Newton’s 2Newton’s 2ndnd Law LawWe discussed objects in mechanical We discussed objects in mechanical equilibrium—at rest or moving at constant equilibrium—at rest or moving at constant velocity. velocity.

Most things, however, do not move at Most things, however, do not move at constant velocity, but undergo changes in constant velocity, but undergo changes in motion. motion.

We say they undergo accelerated motion.We say they undergo accelerated motion.

acceleration describes how fast motion acceleration describes how fast motion changes. changes.

Force causes AccelerationForce causes Acceleration

Any object that accelerates is acted on by Any object that accelerates is acted on by a push or a pull—a force of some kind. a push or a pull—a force of some kind.

It may be a sudden push, like that on a It may be a sudden push, like that on a kicked soccer ball, or the steady pull of kicked soccer ball, or the steady pull of gravity. gravity.

Acceleration is caused by force.Acceleration is caused by force.

Acceleration and Net ForceAcceleration and Net ForceOften more than a single force acts on an object. Often more than a single force acts on an object. Recall that the combination of forces that act on Recall that the combination of forces that act on an object is the an object is the net forcenet force. . Acceleration depends on the net force. For Acceleration depends on the net force. For example, if you push with twice as much force example, if you push with twice as much force on an object and the net force is twice, the on an object and the net force is twice, the object will pick up speed at twice the rate. object will pick up speed at twice the rate. Acceleration will double when the net force Acceleration will double when the net force doubles. doubles. We say that the acceleration produced is directly We say that the acceleration produced is directly proportional to the net force. We write: The proportional to the net force. We write: The symbol ~ stands for “is directly proportional to.” symbol ~ stands for “is directly proportional to.” That means any change in one is the same That means any change in one is the same amount of change in the other.amount of change in the other.

Check YourselfCheck Yourself

Check YourselfCheck Yourself1. You push on a crate that sits on a smooth 1. You push on a crate that sits on a smooth floor and it accelerates. If you apply four times floor and it accelerates. If you apply four times the net force, how much greater will be the the net force, how much greater will be the acceleration?acceleration?2. If you push with the same increased force on 2. If you push with the same increased force on the same crate, but it is on a very rough floor, the same crate, but it is on a very rough floor, how will the acceleration compare to pushing on how will the acceleration compare to pushing on a smooth floor? a smooth floor?

Mass Resists AccelerationMass Resists AccelerationPush your friend on a skateboard and your friend Push your friend on a skateboard and your friend accelerates. accelerates. Now push equally hard on an elephant on the Now push equally hard on an elephant on the skateboard and acceleration is much less. skateboard and acceleration is much less. You'll see that the amount of acceleration depends not You'll see that the amount of acceleration depends not only on the force, but on the mass being pushed.only on the force, but on the mass being pushed.The same force applied to twice the mass produces half The same force applied to twice the mass produces half the acceleration. the acceleration. For three times the mass, one-third the acceleration. For three times the mass, one-third the acceleration. We say that for a given force, the acceleration produced We say that for a given force, the acceleration produced is inversely proportional to the mass. is inversely proportional to the mass. That is, By inversely we mean that the two values That is, By inversely we mean that the two values change in opposite directions. change in opposite directions. As the denominator increases, the whole quantity As the denominator increases, the whole quantity decreases. For example, the quantity is less than .decreases. For example, the quantity is less than .

Newton’s 2Newton’s 2ndnd Law LawEvery day we see things that do not continue in Every day we see things that do not continue in a constant state of motion: objects initially at rest a constant state of motion: objects initially at rest later may move; moving objects may follow later may move; moving objects may follow paths that are not straight lines; things in motion paths that are not straight lines; things in motion may stop. may stop. Most of the motion we observe undergoes Most of the motion we observe undergoes changes and is the result of one or more applied changes and is the result of one or more applied forces. forces. The overall net force, whether it be from a single The overall net force, whether it be from a single source or a combination of sources, produces source or a combination of sources, produces acceleration. The relationship of acceleration to acceleration. The relationship of acceleration to force and inertia is given in Newton's second force and inertia is given in Newton's second law.law.

Newton’s 2Newton’s 2ndnd Law Law

The acceleration of an object is directly The acceleration of an object is directly proportional to the net force acting on proportional to the net force acting on the object, is in the direction of the net the object, is in the direction of the net force, and is inversely proportional to force, and is inversely proportional to the mass of the object.the mass of the object.

FrictionFrictionWhen surfaces slide or tend to slide over one another, a When surfaces slide or tend to slide over one another, a force of force of frictionfriction acts. acts. When you apply a force to an object, a force of friction When you apply a force to an object, a force of friction usually reduces the net force and the resulting usually reduces the net force and the resulting acceleration. acceleration. Friction is caused by the irregularities in the surfaces in Friction is caused by the irregularities in the surfaces in mutual contact, and depends on the kinds of material mutual contact, and depends on the kinds of material and how much they are pressed together. and how much they are pressed together. Even surfaces that appear to be very smooth have Even surfaces that appear to be very smooth have microscopic irregularities that obstruct motion.microscopic irregularities that obstruct motion. Atoms cling together at many points of contact. Atoms cling together at many points of contact. When one object slides against another, it must either When one object slides against another, it must either rise over the irregular bumps or else scrape atoms off. rise over the irregular bumps or else scrape atoms off. Either way requires force.Either way requires force.

Friction ForceFriction ForceThe direction of the friction force is always in a The direction of the friction force is always in a direction opposing motion. An object sliding direction opposing motion. An object sliding downdown an incline experiences friction directed an incline experiences friction directed upup the incline; the incline;

an object that slides to the an object that slides to the rightright experiences experiences friction toward the friction toward the leftleft. .

Thus, if an object is to move at constant velocity, Thus, if an object is to move at constant velocity, a force equal to the opposing force of friction a force equal to the opposing force of friction must be applied so that the two forces exactly must be applied so that the two forces exactly cancel each other. cancel each other.

The zero net force then results in zero The zero net force then results in zero acceleration.acceleration.

How Much Friction?How Much Friction?No friction exists on a crate that sits at rest on a level No friction exists on a crate that sits at rest on a level floor. floor. But disturb the contact surfaces by pushing horizontally But disturb the contact surfaces by pushing horizontally on the crate and friction is produced. How much?on the crate and friction is produced. How much? If the crate is still at rest, then the friction that opposes If the crate is still at rest, then the friction that opposes motion is just enough to cancel your push.motion is just enough to cancel your push. If you push horizontally with, say, 70 newtons, the If you push horizontally with, say, 70 newtons, the friction becomes 70 newtons. friction becomes 70 newtons. If you push harder, say 100 newtons, and the crate is on If you push harder, say 100 newtons, and the crate is on the verge of sliding—the friction between the crate and the verge of sliding—the friction between the crate and floor opposes your push with 100 newtons. If 100 floor opposes your push with 100 newtons. If 100 newtons is the most the surfaces can muster, then when newtons is the most the surfaces can muster, then when you push a bit harder the clinging gives way and the you push a bit harder the clinging gives way and the crate slides. crate slides.

Friction of Sliding Friction of Sliding Interestingly, the friction of sliding is somewhat less than the friction Interestingly, the friction of sliding is somewhat less than the friction that builds up before sliding takes place.that builds up before sliding takes place. Physicists and engineers distinguish between static friction and Physicists and engineers distinguish between static friction and sliding friction. To avoid information overload we won't pursue this sliding friction. To avoid information overload we won't pursue this distinction further, distinction further, except to cite an important example—braking a car in an emergency except to cite an important example—braking a car in an emergency stop. stop. It is very important that you not jam on the brakes so as to make the It is very important that you not jam on the brakes so as to make the tires lock in place. tires lock in place. When tires lock, they slide, providing less friction than if they are When tires lock, they slide, providing less friction than if they are made to roll to a stop. While the tire is rolling, its surface does not made to roll to a stop. While the tire is rolling, its surface does not slide along the road surface, and friction is static friction—and slide along the road surface, and friction is static friction—and therefore greater than sliding friction. therefore greater than sliding friction. The difference between static and sliding friction is also apparent The difference between static and sliding friction is also apparent when your car takes a corner too fast. when your car takes a corner too fast. Once the tires start to slide, the frictional force is reduced and off Once the tires start to slide, the frictional force is reduced and off you go! A skilled driver (or an anti-lock brake system) keeps the you go! A skilled driver (or an anti-lock brake system) keeps the tires below the threshold of breaking loose into a slide.tires below the threshold of breaking loose into a slide.

Friction and SpeedFriction and Speed

It's also interesting that the force of friction does It's also interesting that the force of friction does not depend on speed. not depend on speed. A car skidding at low speed has approximately A car skidding at low speed has approximately the same friction as at high speed.the same friction as at high speed. If the friction force of a crate that slides against If the friction force of a crate that slides against a floor is 90 newtons at low speed, to a close a floor is 90 newtons at low speed, to a close approximation it is 90 newtons at a greater approximation it is 90 newtons at a greater speed. speed. It may be more when the crate is at rest and on It may be more when the crate is at rest and on the verge of sliding, but once sliding the friction the verge of sliding, but once sliding the friction force remains approximately the same.force remains approximately the same.

Friction-Area of ContactFriction-Area of ContactMore interesting still, friction does not depend on the More interesting still, friction does not depend on the area of contact. area of contact. Slide the crate on its smallest surface and all you do is Slide the crate on its smallest surface and all you do is concentrate the same weight on a smaller area with the concentrate the same weight on a smaller area with the result that the friction is the same. result that the friction is the same. So those extra-wide tires you see on some cars provide So those extra-wide tires you see on some cars provide no more friction than narrower tires. no more friction than narrower tires. The wider tire simply spreads the weight of the car over The wider tire simply spreads the weight of the car over more surface area to reduce heating and wear. more surface area to reduce heating and wear. Similarly, the friction between a truck and the ground is Similarly, the friction between a truck and the ground is the same whether the truck has 4 tires or 18! More tires the same whether the truck has 4 tires or 18! More tires spreads the load over more ground area and reduces spreads the load over more ground area and reduces the pressure per tire.the pressure per tire. Interestingly, stopping distance when brakes are applied Interestingly, stopping distance when brakes are applied is not affected by the number of tires. But the wear that is not affected by the number of tires. But the wear that tires experience very much depends on the number of tires experience very much depends on the number of tires.tires.

Friction and FluidsFriction and Fluids

Friction is not restricted to solids sliding over one another.Friction is not restricted to solids sliding over one another. Friction occurs also in liquids and gases, collectively called Friction occurs also in liquids and gases, collectively called fluidsfluids (because they flow).(because they flow). Fluid friction is called Fluid friction is called dragdrag. Just as the friction between solid . Just as the friction between solid surfaces depends on the nature of the surfaces, drag in a fluid surfaces depends on the nature of the surfaces, drag in a fluid depends on the nature of the fluid; depends on the nature of the fluid; for example, drag is greater in water than it is in air. for example, drag is greater in water than it is in air. But unlike the friction between solid surfaces, such as the crate But unlike the friction between solid surfaces, such as the crate sliding across the floor, drag sliding across the floor, drag doesdoes depend on speed and area of depend on speed and area of contact. contact. This makes sense, for the amount of fluid pushed aside by a boat or This makes sense, for the amount of fluid pushed aside by a boat or airplane depends on the size and the shape of the craft. airplane depends on the size and the shape of the craft.

Friction and FluidsFriction and FluidsA slow-moving boat or airplane encounters less drag A slow-moving boat or airplane encounters less drag than faster boats or airplanes. than faster boats or airplanes. And wide boats and airplanes must push aside more And wide boats and airplanes must push aside more fluid than narrow crafts. fluid than narrow crafts. For slow motion through water, drag is approximately For slow motion through water, drag is approximately proportional to the speed of the object.proportional to the speed of the object. In air, drag at most speeds is proportional to the square In air, drag at most speeds is proportional to the square of the speed. of the speed. So if an airplane doubles its speed it encounters four So if an airplane doubles its speed it encounters four times as much drag. times as much drag. At very high speed, however, the simple rules break At very high speed, however, the simple rules break down when the fluid flow becomes erratic and such down when the fluid flow becomes erratic and such things as vortices and shock waves develop.things as vortices and shock waves develop.

Check YourselfCheck Yourself

1. What net force does a sliding crate 1. What net force does a sliding crate experience when you exert a force of 110 experience when you exert a force of 110 N and friction between the crate and the N and friction between the crate and the floor is 100 N?floor is 100 N?2. A jumbo jet cruises at constant velocity 2. A jumbo jet cruises at constant velocity of 1000 km/h when the thrusting force of of 1000 km/h when the thrusting force of its engines is a constant 100,000 N. What its engines is a constant 100,000 N. What is the acceleration of the jet? What is the is the acceleration of the jet? What is the force of air resistance on the jet?force of air resistance on the jet?

Applying Force-PressureApplying Force-PressurePressure is defined as force per unit area and is Pressure is defined as force per unit area and is obtained by dividing the force by the area on which the obtained by dividing the force by the area on which the force acts: force acts:

As an illustration of the distinction between pressure and As an illustration of the distinction between pressure and force, consider the two blocks. The blocks are identical, force, consider the two blocks. The blocks are identical, but one stands on its end and the other on its side. but one stands on its end and the other on its side. Both blocks are of equal weight and therefore exert the Both blocks are of equal weight and therefore exert the same force on the surface (put them on a bathroom same force on the surface (put them on a bathroom scale and each registers the same), but the upright block scale and each registers the same), but the upright block exerts a greater exerts a greater pressurepressure against the surface. If the against the surface. If the block were tipped up so contact is on a single corner, the block were tipped up so contact is on a single corner, the pressure would be greater still .pressure would be greater still .

Bed of Nails HewittBed of Nails Hewitt

Free Fall ExplainedFree Fall ExplainedAlthough Galileo founded both the concepts of Although Galileo founded both the concepts of inertia and acceleration, and was the first to inertia and acceleration, and was the first to measure the acceleration of falling objects, measure the acceleration of falling objects, Galileo could not explain why objects of various Galileo could not explain why objects of various masses fall with equal accelerations. Newton's masses fall with equal accelerations. Newton's second law provides the explanation.second law provides the explanation.We know that a falling object accelerates toward We know that a falling object accelerates toward the Earth because of the gravitational force of the Earth because of the gravitational force of attraction between the object and the Earth. attraction between the object and the Earth. When the force of gravity is the only force—that When the force of gravity is the only force—that is, when friction such as air resistance is is, when friction such as air resistance is negligible— we say that the object is in a state of negligible— we say that the object is in a state of free fallfree fall..

The greater the mass of an object, the greater is the The greater the mass of an object, the greater is the gravitational force of attraction force between it and the gravitational force of attraction force between it and the Earth. Earth. The double brick for example, has twice the gravitational The double brick for example, has twice the gravitational attraction as the single brick.attraction as the single brick. Why then, as Aristotle supposed, doesn't the double Why then, as Aristotle supposed, doesn't the double brick fall twice as fast? The answer is that the brick fall twice as fast? The answer is that the acceleration of an object depends not only on the force—acceleration of an object depends not only on the force—in this case, the weight—but on the object's resistance to in this case, the weight—but on the object's resistance to motion, its inertia. motion, its inertia. Whereas a force produces an acceleration, inertia is a Whereas a force produces an acceleration, inertia is a resistanceresistance to acceleration. to acceleration. So twice the force exerted on twice the inertia produces So twice the force exerted on twice the inertia produces the same acceleration as half the force exerted on half the same acceleration as half the force exerted on half the inertia. Both accelerate equally. the inertia. Both accelerate equally. The acceleration due to gravity is symbolized by The acceleration due to gravity is symbolized by gg. We . We use the symbol g, rather than use the symbol g, rather than aa, to denote that , to denote that acceleration is due to gravity alone.acceleration is due to gravity alone.

The ratio of weight to mass for freely The ratio of weight to mass for freely falling objects equals a constant—falling objects equals a constant—gg. The . The ratio of weight to mass is the same for ratio of weight to mass is the same for both heavy and light objects.both heavy and light objects.

We now understand that the acceleration of free We now understand that the acceleration of free fall is independent of an object's mass. fall is independent of an object's mass.

A boulder 100 times more massive than a A boulder 100 times more massive than a pebble falls at the same acceleration as the pebble falls at the same acceleration as the pebble because although the force on the pebble because although the force on the boulder (its weight) is 100 times greater than the boulder (its weight) is 100 times greater than the force (or weight) on the pebble, its resistance to force (or weight) on the pebble, its resistance to a change in motion (mass) is 100 times that of a change in motion (mass) is 100 times that of the pebble. the pebble.

The greater force offsets the equally greater The greater force offsets the equally greater mass.mass.

Nonfree FallNonfree Fall

What of the practical cases of objects falling in air? What of the practical cases of objects falling in air? Although a feather and a coin will fall equally fast in a Although a feather and a coin will fall equally fast in a vacuum, they fall quite differently in air. vacuum, they fall quite differently in air. How do Newton's laws apply to objects falling in air? How do Newton's laws apply to objects falling in air? The answer is that Newton's laws apply for The answer is that Newton's laws apply for allall objects, objects, whether freely falling or falling in the presence of whether freely falling or falling in the presence of resistive forces. resistive forces. The accelerations, however, are quite different for the The accelerations, however, are quite different for the two cases. The important thing to keep in mind is the two cases. The important thing to keep in mind is the idea of idea of netnet force. In a vacuum or in cases where air force. In a vacuum or in cases where air resistance can be neglected, the net force is the weight resistance can be neglected, the net force is the weight because it is the only force. because it is the only force. In the presence of air resistance, however, the net force In the presence of air resistance, however, the net force is less than the weight—it is the weight minus air drag, is less than the weight—it is the weight minus air drag, the force arising from air resistance.the force arising from air resistance.

The ratio of weight (The ratio of weight (FF) to mass () to mass (mm) is the ) is the same for the large rock and the small same for the large rock and the small feather; similarly, the ratio of feather; similarly, the ratio of circumference (circumference (CC) to diameter () to diameter (DD) is the ) is the same for the large and the small circle. same for the large and the small circle.

Air DragAir DragThe force of air drag experienced by a falling The force of air drag experienced by a falling object depends on two things. object depends on two things. First, it depends on the frontal area of the falling First, it depends on the frontal area of the falling object—that is, on the amount of air the object object—that is, on the amount of air the object must plow through as it falls. must plow through as it falls. Second, it depends on the speed of the falling Second, it depends on the speed of the falling object; the greater the speed, the greater the object; the greater the speed, the greater the number of air molecules an object encounters number of air molecules an object encounters per second and the greater the force of per second and the greater the force of molecular impact. molecular impact. Air drag depends on the size and the speed of a Air drag depends on the size and the speed of a falling object.falling object.

Terminal VelocityTerminal VelocityIn some cases air drag greatly affects falling; in other In some cases air drag greatly affects falling; in other cases it doesn't. cases it doesn't. Air drag is important for a falling feather. Since a feather Air drag is important for a falling feather. Since a feather has so much area compared to its small weight, it has so much area compared to its small weight, it doesn't have to fall very fast before the upward-acting air doesn't have to fall very fast before the upward-acting air drag cancels the downward-acting weight. drag cancels the downward-acting weight. The net force on the feather is then zero and The net force on the feather is then zero and acceleration terminates. When acceleration terminates, acceleration terminates. When acceleration terminates, we say the object has reached its we say the object has reached its terminal speedterminal speed. . If we are concerned with direction, down for falling If we are concerned with direction, down for falling objects, we say the object has reached its objects, we say the object has reached its terminal terminal velocityvelocity. The same idea applies to all objects falling in . The same idea applies to all objects falling in air.air.

Consider SkydivingConsider SkydivingAs a falling skydiver gains speed, air drag may finally As a falling skydiver gains speed, air drag may finally build up until it equals the weight of the skydiver.build up until it equals the weight of the skydiver. If and when this happens, the If and when this happens, the netnet force becomes zero force becomes zero and the skydiver no longer accelerates; she has reached and the skydiver no longer accelerates; she has reached her terminal velocity. her terminal velocity. For a feather, terminal velocity is a few centimeters per For a feather, terminal velocity is a few centimeters per second, whereas for a skydiver it is about 200 kilometers second, whereas for a skydiver it is about 200 kilometers per hour.per hour. A skydiver may vary this speed by varying position. A skydiver may vary this speed by varying position. Head or feet first is a way of encountering less air and Head or feet first is a way of encountering less air and thus less air drag and attaining maximum terminal thus less air drag and attaining maximum terminal velocity. velocity. A smaller terminal velocity is attained by spreading A smaller terminal velocity is attained by spreading oneself out like a flying squirrel. oneself out like a flying squirrel. Minimum terminal velocity is attained when the Minimum terminal velocity is attained when the parachute is opened.parachute is opened.

ParachutingParachutingConsider a man and woman parachuting together from the same Consider a man and woman parachuting together from the same altitude. altitude. Suppose that the man is twice as heavy as the woman and that their Suppose that the man is twice as heavy as the woman and that their same-size chutes are initially opened. same-size chutes are initially opened. The same-size chute means that at equal speeds the air resistance The same-size chute means that at equal speeds the air resistance is the same on each. Who gets to the ground first—the heavy man is the same on each. Who gets to the ground first—the heavy man or the lighter woman? The answer is the person who falls fastest or the lighter woman? The answer is the person who falls fastest gets to the ground first— that is, the person with the greatest gets to the ground first— that is, the person with the greatest terminal speed. At first we might think that because the chutes are terminal speed. At first we might think that because the chutes are the same, the terminal speeds for each would be the same, and the same, the terminal speeds for each would be the same, and therefore both would reach the ground together. This doesn't therefore both would reach the ground together. This doesn't happen because air drag also depends on speed. Greater speed happen because air drag also depends on speed. Greater speed means greater force of impact of the air. The woman will reach her means greater force of impact of the air. The woman will reach her terminal speed when air drag against her chute equals her weight. terminal speed when air drag against her chute equals her weight. When this happens, the air drag against the chute of the man will When this happens, the air drag against the chute of the man will not yet equal his weight. He must fall faster than she does for air not yet equal his weight. He must fall faster than she does for air drag to match his greater weight. drag to match his greater weight. Terminal velocity is greater for the heavier person, with the result Terminal velocity is greater for the heavier person, with the result that the heavier person reaches the ground first.that the heavier person reaches the ground first.

The heavier parachutist The heavier parachutist

must fall faster than the must fall faster than the

lighter parachutist for lighter parachutist for

air resistance to cancel air resistance to cancel

his greater weight. his greater weight.

When weight When weight mgmg is greater than air is greater than air resistance resistance RR, the falling sack accelerates. , the falling sack accelerates. At higher speeds, At higher speeds, RR increases. When increases. When RR = = mgmg, acceleration reaches zero, and the , acceleration reaches zero, and the sack reaches its terminal velocity sack reaches its terminal velocity