is

on C G on C T on C C C( )xF F F m a∑ = − =

The x-component of Newton’s second law for the truck is

on T C on T T T( )xF F m a∑ = =

Using aC = aT = a and FT on C = FC on T, we get

C on G C on TC

1( )F F am

− =

C on T

T

1( )F am

=

Combining these two equations,

C on G C on T C on TC T

1 1( ) ( )F F Fm m

− =

C on T C on G

C T C

1 1 1( )F Fm m m

⇒ + =

TC on T C on G

C T

( ) mF Fm m

⇒ =

+

2000 kg(4500 N)1000 kg 2000 kg

= +

= 3000 N

(b) Due to Newton’s third law, FT on C = 3000 N.


P5.35. Prepare: The blocks are denoted as 1, 2, and 3. The surface is frictionless and along with the earth it is a part of the environment. The three blocks are our three systems of interest. The force applied on block 1 is FA on 1 = 12 N. The acceleration for all the blocks is the same and is denoted by a. A visual overview shows a pictorial representation, a list of known and unknown values, and a free-body diagram for the three blocks.

Solve: Newton’s second law for the three blocks along the x-direction is

on 1 A on 1 2 on 1 1( )xF F F m a∑ = − = on 2 1on 2 3 on 2 2( )xF F F m a∑ = − = on 3 2 on 3 3( )xF F m a∑ = =

Adding these three equations and using Newton’s third law (F2 on 1 = F1 on 2 and F3 on 2 = F2 on 3), we get

A on 1 1 2 3( )F m m m a= + + (12 N) (1 kg 2 kg 3 kg)a⇒ = + + 22 m/sa⇒ =

Using this value of a, the force equation on block 3 gives 2

2 on 3 3 (3 kg)(2 m/s ) 6 NF m a= = =

Substituting into the force equation on block 1, 2

2 on 112 N (1 kg)(2 m/s )F− = 2 on 1 10 NF⇒ =

Assess: Because all three blocks are pushed forward by a force of 12 N, the value of 10 N for the force that the 2 kg block exerts on the 1 kg block is reasonable.

5-26 Chapter 5

P5.36. Prepare: The man (M) and the block (B) are interacting with each other through a rope (R). We will assume the pulley to be frictionless and the rope to be massless. This assumption implies that the tension in the rope is the same on both sides of the pulley. The two systems are the man and the block. A visual overview shows below a pictorial representation, a list of known and unknown values, and a free-body diagram for both the man and the block. Clearly the entire system remains in equilibrium since mB > mM. The block would move downward but it is already on the ground.

Solve: From the free-body diagrams, we can write down Newton’s second law in the vertical direction as

on M R on M M( ) 0NyF T w∑ = − = 2R on M M (60 kg)(9.8 m/s ) 590 NT w⇒ = = =

Since the tension is the same on both sides, B on R M on R 590 N.T T T= = =

P5.37. Prepare: A visual overview shows below a pictorial representation, a list of known and unknown values, and a free-body diagram for both the ice (I) and the rope (R). The force extF

acts only on the rope. Since the rope and the ice block move together, they have the same acceleration. Also because the rope has mass, Fext on the front end of the rope is not the same as FI on R that acts on the rear end of the rope.

Solve: (a) Newton’s second law along the x-axis for the ice block is

2on I R on I I( ) (10 kg)(2.0 m/s ) 20 NxF F m a∑ = = = =

(b) Newton’s second law along the x-axis for the rope is 2

on R ext I on R R ext R on I R ext R on I R( ) 20 N (0.5 kg)(2.0 m/s ) 21 NxF F F m a F F m a F F m a∑ = − = ⇒ − = ⇒ = + = + =


P5.38. Prepare: Please refer to Figure P5.38. We show below a free-body diagram for the two ropes (1 and 2) and the two blocks (A and B).

7

Solve: (a) The two blocks and two ropes form a combined system of total mass M = 2.5 kg. This combined system is accelerating upward at a = 3.0 m/s2 under the influence of a force F and the weight Mg. Newton’s second law applied to the combined system is

net( ) ( ) 32.0 NyF F Mg Ma F M a g= − = ⇒ = + =

(b) The ropes are not massless. We must consider both the blocks and the ropes as systems. The force F acts only on block A because it does not contact the other objects. We can proceed to apply the y-component of Newton’s second law to each system, starting at the top. Each has an acceleration a = 3.00 m/s2. For block A:

net on A A 1 on A A( ) yF F m g T m a= − − = 1on A A ( ) 19.2 NT F m a g⇒ = − + =

(c) Applying Newton’s second law to rope 1:

net on 1 A on 1 1 B on 1 1( ) yF T m g T m a= − − =

A on 1T

and 1on AT

are an action/reaction pair. But, because the rope has mass, the two tension forces A on 1T

and

B on 1T

are not the same. The tension at the lower end of rope 1, where it connects to B, is

B on 1 A on 1 1( ) 16 NT T m a g= − + =

(d) We can continue to repeat this procedure, noting from Newton’s third law that

1on B B on 1T T= and 2 on B B on 2T T=

Newton’s second law applied to block B is

net on B 1on B B 2 on B B( ) yF T m g T m a= − − = 2 on B 1on B B( ) 3.2 NT T m a g⇒ = − + =

P5.39. Prepare: Since each block has the same acceleration as all the others they must each experience the same net force. Each block will have one more newton pulling forward than the force pulling back on it from the blocks behind. Solve: (a) 1 N (b) 50 N Assess: Since 100 N accelerates 100 blocks then n newtons accelerates n blocks.

5-28 Chapter 5

P5.40. Prepare: Look at each block as the trailing block in turn. For the trailing block the net force on it is the tension in the string since there is no retarding friction force. Both blocks will have the same acceleration a regardless of which is the leading and which is the trailing block because on the whole system A B= ( ) .F m m a+ Solve: Use Newton’s second law separately on each trailing block.

Ablock A: = 18Nm a

Bblock B: = 24Nm a

Divide the first equation by the second and cancel .a

A

B

18N 3= =24N 4

mm

Assess: The ratio of the masses is the ratio of the tensions (net forces) since they have the same acceleration.

P5.41. Prepare: Because the piano is to descend at a steady speed, it is in dynamic equilibrium. The following shows a free-body diagram of the piano and a list of values.

Solve: (a) Based on the free-body diagram, Newton’s second law is

net 1 2 2 1 1

net 1 2 3 3 1 1 2 2

( ) 0 N cos cos( ) 0 N sin sin

x x x

y y y y y

F T T T TF T T T w T T T mg

θ θθ θ

= = + = −= = + + + = − − −

Notice how the force components all appear in the second law with plus signs because we are adding vector forces. The negative signs appear only when we evaluate the various components. These are two simultaneous equations in the two unknowns T2 and T3. From the x-equation we find

1 12

2

cos (500 N)cos15 530 Ncos cos 25

TT θθ

°= = =

°

(b) Now we can use the y-equation to find

3 1 1 2 2sin sin 5300 NT T T mgθ θ= + + =


P5.42. Prepare: Note that the medal would hang straight down if the car were going at a constant velocity, so the deviation from vertical only occurs while the car is accelerating. We apply Newton’s second law.

Solve: (a) Because she accelerates onto the highway we assume she is accelerating forward so the medal hangs away from the windshield. (b) Use Newton’s law in vertical and horizontal directions. In the horizontal direction there is only one (component of) force, but there is an acceleration.

= sin =x xF T maθΣ

There is no acceleration in the y direction.

= cos = 0 cos =yF T mg T mgθ θΣ − ⇒

Divide the first equation by the second and cancel m and .T

2 2sin = = tan = (9.8 m/s )(tan 10 ) = 1.7 m/scos

xx

a a gg

θ θθ

⇒ °

Assess: This is a reasonable acceleration for a car.

P5.43. Prepare: Please refer to Figure P5.43. To find the net force at a given time, we need the acceleration at that time. Because the times where we are asked to find the net force fall on distinct slopes of the velocity versus time graph, we can use the constant slopes of the three segments of the graph to calculate the three accelerations. Solve: For t between 0 s and 3 s,

212 m/s 0 s 4 m/s3 s

xx

vat

∆ −= = =

∆

For t between 3 s and 6 s, ∆vx = 0 m/s, so ax = 0 m/s2. For t between 6 s and 8 s,

20 m/s 12 m/s 6 m/s2 s

xx

vat

∆ −= = = −

∆

From Newton’s second law, at t = 1 s we have

Fnet = max = (2.0 kg)(4 m/s2) = 8 N

At t = 4 s, ax = 0 m/s2, so Fnet = 0 N. At t = 7 s,

Fnet = max = (2.0 kg)(–6.0 m/s2) = –12 N

Assess: The magnitudes of the forces look reasonable, given the small mass of the object. The positive and negative signs are appropriate for an object first speeding up, then slowing down.

P5.44. Prepare: Please refer to Figure P5.44. Positive forces result in the object gaining speed and negative forces result in the object slowing down. The final segment of zero force is a period of constant speed. We have the mass and net force for all the three segments. This means we can use Newton’s second law to calculate the accelerations. Kinematics equations then allow us to find velocity.

5-30 Chapter 5

Solve: The acceleration from t = 0 s to t = 3 s is

24 N 2 m/s2.0 kg

xx

Fam

= = =

The acceleration from t = 3 s to t = 5 s is

22 N 1 m/s2.0 kg

xx

Fam

−= = = −

The acceleration from t = 5 s to 8 s is ax = 0 m/s2. In particular, 2(at 6 s) 0 m/s .xa t = = We can now use one-dimensional kinematics to calculate v at t = 6 s in three steps as follows:

23 0 0 3 3 0

25 3 3 5 5 3

6 5 5 6 6 5

( ) 0 (2 m/s )(3 s) 6 m/s( ) 6 m/s ( 1 m/s )(2 s) 4 m/s( ) 4 m/s (0)(3 s) 4.0 m/s

v v a t tv v a t tv v a t t

−

−

−

= + − = + =

= + − = + − == + − = + =

Assess: The positive final velocity makes sense, given the greater magnitude and longer duration of the positive 1.F

A velocity of 4 m/s also seems reasonable, given the magnitudes and directions of the forces and the mass involved.

P5.45. Prepare: The box is acted on by two forces: the tension in the rope and the pull of gravity. Both the forces act along the same vertical line which is taken to be the y-axis. The free-body diagram for the box is shown below.

Solve: (a) Since the box is at rest, ay = 0 m/s2 and the net force on it must be zero:

2net 0 N (50 kg)(9.8 m/s ) 490 NF T w T w mg= − = ⇒ = = = =

(b) The velocity of the box is irrelevant, since only a change in velocity requires a nonzero net force. Since 25.0 ,ya = m/s

2net (50 kg)(5.0 m/s ) 250 N 250 N 250 N 490 N 740 NyF T w ma T w= − = = = ⇒ = + = + =

Assess: For part (a) the zero acceleration immediately implies that the box’s weight must be exactly balanced by the upward tension in the rope. For part (b) the tension not only has to support the box’s weight but must also accelerate it upward, hence, T must be greater than w.

P5.46. Prepare: The box is acted on by two forces: the tension in the rope and the pull of gravity. Both the forces act along the same vertical line which is taken to be the y-axis. The following shows the free-body diagram for the box.

Solve: (a) Since the box is rising at a constant speed, ay = 0 m/s2 and the net force on it must be zero:


2net 0 N (50 kg)(9.8 m/s ) 490 NF T w T w mg= − = ⇒ = = = =

(b) Since the box is slowing down, ay = –5.0 2m/s and we have

2net (50 kg)( 5.0 m/s ) 250 N

250 N 250 N 490 N 240 NyF T w ma

T w= − = = − = −

⇒ = − + = − + =

Assess: For part (a) the zero acceleration immediately implies that the box’s weight must be exactly balanced by the upward tension in the rope. For part (b), when the box accelerates downward, the rope need not support the entire weight, hence, T is less than w.

P5.47. Prepare: We will assume constant acceleration so we can use the equations of Table 2.4. Assume the baseball is initially moving in the positive x-direction. We list the known quantities:

Known

i

0.14 kg( ) 30 m/s

0.0015 sx

mv

t

==

∆ =

Find a F

Solve: (a) With f( ) 0 m/s,xv = we solve for a from f i( ) ( ) .x x xv v a t= + ∆

i 2( ) (30 m/s) 20,000 m/s0.0015 s

xx

vat

− −= = = −

∆

The magnitude of this is 220,000 m/s or 4 22.0 10 m/s .× (b) Apply Newton’s second law: xxF maΣ = where the force of the body on the ball is the only force (and is therefore the net force).

2(0.14 kg)( 20,000 m/s ) 2800 Nx xF ma∑ = = − = −

The magnitude of this is 2800 N. This force is exerted by the body the ball hits. (c) By Newton’s third law if the body exerts a force on the ball, then the ball exerts a force equal in magnitude and opposite in direction on the body. Therefore the ball applies a force of 2800 N to the object it hits. (d) This force of 2.8 kN is less than 6.0 kN, so the forehead is not in danger (although it would still hurt and maybe raise a lump). This force of 2.8 kN is greater than 1.3 kN, so the cheek is in danger of fracture. Assess: This is a nice real-life problem that employs the definition of acceleration and Newton’s second and third laws. The data provided are typical of real baseballs and real pitching speeds, so the conclusion is also true-to-life. Catchers, whose faces are in the line of fire, wear masks for this reason.

5-32 Chapter 5

P5.48. Prepare: We can assume the person is moving in a straight line under the influence of the combined decelerating forces of the air bag and seat belt or, in the absence of restraints, the dashboard or windshield. The following is an overview of the situation in a pictorial representation and the occupant’s free-body diagram is shown below. Note that the occupant is brought to rest over a distance of 1 m in the former case, but only over 5 mm in the latter.

Solve: (a) In order to use Newton’s second law for the passenger, we’ll need the acceleration. Since we don’t have the stopping time,

2 2f i f i2 ( )v v a x x= + −

2 2 2 2 2f i 2

f i

0 m /s (15 m/s) 112.5 m/s2( ) 2(1 m 0 m)

v vax x

− −⇒ = = = −

− −

2net (60 kg)( 112.5 m/s ) 6750 NF F ma⇒ = = = − = −

The net force is 6800 N to the left. (b) Using the same approach as in part (a),

2 2 2 2 2f i

f i

0 m /s (15 m/s)(60 kg) 1,350,000 N2( ) 2(0.005 m)

v vF ma mx x

− −= = = = −

−

The net force is 1.4 × 106 N to the left. (c) The passenger’s weight is mg = (60 kg)(9.8 m/s2) = 590 N. The force in part (a) is 11.5 times the passenger’s weight. The force in part (b) is 2300 times the passenger’s weight. Assess: An acceleration of 11.5g is well within the capability of the human body to withstand. A force of 2300 times the passenger’s weight, on the other hand, would surely be catastrophic.

P5.49. Prepare: The rock (R) and Bob (B) are two systems of our interest. We give a pictorial representation, a list of values, and free-body diagrams for Bob and the rock. Motion of the rock is assumed to be along the x-axis. We realize that Bob must accelerate the rock forward until he releases the rock with a speed of 30 m/s. From the given information we can find this acceleration using kinematics. Newton’s second law will then yield the force exerted on the rock by Bob. This is also the force that is exerted by the rock on Bob (Newton’s third law). We can then calculate Bob’s acceleration using his mass in Newton’s second law. Kinematics once again can be used to find Bob’s recoil speed.


Solve: (a) Bob exerts a forward force B on RF

on the rock to accelerate it forward. The rock’s acceleration is calculated as follows:

2 2f R2 2 2

f R i R R R( ) (30 m/s)( ) ( ) 2 450 m/s2 2(1 m)vv v a x a

x= + ∆ ⇒ = = =

∆

The force is calculated from Newton’s second law: 2

B on R R R (0.5 kg)(450 m/s ) 225 N or 230 N to two significant figures.F m a= = =

(b) Because Bob pushes on the rock, the rock pushes back on Bob with a force R on B.F

Forces R on BF

and R on BF

are

an action/reaction pair, so R on B B on R 225 N.F F= =

The force causes Bob to accelerate backward with an acceleration equal to

net on B R on B 2B

B B

( ) 225 N 3.0 m/s75 kg

xF Fa

m m= = − = − = −

This is a rather large acceleration, but it lasts only until Bob releases the rock. We can determine the time interval by returning to the kinematics of the rock:

f Rf R i R R

R

( )( ) ( ) 0.0667 svv v a t ta

= + ∆ ⇒ ∆ = =

At the end of this interval, Bob’s velocity is

f B i B B B( ) ( ) 0.20 m/sv v a t a t= + ∆ = ∆ = −

Thus his recoil speed is 0.20 m/s.

P5.50. Prepare: The astronaut and the satellite are the two systems of our interest. The astronaut and the satellite accelerate in opposite directions for 0.5 s. The force on the satellite and the force on the astronaut are an action/reaction pair, so both are 100 N. A pictorial representation and a list of values are given below. The motion is assumed to be along the x-direction.

Solve: Newton’s second law for the satellite along the x-direction is

on S A on S S S( )xF F m a∑ = = A on S 2S

S

(100 N) 0.156 m/s640 kg

Fa

m−

⇒ = = = −

Newton’s second law for the astronaut along the x-direction is

S on A A on S 2on A S on A A A A

A A

100 N( ) 1.25 m/s80 kgx

F FF F m a a

m m∑ = = ⇒ = = = =

5-34 Chapter 5

Let us first calculate the positions and velocities of the astronaut and the satellite at t1 = 0.5 s under the accelerations aA and aS:

2 2 2f A f A i A f i A f i

2 2 2f S f S i S f i S f i

2f A f A A f i

1 1( ) ( ) ( ) ( ) ( ) 0 m 0 m (1.25 m/s )(0.5 s 0 s) 0.156 m2 2

1 1( ) ( ) ( ) ( ) ( ) 0 m 0 m ( 0.156 m/s )(0.5 s 0 s) 0.02 m2 2

( ) ( ) ( ) 0 m/s (1.25 m/s )(0.5 s 0 s) 0.625

x x v t t a t t

x x v t t a t t

v v a t t

= + − + − = + + − =

= + − + − = + + − − = −

= + − = + − =2

f S f S S f i

m/s( ) ( ) ( ) 0 m/s ( 0.156 m/s )(0.5 s 0 s) 0.078 m/sv v a t t= + − = + − − = −

With (xf )A and (xf )S as initial positions, (vf)A and (vf)S as initial velocities, and zero accelerations, we can now obtain the new positions at 2 1( ) 59.5 s:t t− =

f A f A i A f i( ) ( ) ( ) ( )x x v t t= + − 0.156 m (0.625 m/s)(59.5 s)= + = 37.34 m

f S f S i S f i( ) ( ) ( ) ( )x x v t t= + − 0.02 m ( 0.078 m/s)(59.5 s) 4.66 m= − + − = −

Thus the astronaut and the satellite are (37.34 m) ( 4.66 m) 42 m− − = apart.

P5.51. Prepare: We assume the rocket is moving in a vertical straight line along the y-axis under the influence of only two forces: gravity and its own thrust. The free-body diagram for the model rocket is shown later.

Solve: (a) Using Newton’s second law and reading the forces from the free-body diagram,

thrust thrust earthF w ma F ma mg− = ⇒ = + 2 2(0.200 kg)(10 m/s 9.8 m/s ) 4.0 N= + =

(b) Likewise, the thrust on the moon is (0.200 kg)(10 m/s2 + 1.62 m/s2) = 2.3 N. Assess: The thrust required is smaller on the moon, as it should be, given the moon’s weaker gravitational pull. The magnitude of a few newtons seems reasonable for a small model rocket.

P5.52. Prepare: The rocket is moving along the y-axis under the influence of two forces: the rocket’s thrust and the force of gravity. Its free-body diagram is shown, which, along with Newton’s second law (and using rocket’s initial mass), will help us find its initial acceleration. At 5000 m the acceleration has increased because the rocket mass has decreased. Utilizing the free-body diagram, Newton’s second law, and the increased acceleration, we can determine the decreased mass of the rocket.

Solve: (a) The y-component of Newton’s second law is

net thrust 2 2( ) 300,000 N 9.8 m/s 5.2 m/s

20,000 kgy

y

F F mga am m

−= = = = − =

(b) Solving the equation of part (a) for m gives


thrust5000 m 2 2

5000 m

300,000 N 18,990 kg6.00 m/s 9.8 m/s

Fma g

= = =+ +

The mass of fuel burned is mfuel = minitial – m5000 = 1010 kg or 1000 kg to two significant figures.

P5.53. Prepare: Your body is moving in a straight line along the y-direction under the influence of two forces: gravity and the support force of the scale. The free-body diagrams for you for the following three cases are shown below: no acceleration, upward acceleration, and downward acceleration. The apparent weight (see Equation 5.8) of an object moving in an elevator is app

app (1 ) ( 1) .wag ww w a g= + ⇒ = −

Solve: (a) When accelerating upward, the acceleration is

2 2170 lb 1 (9.8 m/s ) 1.3 m/s150 lb

a

= − =

(b) When braking, the acceleration is

2 2120 lb 1 (9.8 m/s ) 2.0 m/s150 lb

a

= − = −

Assess: A 10–20% change in apparent weight seems reasonable for a fast elevator, as the one in the Empire State Building must be. Also note that we did not have to convert the units of the weights from pounds to newtons because the weights appear as a ratio.

P5.54. Prepare: See P5.24. The child is not accelerating in the y-direction, so we can use Equation 5.1 for the forces perpendicular to the incline. Solve:

There are three forces with components in the y-direction, the normal force, the weight of the child, and the force of the rope. Equation 5.1 gives

sin( ) cos( ) 0n F wθ θ+ − =

Solving for the normal force, we have 2cos( ) sin( ) cos( ) sin( ) (23 kg)(9.80 m/s )cos(38 ) (30 N)sin(38 ) 160 Nn w F mg Fθ θ θ θ= − = − = ° − ° =

Assess: This is less than the child’s weight, as expected. Note that the force from the rope acts to decrease the normal force on the child since it tends to pull the child away from the incline. Compare to Problem 5.24, which is an identical situation except for the rope.

P5.55. Prepare: The length of the hill is = / sin .x h θ∆ The acceleration is sin .g θ Solve: First use the kinematic equation, with i = 0 m/sv at the top of the hill, to determine the speed at the bottom of the hill.

5-36 Chapter 5

2 2 2f 1 i 1 f 1( ) = ( ) 2 ( ) = 2( sin )( / sin ) = 2v v a x v g h ghθ θ+ ∆ ⇒

Now apply the same kinematic equation to the horizontal patch of snow, only this time we want .x∆ To connect the two parts f 1 i 2( ) = ( ) .v v The final speed is zero: f 2( ) = 0.v

2 2 2f 2 i 2 f 1( ) = ( ) 2 = ( ) 2 = 2 2 = 0v v a x v a x gh a x+ ∆ + ∆ + ∆

The friction force is the net force, so k= / .a f m− Note k k k= = .f n mgµ µ Solve for .x∆

k k k

2 3.0 m= = = = = = 60 m2 / / 0.05

gh gh gh hxa f m mg mµ µ

− −∆

−

Assess: It seems reasonable to glide 60 m with such a low coefficient of friciton. It is interesting that we did not need to know the angle of the (frictionless) slope; this will become clear in the chapter on energy. The answer is also independent of Josh’s mass.

P5.56. Prepare: We will use Newton’s second law with k = 0.20µ given in Table 5.1. We also orient our coordinate system so the x-axis runs down the ramp.

Solve:

k= sin = 0xF mg nθ µΣ −

= cos = 0yF n mg θΣ −

Solve the second equation for n and insert into the first.

ksin cos = 0mg mgθ µ θ−

1 1k ktan = = ( ) = (0.20) = 11.3 11tan tanθ µ θ µ− −⇒ ° ≈ °

Assess: The slope of the ramp seems shallow, but probably OK for the low coefficient of friction given.

P5.57. Prepare: The force plate reads the normal force n of the plate on the woman. The other force on the woman is her weight (the gravitational force of the earth down on her). The net force will be the sum of these two and will be different at different times as the normal force of the force plate changes according to the graph. Since the graph is piece-wise constant, the acceleration will be constant during (within) each phase of the jump. We can do a preliminary calculation to find the woman’s mass. During the standing still phases, the force plate reads 500 N. During these equilibrium phases the force plate reads the same magnitude as her weight (so w = 500 N); hence her mass must be m = w/g = (500 N)/(9.8 m/s2) = 51 kg. We will assume air resistance is negligible during all portions of the problem. Solve: (a) We now know m and the normal force during push-off (1000 N). Apply the second law:

21000 N 500 N 500 N 9.8 m/s51 kg 51 kg

y y

y

F n w man wa

m

∑ = − =

− −= = = =

This result looks familiar, but it is not the acceleration of an object in free fall for two reasons: (1) she is not in free fall, and (2) this acceleration is up while objects in free fall accelerate down. (b) After she leaves the force plate she is in free fall, so her acceleration is a = –g or, in other words, 9.8 m/s2, down. (c) During the landing phase the normal force of the plate on her is 1500 N while her weight is still 500 N. Apply the second law:


21500 N 500 N 1000 N 20 m/s51 kg 51 kg

y y

y

F n w man wa

m

∑ = − =

− −= = = =

This acceleration is positive, or up (opposite the direction of motion, as she is slowing down). (d) We’ll assume she accelerates from rest. We are given that the push-off phase lasts 0.25 s.t∆ = We’ll use the answer from part (a) for .ya

2f i( ) ( ) 0.0 m/s (9.8 m/s )(0.25 s) 2.45 m/s 2.5 m/sy y yv v a t= + ∆ = + = ≈

(e) After she leaves the force table she is in free fall (see part (b)). What was the final velocity in part (d) becomes the initial velocity now. Use Equation 2.13 with f( ) 0.0 m/s.yv =

2 2f i( ) ( ) 2y y yv v a x= + ∆

Solve for :y∆

2 2i

2

( ) (2.45 m/s) 0.31 m2 2( 9.8 m/s )

y

y

vy

a− −

∆ = = =−

Assess: All of these results appear reasonable. The accelerations are within the expectations of daily life experience. The result of part (d) is used in part (e), so we kept a third significant figure to use in the last calculation, but still reported the answers to two significant figures. The last answer is kind of a check on the previous ones, and it is quite reasonable: 31 cm is just over one foot. Also review each calculation to verify that the units work out.

P5.58. Prepare: The sprinter is accelerating in the horizontal direction, so we can apply Equation 5.2. Since the sprinter’s acceleration is parallel to the ground, the net force in the vertical direction is zero. Solve: (a) Refer to the free-body diagram below. We consider the sprinter to be the system.

The sprinter’s acceleration in the horizontal direction is entirely due to the force of static friction on his foot. We can find the magnitude of that force by using Newton’s second law.

2s (77 kg)(4.7 m/s ) 362 Nxf ma= = =

The track also exerts a normal force on his foot. From the diagram above, 755 N.n w mg= = = An additional significant figure has been kept in these intermediate results for use later. The total force that the track exerts on his foot is the resultant of the friction force and the normal force. The magnitude of the total force is

2 2 2 2track on sprinter s( ) ( ) (362 N) (755 N) 837 NF f n= + = + =

This should be reported as 840 N to two significant figures. An additional figure has been kept for use in part (b). The angle this force makes with the horizontal is

1 755 Ntan 64.4362 N

θ − = = °

An addition figure has been kept for use in part (b). This should be reported as 64 .°

5-38 Chapter 5

(b) In part (a), we found the force of the track on the sprinter. Now consider the foot to be the system. The force of the track on the foot is equal and opposite to the force of the lower leg on the foot since the foot is in equilibrium. So the force of the lower leg on the foot has a magnitude of 837 N and is in the direction shown in the figure later.

To calculate the components of the force along and perpendicular to the leg, we will use coordinate axes such that the x-axis is along the leg, and the y-axis is perpendicular to the leg. These axes are shown solid in the figure. The angle between the force and these axes is 64.4 60 4.4 .° − ° = ° The component of the force along the leg is

leg on foot( ) (837 N)cos(4.4 ) 830 NxF = − ° = −

The component of this force perpendicular to the leg is

leg on foot( ) (837 N)sin(4.4 ) 60 NyF = − ° = −

Assess: Carefully choosing what to consider as the system is essential to understanding this problem. Note that by adjusting his stride, the sprinter can minimize the component of the force perpendicular to the leg to avoid dislocation.

P5.59. Prepare: Sam is moving along the x-axis under the influence of two forces: the thrust of his jet skis and the resisting force of kinetic friction on the skis. A visual overview of Sam’s motion is shown below in a pictorial representation, motion diagram, free-body diagrams in the accelerating and coasting periods, and a list of values. To find Sam’s top speed using kinematics, we will first find his acceleration from Newton’s second law. Kinematic equations will then allow us to find his displacement during both accelerating and coasting intervals.


Solve: (a) The friction force of the snow can be found from the free-body diagram and Newton’s first law, since there’s no acceleration in the vertical direction:

2(75 kg)(9.8 m/s ) 735 Nn w mg= = = = k k (0.1)(735 N) 73.5 Nf nµ⇒ = = =

Then, from Newton’s second law,

thrust k 2net thrust k 0 0

200 N 73.5 N( ) 1.687 m/s75 kgx

F fF F f ma am

− −= − = ⇒ = = =

From kinematics: 2

1 0 0 1 0 m/s (1.687 m/s )(10 s) 16.87 m/sv v a t= + = + = or 17 m/s to two significant figures.

(b) During the acceleration, Sam travels to

2 2 21 0 0 1 0 1

1 1 (1.687 m/s )(10 s) 84 m2 2

x x v t a t= + + = =

After the skis run out of fuel, Sam’s acceleration can again be found from Newton’s second law:

net 2net k 1

73.5 N( ) 73.5 N 0.98 m/s75 kgx

FF f am

−= − = − ⇒ = = = −

Since we don’t know how much time it takes Sam to stop,

2 22 1 1 2 12 ( )v v a x x= + −

2 2 2 2 22 1

2 1 21

0 m /s (16.87 m/s) 145 m2 2( 0.98 m/s )

v vx xa− −

⇒ − = = =−

The total distance traveled is 2 1 1( ) 145 m 84 m 229 mx x x− + = + = or 230 m to two significant figures. Assess: A top speed of 16.9 m/s (roughly 40 mph) seems quite reasonable for this acceleration, and a coasting distance of nearly 150 m also seems possible, starting from a high speed, given that we’re neglecting air resistance.

P5.60. Prepare: The book is in static equilibrium so Equation 5.1 can be applied. The maximum static frictional force the person can exert will determine the heaviest book he can hold. Solve: Consider the free-body diagram below. The force of the fingers on the book is the reaction force to the normal force of the book on the fingers, so is exactly equal and opposite the normal force on the fingers.

The maximal static friction force will be equal to s max s (0.80)(6.0 N) 4.8 N.f nµ= = = The frictional force is exerted on both sides of the book. Considering the forces in the y-direction, we have that the weight supported by the maximal frictional force is

s max s max s max2 9.6 Nw f f f= + = =

Assess: Note that the force on both sides of the book are exactly equal also because the book is in equilibrium.

P5.61. Prepare: We show below the free-body diagram of the 1 kg block. The block is initially at rest, so initially the friction force is static friction. If the 12 N pushing force is too strong, the box will begin to move up the wall. If it is too weak, the box will begin to slide down the wall. And if the pushing force is within the proper range, the box will remain stuck in place.

5-40 Chapter 5

Solve: First, let’s evaluate the sum of all the forces except friction:

push pushcos 30 0 N cos 30xF n F n F∑ = − ° = ⇒ = ° 2

push pushsin 30 sin 30 (12 N)sin 30 (1 kg)(9.8 m/s ) 3.8 NyF F w F mg∑ = ° − = ° − = ° − = −

In the first equation we have utilized the fact that any motion is parallel to the wall, so ax = 0 m/s2. The two forces in the second y-equation add up to 3.8 N.− This means the static friction force will be able to prevent the box from moving if s 3.8 N.f = + Using the x-equation we get

fs max = µsn = µsFpush cos 30° = 5.2 N

where we used µs = 0.5 for wood on wood. The static friction force sf

needed to keep the box from moving is less than fs max. Thus the box will stay at rest.

P5.62. Prepare: The locomotive is subject to rolling friction but not to drag (because of its slow speed and large mass). So, the locomotive is decelerating in a straight line under the influence of the rolling friction of the wheels on the runway. We let the x-axis run to the right, and show below a visual overview of the situation that includes a pictorial representation, a motion diagram, free-body diagram, and a list of values. We will first find the acceleration a from Newton’s second law and then apply kinematics to find the locomotive’s displacement.

Solve: The locomotive is not accelerating in the vertical direction, so the free-body diagram shows us that n = w = mg. Thus,

2r r (0.002)(50,000 kg)(9.8 m/s ) 980 Nf mgµ= = =

From Newton’s second law for the decelerating locomotive,

2980 N 0.0196 m/s50,000 kgx

fam

τ− −= = = −

Since we’re looking for the distance the train rolls, but we don’t have the time:


2 2 2 22 2 f if i 2

(0 m/s) (10 m/s)2 ( ) 2600 m2 2( 0.0196 m/s )x

x

v vv v a x xa− −

− = ∆ ⇒ ∆ = = =−

Assess: The locomotive’s enormous inertia (mass) and the small coefficient of rolling friction make this long stopping distance seem reasonable.

P5.63. Prepare: We assume that the plane is accelerating in a straight line under the influence of two forces: the thrust of its engines and the rolling friction of the wheels on the runway. We let x-axis run to the right, and show below a visual overview of the situation that includes a pictorial representation, a motion diagram, a free-body diagram and a list of values. We will use one-dimensional kinematics to find acceleration a, and then apply Newton’s second law.

Solve: We obtain

2

net thrust r thrust r

82 m/s 0 m/s 2.34 m/s35 s

( ) x

vat

F F F f ma F f ma

∆ −= = =

∆= ∑ = − = ⇒ = +

For rubber rolling on concrete, r 0.02µ = (Table 5.1), and since the runway is horizontal, n = w = mg. Thus

thrust r r r2 2 5

( )(75,000 kg)[(0.02)(9.8 m/s ) 2.34 m/s ] 1.9 10 N

F w ma mg ma m g aµ µ µ= + = + = +

= + = ×

Assess: It’s hard to evaluate such an enormous thrust, but comparison with the plane’s mass suggests that 190,000 N is enough to produce the required acceleration.

P5.64. Prepare: We let the x-axis run along the ramp and show below a visual overview of the situation that includes a pictorial representation, a motion diagram, free-body diagrams, and a list of values. The motion diagram shows decreasing velocity vectors as the block moves uphill, so the acceleration vector is opposite the direction of the velocity vector. But, for the downhill motion, the acceleration and the velocity vector are in the same direction. The block ends where it starts, so x2 = x0 = 0 m. We expect v2 to be negative, because the block will be moving in the –x-direction, so we’ll want to take |v2| as the final speed. Because of friction, we expect to find |v2| < v0.

5-42 Chapter 5

Solve: (a) The friction force is opposite to ,v→ so kf

points down the slope during the first half of the motion and up the slope during the second half; w and n are the only other forces. Newton’s second law for the upward motion is

net k k0

net2

( ) sin sin

( ) cos cos0 m/s

xx

yy

F w f mg fa am m m

F n w n mgam m m

θ θ

θ θ

− − − −= = = =

− −= = = =

First solve the y-equation to give n = mgcosθ. Use this in k kf nµ= to get k k cos .f mgµ θ= Now substitute this result for fk into the x-equation:

2 2k0 k

sin cos (sin cos ) (9.8 m/s )(sin 35 0.20 cos 35 ) 7.23 m/smg mga gm

θ µ θ θ µ θ− −= = − + = − ° + ° = −

Kinematics now gives 2 2 2 2 21 02 2

1 0 0 1 0 1 20

0 m /s (10 m/s)2 ( ) 6.92 m2 2( 7.23 m/s )

v vv v a x x xa− −

= + − ⇒ = = =−

The block’s height is then h = x1sin θ = (6.92 m)sin 35° = 4.0 m. (b) For the return trip, kf

points up the slope, so the x-component of the second law is

net k k1

( ) sin sinxx

F w f mg fa am m m

θ θ− + − += = = =

Note the sign change. The y-equation and the friction equation k kf nµ= are unchanged, so we have

21 k(sin cos ) 4.02 m/sa g θ µ θ= − − = −

The kinematics for the return trip are

2 2 22 1 1 2 1 2 1 12 ( ) 2 2( 4.02 m/s )(6.92 m) 7.5 m/sv v a x x v a x= + − ⇒ = − = − − = −

Notice that we used the negative square root because v2 is a velocity with the vector pointing in the –x-direction. The final speed is |v2| = 7.5 m/s.

P5.65. Prepare: With no friction, the only forces along the incline are the tension T and a component of the weight, sin .mg θ Since the blocks aren’t accelerating then = sin .T mg θ Solve: For tension 1:

21 = (5 kg)(9.8 m/s )sin 20 = 16.76 N 17 NT ° ≈


For tension 2 the mass is the sum of both blocks: 2

2 = (5 kg 3 kg)(9.8 m/s )sin 20 = 26.81 N 27 NT + ° ≈

Assess: We would expect 2 1>T T .

P5.66. Prepare: Call the force we seek ,F and the mass of one block .m The maximum force without slippage is when the friction force between the blocks is maximum: s s s= = .f n mgµ µ Solve: For the two-block system F is the net force.

= (2 )F m a

Considering only the top block, sf is the net force, so s = .f ma

s s s= = =f n mg maµ µ

Now insert our latest expression for ma into the equation for the two-block system. 2

s s= 2 = 2( ) = 2( ) = 2(0.35)(9.8 m/s ) = 6.86 NF ma mg mg m mµ µ

Assess: We need to know m to get a numerical solution.

P5.67. Prepare: Since the block comes to rest for an instant, we use the coefficient of static friction for wood on wood: s = 0.50.µ Solve:

s= sin = 0xF mg nθ µΣ −

= cos = 0yF n mg θΣ −

Solve the second equation for n and insert into the first.

ssin cos = 0mg mgθ µ θ−

1 1s stan = = ( ) = (0.50) = 26.57 27tan tanθ µ θ µ− −⇒ ≈

Assess: From experience, 27° seems like a reasonable tilt for the block to slide back down.

P5.68. Prepare: Assume the 614 mph is the terminal speed at 90,000 ft, and 125 mph is the terminal speed at low altitude where we know the density to be 3= 1.22kg/m ,ρ and all other variables remain unchanged. We can use Equation 5.16 and ratios to solve this problem simply. Solve: Solve Equation 5.16 for ρ : 2= 4 / .mg v Aρ

2 2 223 3hi lo lohi

hi lo2 2 2l hi hi

2lo

4(125 mph)= = = = (1.22 kg/m ) = 0.051 kg/m4 (614 mph)o

mgv vv A

mg v vv A

ρ ρ ρρ

⇒

Assess: The answer is very small, but at 17 miles (90,000 ft) up one is above most of the air, and the density really is small. Commercial airplanes genrally cruise at 30,000 ft to 37,000 ft.

P5.69. Prepare: The Ping-Pong ball when shot straight up is subject to a net force that is the resultant of the weight and drag force vectors, both acting vertically downward. On the other hand, for the ball’s motion straight down, the ball is subject to a net force that is the resultant of the weight and drag force vectors, the former in the downward and the latter in the upward direction. An overview of a pictorial representation and a free-body diagram are shown below. The Ping-Pong ball experiences a drag force equal to 21

4 ,Avρ as modeled in the text

with termv as the terminal velocity.

5-44 Chapter 5

Solve: (a) Imagine the ball falling at its terminal speed. The ball’s weight is directed down and the resistive drag force is directed up. The net force is zero because the magnitude of the drag force is equal to the magnitude of the weight, D = w. When the ball is shot upward at twice the terminal speed, the drag force is four times the terminal drag force. That is, 4 4 .D D w′ = = Since all the forces are down, the y-component of Newton’s second law is

4 5yF D w w w mg ma′∑ = − − = − − = − = ⇒ 5a g= −

(b) The ball is initially shot downward. Therefore D′′ is upward but w is down. Again 4D D′′ = and the y-component of Newton’s second law is

4 3yF D w w w mg ma′′∑ = − = − = = ⇒ 3a g=

That is, the ball initially decelerates at 3g but as v becomes smaller, the drag force approaches the weight so the deceleration goes to zero and v approaches vterm. Assess: D′ is very large and with w yields a large initial deceleration when the ball is shot up. When the ball is shot down w opposes D′′ so the ball decelerates at a lesser rate.

P5.70. Prepare: Call the 2.0 kg block 2m and the 1.0 kg block 1.m Assume the pulley is massless and frictionless. Solve: On block 2 k k k 2= = .f n m gµ µ

k k 2 2 2= = =xF T n T m g m aµ µΣ − −

Block 1 is also accelerating.

1 1 1= =yF T m g m aΣ −

The acceleration constraint is 2 1( ) = ( ) = .x ya a a− Solve for T in the second equation and insert in the first.

1= ( ).T m g a−

1 k 2 2( ) =m g a m g m aµ− −

1 k 2 2 1=m g m g m a m aµ− +

22 21 k 2

1 2

( ) (9.8 m/s )(1.0 kg (0.20)(2.0 kg))= = = 1.96 m/s 2.0 m/s1.0 kg 2.0 kg

g m mam m

µ− −≈

+ +

Assess: The answer seems reasonable.

P5.71. Prepare: Call the 10 kg block 2m and the 5.0 kg block 1.m Assume the pulley is massless and frictionless. Solve: On block 2 use tilted axes.

2 2 2= = sin =xF T m g m aθΣ −

Block 1 is also accelerating.


1 1 1= =yF T m g m aΣ −

The acceleration constraint is 2 1( ) = ( ) =x ya a a− . Solve for T in the second equation and insert in the first.

1= ( ).T m g a−

1 2 2( ) sin =m g a m g m aθ− −

1 2 2 1sin =m g m g m a m aθ− +

22 2 21 2

1 2

( sin ) (9.8 m/s )(1.0 kg (2.0 kg)sin 40 )= = = 1.96 m/s 2.0 m/s = 0.93 m/s1.0 kg 2.0 kg

g m mam m

θ− − °≈ −

+ +

Or 20.93 m/s , down the ramp. Assess: The answer depends on ;θ for a shallow angle the block accelerates up the ramp, for a steep angle the block accelerates down the ramp. This is expected behavior.

P5.72. Prepare: Apply Newton’s second law to each block in turn. The blocks aren’t accelerating, so = 0.FΣ Call the force we seek .F

Solve: On block 1:

k 1= ( ) = 0xF F T fΣ − −

On block 2:

k 2= ( ) = 0F T fΣ −

Solve for T and insert in the first equation. 2

k 2 k 1 k 2 1= ( ) ( ) = ( ) = (0.20)(9.8 m/s )(5 kg) = 9.8 NF f f g m mµ+ +

Assess: It would be even easier to consider the two blocks as one system with one combined friction force and one forward force F which we seek. Then one doesn’t need to ever solve for .T You only need one application of Newton’s second law. The answer is the same.

P5.73. Prepare: We assume the dishes have a constant acceleration during the 0.25 s, and we’ll use a kinematic equation as well as Newton’s second law. The friction force is the net force, net k k= = .F f mgµ We also know net= / .a F m Solve: Use the kinematic equation (where i = 0v ):

2 2 2 2 2 2net kk

1 1 1 1 1= ( ) = ( ) = ( ) = ( ) = (0.12)(9.8 m/s )(0.25 s) = 3.7 cm2 2 2 2 2

F mgx a t t t g tm m

µ µ∆ ∆ ∆ ∆ ∆

Assess: This seems to be a reasonable distance the dishes could travel without falling off the edge.

P5.74. Prepare: We will assume that the rope is massless and the pulley is both massless and frictionless. We will use the constant-acceleration kinematic equations for m and M (=100 kg) because the two masses move together with the same magnitude of acceleration. A visual overview shows a pictorial representation, list of values, and free-body diagrams for the two masses.

5-46 Chapter 5

Solve: Using 21

f i i f i M f i2( ) ( ) ( ) ,yy y v t t a t t= + − + −

21( 1 m) 0 m 0 m (6.0 s 0 s)2 Ma− = + + − 20.0556 m/sMa⇒ = −

Newton’s second law for m and M is

on R on( )m y m m mF T w ma∑ = − = on R on( )M y M M MF T w Ma∑ = − =

The acceleration constraint is .m Ma a= − Also, the tensions are an action/reaction pair, thus R on R on .m MT T= With these, the second law equations are

R on

R on

M M

M M

T Mg MaT mg ma

− =− = −

Subtracting the second from the first gives

9.8 0.556(100 kg) 99 kg9.8 0.556

MM M

M

g aMg mg Ma ma m Mg a

+ − − + = + ⇒ = = = − +

Assess: Note that am = − aM = .0556 m/s2. For such a small acceleration, a mass of 99 kg for m compared to 100 kgM = is understandable.

P5.75. Prepare: Please refer to Figure P5.75. The free-body diagram implies accelerating motion along the x-direction, as the y-forces cancel out. Solve: (a) A 1 kg block is pulled across a level surface by a string, starting from rest. The string has a tension of 20 N, and the block’s coefficient of kinetic friction is 0.50. How long does it take the block to move 1 m? (b) Newton’s second law for the block is

net2net k k ( )( ) 0 m/s yxx y

FF T f T n n w n mga a am m m m m m

µ− − − −= = = = = = = =

where we have incorporated the kinetic friction Equation 5.13 into the first equation. The second equation gives n = mg. Substituting this into the first equation gives

k 220 N 4.9 N 15.1 m/s1 kg

T mgamµ− −

= = =

Finally, constant acceleration kinematics gives

2 2 ff i i 2

1 1 2 2(1 m)( ) ( ) 0.36 s2 2 15.1 m/s

xx x v t a t a t ta

= + ∆ + ∆ = ∆ ⇒ ∆ = = =

P5.76. Solve: (a) A 15,000 N truck starts from rest and moves down a 15° hill with the engine providing an 12,000 N force in the direction of the motion. Assume the frictional force between the truck and the road is very small. If the hill is 50 m long, what will be the speed of the truck at the bottom of the hill? (b) Newton’s second law is


22

0

0 N sin 0 N 12,000 Nsin 12,000 N (15,000 N) sin 15 12,000 N 10.4 m/s

(15,000 N/9.8 m/s )

y y y y y y

xx x x x x x

x

F n w f E maF n w f E ma w ma

wam

θθ

∑ = + + + = =

∑ = + + + = ⇒ + + + =+ ° +

⇒ = = =

where we have calculated the mass of the truck from its weight. Using the constant-acceleration kinematic equation 2 2f i 2 ,xv v a x− = ∆

2 2f 2 2(10.4 m/s )(50 m)xv a x= ∆ = ⇒ vx = 32 m/s

Assess: Note that the normal force of 14,500 N cancels with the vertical component of the weight, so there is no motion along the y-direction.

P5.77. Solve: (a) A driver traveling at 40 m/s in her 1500 kg auto slams on the brakes and skids to rest. How far does the auto slide before coming to rest? (b)

(c) Newton’s second law is

0 N 0.8y y y y x xF n w n mg ma F n ma∑ = + = − = = ∑ = − =

The y-component equation gives 2(1500 kg)(9.8 m/s ).n mg= = Substituting this into the x-component equation yields

2(1500 kg) 0.8(1500 kg)(9.8 m/s )xa = − 2 2( 0.8)(9.8 m/s ) 7.8 m/sxa⇒ = − = −

Using the constant-acceleration kinematic equation 2 2f i 2 ,xv v a x= + ∆ we find

2 2i

2

(40 m/s) 100 m2 2( 7.8 m/s )x

vxa

∆ = − = − =−

P5.78. Prepare: The three given equations imply an accelerating object on an incline because the third term in the first equation and the second term in the second equation are the two components of the weight .w We also see a force T

applied to the object through a rope along the incline. Solve: (a) A 20.0 kg wooden crate is being pulled up a 20° wooden incline by a rope that is connected to an electric motor. The crate’s acceleration is measured to be 2 m/s2. The coefficient of kinetic friction between the crate and the incline is 0.2. Find the tension T in the rope. (b)

(c) Newton’s second law for this problem in the component form is

5-48 Chapter 5

2 2net

2net

( ) 0.2 (20 kg)(9.80 m/s )sin 20 (20 kg)(2 m/s )

( ) (20 kg)(9.80 m/s )cos 20 0 Nx x

y y

F F T n

nF F

= ∑ = − − ° =

= = − ° =

Solving the y-component equation, 184 N.n = Substituting this value for n in the x-component equation yields 144 N.T = The tension should be reported as 140 N to two significant figures.

P5.79. Prepare: The three given equations imply an accelerating object on a flat surface because the third term in the second equation is simply the weight w = mg. We also see a force applied to the object at an angle of 30° relative to the horizontal. Solve: (a) You wish to pull a 20 kg wooden crate across a wood floor (µk = 0.2) by pulling on a rope attached to the crate. Your pull is 100 N at an angle of 30° above the horizontal. What will be the acceleration of the crate? (b)

(c) Newton’s equations and the model of kinetic friction are

k k

k k

0 N (100 N)cos 30 0 N (100 N)cos 30(100 N)sin 30 0 N 0 N

x x x x x x

y y y y y y

F n P w f f f maF n P w f n mg maf nµ

∑ = + + + = + ° + − = ° − =∑ = + + + = + ° − − = =

=

From the y-component equation, 150 N.n = From the x-component equation and using the model of kinetic friction with µk = 0.2,

(100 N) cos 30° – (0.2)(150 N) = (20 kg)ax 22.8 m/sxa⇒ =

P5.80. Prepare: We can use Newton’s second law to calculate the force needed to bring the stone up to speed. Solve: Ignoring friction, the only force on the stone is the force the curler applies. Using Newton’s second law, we can calculate the force given the acceleration and mass of the stone. The acceleration of the stone is given by the definition of acceleration, Equation 2.8.

23.0 m/s 1.5 m/s2.0 s

xx

vat

∆= = =

∆

The force on the stone is then 2(20 kg)(1.5 m/s ) 30 Nx xF ma= = =

The correct choice is C. Assess: We will check that ignoring friction is a reasonable assumption in Problem 5.82.

P5.81. Prepare: While the coefficient of friction is low, it is not zero, or the stone would never stop. It is the kinetic friction force that slows the stone. Solve: Since sweeping can in fact lengthen the travel of the stone it must be that sweeping decreases the kinetic friction force (by decreasing the coefficient of kinetic friction). The correct choice is A. Assess: Examine the alternate choices. If sweeping increased the coefficient of friction then the stone would stop sooner, not later. As for C and D, sweeping the ice does nothing to change the fact that the stone is in motion. As long as the stone is moving we are talking about kinetic friction, not static friction. Since sweeping decreases the coefficient of friction, it is probably accomplished by making the ice smoother.


P5.82. Prepare: We can use the kinematic equations to find the acceleration of the stone and then use Newton’s second law to find the approximate magnitude of the force of friction on the stone. Solve: Using Equation 2.13 with f 0 m/s,v = i 3 m/s,v = and 40 m,x∆ = we have

2 2i 2( ) (3 m/s) 0.1125 m/s

2 2(40 m)xva

x= − = =

∆

Where additional significant figures have been kept in this intermediate result. The force on the stone is then

2(20 kg)(0.1125 m/s ) 2 Nx xF ma= = =

The correct choice is B. Assess: Compare to Problem 5.80. The force of friction is much less than the force the curler applies, so ignoring friction in Problem 5.80 is reasonable.

P5.83. Prepare: Let’s draw a free-body diagram. Assume the stone is moving in the positive x-direction on level ice.

Solve: Newton’s second law in the vertical direction tells us that n w= so .n mg= Now apply the second law in the horizontal direction.

k

k

k

k

cancel

x x

x

x

x

x

F maf man ma

mg ma mg a

µµ

µ

Σ =− =

− =− =

− =

Since m canceled we see that the deceleration is independent of m. Therefore the distance the stone travels will be the same as before if it has the same initial velocity (which it does). This eliminates choices A and B. The coefficient of friction between two substances only depends on the microscopic makeup of the substances and does not depend on the normal force or the speed (in our simple model) of the two surfaces relative to each other. If it were otherwise, then tables of the coefficient of friction would have to have entries for different normal forces and speeds. This eliminates choice C. The only choice left (and the correct one) is D; however we actually already knew that when we first wrote fk = µkn = µkmg. The mass appears in the equation and tells us that in this scenario the friction force is proportional to the mass. Assess: Although the friction force is greater in the 40 kg case, so is the inertia (which tends to keep the stone moving), so the acceleration remains the same.

APPLYING NEWTON S LAWS

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