Dynamics IMotion Along a

Line

• Example: consider the situation below where two ropes hold up a weight:

qleft =30o qright = 55o

Tleft Tright

W = 100 Nt

Static Equilibrium

Dynamic Equilibrium

An elevator that is moving upward at 2 m/s. Find the tension T in the cable, if elevator has mass 400 kg

RampsWhy is it easier to push something up a ramp

than it is to lift it?

P

RampsF// = P - mg sin(q) = mg//

F = FN - mg cos(q) = mg = 0

W=mg

P

q F

q

F//

FN

Pbalance = mg sin(q)

Ramps

F// = P – F//- Ff = ma//

W=mg

FN

P

Ff =Fc

F//

Ramps

W=mg

FN

PFf =FN

FN = mg cos(q)P = mg sin(q) + mg cos(q) = mg[sin(q) + cos(q)]

sin(q) + cos(q) < 1 → P < mg

Pulleys

Pulleys

P

W = mg

P = Tension = W

Pulleys

P

W

T T

P=T, 2T = W; so P=W/2

Pulleys

P

Pulleys

P

W

Newton’s law of gravitation

g 1 2F m m

g 2

1Fr

1 2g 2

Gm mFr

G = 6.67 x 10-11 Nm2/kg2

Problem 1• Three masses are each at a vertex of an

isosceles right triangle as shown. Write an expression for the force on mass three due to the other two.

r

r

m3

m1

m2

3 1 2n

Gm m mF

r

Gravity at earth’s surface

E 2G 2

Gm mFr

E2

Gm mmgr

E2

Gmgr

Gravity:• What is the force of gravity exerted by the earth on a

typical physics student?

– Typical student mass m = 55kg– g = 9.81 m/s2.– Fg = mg = (55 kg)x(9.81 m/s2 )– Fg = 540 N = WEIGHT

Fg = mg

Test your Understanding• An astronaut on Earth kicks a bowling ball and hurts

his foot. A year later, the same astronaut kicks a bowling ball on the moon with the same force. His foot hurts...

(a) more

(b) less (c) the same

Ouch!

• However the weights of the bowling ball and the astronaut are less:

• Thus it would be easier for the astronaut to pick up the bowling ball on the Moon than on the Earth.

W = mgMoon gMoon < gEarth

Test your Understanding

Friction

The forces shown are an action-reaction pair.

Acme Hand

Grenades

f (force on table due to box)

f v

(force on box due to table)

Friction...• Friction is caused by the “microscopic”

interactions between the two surfaces:

Two Kinds of Friction

• Static friction– Must be overcome in order to

budge an object– Present only when there is no

relative motion between the bodies, e.g., the box & table top

• Kinetic friction– Weaker than static friction– Present only when objects are

moving with respect to each other (skidding)

FAfk

Fnet is to the right.a is to the right.v is left or right.

FAfs

Objects are still or moving together. Fnet = 0.

Friction Facts

• Friction is due to electrostatic attraction between the atoms of the objects in contact.

• It can speed you up, slow you down, or make you turn.• It allows you to walk, turn a corner on your bike, warm

your hands in the winter, and see a meteor shower.• Friction often creates waste heat.• It makes you push harder / longer to attain a given

acceleration.• Like any force, it always has an action-reaction

pair.

Friction Strength

The magnitude of the friction force is proportional to:• how hard the two bodies are pressed together (the

normal force, N ).• the materials from which the bodies are made (the

coefficient of friction, ).

Attributes that have little or no effect:• sliding speed• contact area

Coefficients of Friction

• Static coefficient … s.• Kinetic coefficient … k.• Both depend on the materials in contact.

– Small for steel on ice or scrambled egg on Teflon frying pan

– Large for rubber on concrete or cardboard box on carpeting

• The bigger the coefficient of friction, the bigger the frictional force.

Friction...Force of friction acts to oppose motion:

Parallel to surface.Perpendicular to Normal force.

maF

fF mg

N

i

j

Model for Sliding Friction

The “heavier” something is, the greater the friction will be...makes sense!

fF N

fF N

Dynamicsi : F KN = maj : N = mg

so F Kmg = ma

maF

mg

Ni

j

K mg

Test your Understanding

• A box of mass m1 = 1.5 kg is being pulled by a horizontal string having tension T = 90 N. It slides with friction (k = 0.51) on top of a second box having mass m2 = 3 kg, which in turn slides on a frictionless floor.– What is the acceleration of the second box ?

(a) a = 0 m/s2 (b) a = 2.5 m/s2 (c) a = 3.0 m/s2

m2

Tm1 slides with friction (k=0.51 )

slides without frictiona = ?

Solution

• First draw FBD of the top box:

m1

N1

m1g

T f = KN1 = Km1g

Newtons 3rd law says the force box 2 exerts on box 1 is equal and opposite to the force box 1 exerts on box 2.

m1 f1,2

m2 f2,1

= Km1g

Solution

• Now consider the FBD of box 2:

m2 f2,1 = km1g

m2g

N2

m1g

Solution

Finally, solve F = ma in the horizontal direction:

m2

f2,1 = Km1g

Km1g = m2a1

2k

ma gm

21.5 0.51 9.813

kg m skg

a = 2.5 m/s2

Solution

Inclined Plane with Friction:• Draw free-body diagram:

q

i

jmg

N

KNma

q

Inclined plane...Consider i and j components of

FNET = ma : i mg sin q KN = ma j N = mg cos q

q

i

j

mg

N

q

KNma

mg sin q

mg cos q

mg sin q Kmg cos q = ma

a / g = sin q Kcos q

Static Friction...

F

mg

N

i

j

fF

• So far we have considered friction acting when something moves.– We also know that it acts in un-moving “static”

systems:• In these cases, the force provided by friction will

depend on the forces applied on the system.

Static Friction...• Just like in the sliding case except a = 0.

i :F fF = 0j :N = mg

F

mg

Ni

j

fF

While the block is static: fF F

Static Friction...

F

mg

Ni

j

fF

Static Friction...S is discovered by increasing F until the

block starts to slide:i : FMAX SN = 0j : N = mg S FMAX / mg

FMAX

mg

N

i

j

Smg

• A box of mass m =10.21 kg is at rest on a floor. The coefficient of static friction between the floor and the box is s = 0.4.

• A rope is attached to the box and pulled at an angle of q = 30o above horizontal with tension T = 40 N.– Does the box move?

(a) yes (b) no (c) too close to callT

m

static friction (s = 0.4 ) q

Test Your Understanding

Pick axes & draw FBD of box:Apply FNET = ma

y: N + T sin q - mg = maY = 0

N = mg - T sin q = 80 N

x: T cos q - fFR = maX

The box will move if T cos q - fFR > 0

y

xT

m q

N

mg

fFR

Solution

x: T cos q - fFR = maX

y: N = 80 N

The box will move if T cos q - fFR > 0

T cos q = 34.6 NfMAX = sN = (.4)(80N) = 32 N

So T cos q > fMAX and the box does move

T

mfMAX = sN

N

mg

y

x

q

Solution

Static Friction:

• We can also consider S on an inclined plane.

• In this case, the force provided by friction will depend on the angle q of the plane.

q

Static Friction...

ma = 0 (block is not moving)

• The force provided by friction, fF , depends on q.

mg sin q ff (Newton’s 2nd Law along x-axis)

qmg

N

q

fF

i

j

Static Friction...• We can find s by increasing the ramp angle until the

block slides:

In this case:

mg sin qM Smg cos qM

S tan qM qM

mgN

SN

qi

j

mg sin q ff

ff SN Smg cos qM

Friction Facts• Since fF = N , the force of friction does

not depend on the area of the surfaces in contact.

• By definition, it must be true that S > K

for any system (think about it...).

Friction vs Applied force

fF

FA

fF = FA

fF = KN

fF = SN

Problem: Box on Truck

A box with mass m sits in the back of a truck. The coefficient of static friction between the box and the truck is S.What is the maximum acceleration a that

the truck can have without the box slipping?

m S

a

Problem: Box on Truck• Draw Free Body Diagram for box:

– Consider case where fF is max...(i.e. if the acceleration were any larger, the box would slip).

N

fF = SN mg

i

j

Problem: Box on TruckUse FNET = ma for both i and j components

I : SN = maMAX

J: N = mgaMAX = S g

N

fF = SN mg

aMAXi

j

Forces and Motion • An inclined plane is accelerating with constant

acceleration a. A box resting on the plane is held in place by static friction. What is the direction of the static frictional force?

(a) (b) (c)

Ff

Ff Ff

S a

Solution

• First consider the case where the inclined plane is not accelerating.

mg

Ff

N

All the forces add up to zero!

mg

NFf

If the inclined plane is accelerating, the normal force decreases and the frictional force increases, but the frictional force still points along the plane:

mg

NFf

a

All the forces add up to ma!F = maThe answer is (a)

mg

Ff

Nma

Solution

Putting on the brakes

• Anti-lock brakes work by making sure the wheels roll without slipping. This maximizes the frictional force slowing the car since S > K .

• The driver of a car moving with speed vo slams on the brakes. The coefficient of static friction between the wheels and the road is S . What is the stopping distance D?

abvo

v = 0

D

Putting on the brakesUse FNET = ma for both i and j components

i : SN = maj: N = mg

a = S gN

fF = SN mg

ai

j

Putting on the brakes

As in the last example, find ab = Sg. Using the kinematic equation:

v2 - v0

2 = 2a( x -x0 ) In our problem: 0 - v0

2 = 2ab( D )

ab

vov = 0

D

Putting on the brakes In our problem: 0 - v0

2 = 2ab( D ) Solving for D:

Putting in ab = Sg

abvo

v = 0

D

20

2 b

vDa

20

2 s

vDg

Dynamics Problem• A box of mass m = 2 kg slides on a horizontal

frictionless floor. A force Fx = 10 N pushes on it in the x direction. What is the acceleration of the box?

y

F = Fx i a = ?m x

Solution• Draw a picture showing all of the forces

FFB,F

FF,BFB,E

FE,B

y

x

Solution• Draw a picture showing all of the forces.• Isolate the forces acting on the block.

FFB,F

FF,BFB,E = mg

FE,B

y

x

Solution• Draw a picture showing all of the

forces.• Isolate the forces acting on the block.• Draw a free body diagram.

FFB,F

mg

y

x

• Draw a picture showing all of the forces.• Isolate the forces acting on the block.• Draw a free body diagram.• Solve Newton’s equations for each component.

– FX = maX

– FB,F - mg = maY

FFB,F

mg

y

x

Solution

• FX = maX

– So aX = FX / m = (10 N)/(2 kg) = 5 m/s2.

• FB,F - mg = maY

– But aY = 0– So FB,F = mg.

• The vertical component of the force of the floor on the object (FB,F ) is called the Normal Force (N).

• Since aY = 0 , N = mg in this case.

FX

N

mg

y

x

Solution

Recap

FX

N = mg

mg

aX = FX / m y

x

Normal Force

• A block of mass m rests on the floor of an elevator that is accelerating upward. What is the relationship between the force due to gravity and the normal force on the block?

m

(a) N > mg(b) N = mg(c) N < mg

a

Solution

All forces are acting in the y direction, so use:

Ftotal = ma

N - mg = maN = ma + mg

therefore N > mg

m

N

mg

a

Tools: Ropes & Strings

• Can be used to pull from a distance.• Tension (T) at a certain position in a rope is the

magnitude of the force acting across a cross-section of the rope at that position.– The force you would feel if you cut the rope and

grabbed the ends.– An action-reaction pair.

cut

TT

T

Tools: Ropes & Strings...

• Consider a horizontal segment of rope having mass m:– Draw a free-body diagram (ignore gravity).

• Using Newton’s 2nd law (in x direction): FNET = T2 - T1 = ma

• So if m = 0 (i.e. the rope is light) then T1 = T2

T1 T2

m

a x

Tools: Ropes & Strings...• An ideal (massless) rope has constant tension along

the rope.

• If a rope has mass, the tension can vary along the rope– For example, a heavy rope

hanging from the ceiling...

• We will deal mostly with ideal massless ropes.

T = TgT = 0

T T

Tools: Ropes & Strings...• The direction of the force provided by a

rope is along the direction of the rope:

mg

T

m

Since ay = 0 (box not moving),

T = mg

Force and acceleration

• A fish is being yanked upward out of the water using a fishing line that breaks when the tension reaches 180 N. The string snaps when the acceleration of the fish is observed to be is 12.2 m/s2. What is the mass of the fish?

(a) 14.8 kg(b) 18.4 kg(c) 8.2 kg

m = ?a = 12.2 m/s2

snap !

Solution:

• Draw a Free Body Diagram!! T

mg

m = ?

a = 12.2 m/s2 Use Newton’s 2nd law

in the upward direction:

FTOT = maT - mg = maT = ma + mg = m(g+a)

Tm=g+a 2180Nm= =8.2kg9.8+12.2 ms

Tools: Pegs & Pulleys• Used to change the direction of forces

– An ideal massless pulley or ideal smooth peg will change the direction of an applied force without altering the magnitude:

F1 ideal peg or pulley

F2

| F1 | = | F2 |

Tools: Pegs & Pulleys• Used to change the direction of forces

– An ideal massless pulley or ideal smooth peg will change the direction of an applied force without altering the magnitude:

mg

T

m T = mg

FW,S = mg

Springs• Hooke’s Law: The force exerted by a spring is

proportional to the distance the spring is stretched or compressed from its relaxed position.– FX = -k x– Where x is the displacement from the relaxed

position and k is the constant of proportionality.

relaxed position

FX = 0x

Springs...• Hooke’s Law: The force exerted by a spring is

proportional to the distance the spring is stretched or compressed from its relaxed position.– FX = -k x– Where x is the displacement from the relaxed

position and k is the constant of proportionality.

relaxed position

FX = -kx > 0

xx 0

Springs...• Hooke’s Law: The force exerted by a spring is

proportional to the distance the spring is stretched or compressed from its relaxed position.– FX = -k x– Where x is the displacement from the relaxed

position and k is the constant of proportionality.

FX = - kx < 0

xx > 0

relaxed position

Scales:• Springs can be calibrated to tell us the applied force.

– We can calibrate scales to read Newtons, or...– Fishing scales usually read weight in kg or lbs.

02468

1 lb = 4.45 N

(a) 0 lbs. (b) 4 lbs. (c) 8 lbs.

m m m

(1) (2)

?

Force and acceleration

• A block weighing 4 lbs is hung from a rope attached to a scale. The scale is then attached to a wall and reads 4 lbs. What will the scale read when it is instead attached to another block weighing 4 lbs?

Solution:• Draw a Free Body Diagram of one of the blocks!!

Use Newton’s 2nd Law in the y direction:

FTOT = 0T - mg = 0T = mg = 4 lbs. mg

T

m T = mg

a = 0 since the blocks are stationary

Solution:

• The scale reads the tension in the rope, which is T = 4 lbs in both cases!

m m m

T T T T

TTT

Car going around a turnWhat are the forces that cause this circular

acceleration?Gravity (weight) acts downContact Force acts up (perpendicular to the

surface)Friction - which way does it act (left or right)?

a=v2/r

W=mg

Fc

Car is heading into the screen.

Car going around a turn

Without friction the car will NOT make the turn, it will continue straight - into the left ditch! Therefore, friction by the road must push on the car to the right. (Friction by the car on the road will be opposite - to the left.)

The fastest the car can go aroundthe turn without sliding is whenthe friction is maximum:Ff = Fc .

a=v2/r

W=mg

Fc Ff Fc

Car going around a turnWe now apply Newton’s Second Law - in

rectangular components:Fx = Ff = ma = mv2/r

To make the car go around the turn fastest, we need the maximum force of friction: Ff = Fc

Fy = Fc - W = 0

which says Fc = mg, so

mg = mv2/r, or vmax = [gr] a=v2/r

W=mg

Fc Ff Fc

Car going around a turnvmax = [gr]

Note that the maximum speed (without slipping) around a turn depends on the coefficient of friction, the amount of gravity (not usually under our control), and the sharpness of the turn (radius).

If we go at a slower speed around the turn, friction will be less than the maximum: Ff < Fc.

There is one other thing we can do to go faster around the turn - bank the road! How does this work?

Banked turnBy banking the road, we have not added any forces,

but we have changed the directions of both the contact force and the friction force!

Have we changed the direction of the acceleration? No - the car is still travelling in a horizontal circle.

W=mg

Fc

q

a=v2/r

Ff

Banked turnSince the acceleration is still in the x direction, we

will again use x and y components (rather than // and

Fx = Fc sin(q) + Ff cos(q) = mv2/rFy = Fc cos(q) - Ff sin(q) - mg = 0If we are looking for the maxspeed, we will need the maxfriction: Ff = Fc .This gives 3 equations for 3 unknowns: Ff, Fc, and v.

Fc

W=mg

q

a=v2/r

Ff

Banked turn

Fx = Fc sin(q) + Ff cos(q) = mv2/rFy = Fc cos(q) - Ff sin(q) - mg = 0Ff = Fc . Using the third equation, we can eliminate Ff in

the first two:Fc sin(q) + Fc cos(q) = mv2/rFc cos(q) - Fc sin(q) - mg = 0We can now use the second equation to find Fc:Fc = mg / [cos(q) - sin(q)], and use this in the first

equation to get: v = [gr {sin(q)+ cos(q)} / {cos(q) - sin(q)} ]1/2

Banked Turn

v = [g r {sin(q)+ cos(q)} / {cos(q) - sin(q)} ]1/2

Notice that the mass cancels out. This means that the mass of the car does not matter! (Big heavy trucks slip on slippery streets just like small cars. When going fast, big heavy trucks flip over rather than slide off the road; little cars don’t flip over like big trucks. But flipping over is not the same as slipping! We’ll look at flipping in Part 4 of the course.)

Note also that when q 0, the above expression reduces to the one we had for a flat road: vmax = [gr]1/2 .

Banked Turn

v = [g r {sin(q)+ cos(q)} / {cos(q) - sin(q)} ]1/2

Note that as sin(q) approaches cos(q), the denominator approaches zero, so the maximum speed approaches infinity!

What force really supports such large speeds (and so large accelerations)? As the angle increases, the contact force begins to act more and more to cause the acceleration. And as the contact force increases, so does friction.

Actually, there is a limit on the maximum speed because there is a limit to the contact force.

Banked turn - Minimum Speed

Is there a minimum speed for going around a banked turn? Consider the case where the coefficient of friction is small and the angle of bank is large. In that case the car, if going too slow, will tend to slide down (to the right) so friction should act to the left.

Can you get an equation for the minimum speed necessary?

What changes in what we did for max speed?

Fc

W=mg

q

a=v2/r

Ff