1. Newton’s 2 nd Law We discussed objects in mechanical equilibrium—at rest or moving at constant velocity. Most things, however, do not move at constant velocity, but undergo changes in motion. We say they undergo accelerated motion. acceleration describes how fast motion changes.

2. Force causes Acceleration Any object that accelerates is acted on by a push or a pull—a force of some kind. It may be a sudden push, like that on a kicked soccer ball, or the steady pull of gravity. Acceleration is caused by force.

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

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

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

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

7. Newton’s 2 nd Law The acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object.

8. Friction When surfaces slide or tend to slide over one another, a force of friction acts. When you apply a force to an object, a force of friction usually reduces the net force and the resulting acceleration. Friction is caused by the irregularities in the surfaces in mutual contact, and depends on the kinds of material and how much they are pressed together. Even surfaces that appear to be very smooth have 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 rise over the irregular bumps or else scrape atoms off. Either way requires force.

9. Friction Force The direction of the friction force is always in a direction opposing motion. An object sliding down an incline experiences friction directed up the incline; an object that slides to the right experiences friction toward the left. Thus, if an object is to move at constant velocity, a force equal to the opposing force of friction must be applied so that the two forces exactly cancel each other. The zero net force then results in zero acceleration.

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

11. Friction of Sliding Interestingly, the friction of sliding is somewhat less than the friction that builds up before sliding takes place. Physicists and engineers distinguish between static friction and sliding friction. To avoid information overload we won't pursue this distinction further, Physicists and engineers distinguish between static friction and sliding friction. To avoid information overload we won't pursue this distinction further, except to cite an important example—braking a car in an emergency stop. It is very important that you not jam on the brakes so as to make the tires lock in place. 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 slide along the road surface, and friction is static friction—and therefore greater than sliding friction. The difference between static and sliding friction is also apparent when your car takes a corner too fast. 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 tires below the threshold of breaking loose into a slide.

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

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

14. Friction and Fluids Friction is not restricted to solids sliding over one another. Friction occurs also in liquids and gases, collectively called fluids (because they flow). Friction occurs also in liquids and gases, collectively called fluids (because they flow). Fluid friction is called drag. Just as the friction between solid surfaces depends on the nature of the surfaces, drag in a fluid depends on the nature of the fluid; Fluid friction is called drag. Just as the friction between solid surfaces depends on the nature of the surfaces, drag in a fluid depends on the nature of the fluid; for example, drag is greater in water than it is in air. But unlike the friction between solid surfaces, such as the crate sliding across the floor, drag does depend on speed and area of contact. 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.

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

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

17. Applying Force-Pressure Pressure is defined as force per unit area and is obtained by dividing the force by the area on which the force acts: As an illustration of the distinction between pressure and force, consider the two blocks. The blocks are identical, but one stands on its end and the other on its side. Both blocks are of equal weight and therefore exert the same force on the surface (put them on a bathroom scale and each registers the same), but the upright block exerts a greater pressure against the surface. If the block were tipped up so contact is on a single corner, the pressure would be greater still.

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

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

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

22. We now understand that the acceleration of free fall is independent of an object's mass. A boulder 100 times more massive than a pebble falls at the same acceleration as the pebble because although the force on the boulder (its weight) is 100 times greater than the force (or weight) on the pebble, its resistance to a change in motion (mass) is 100 times that of the pebble. The greater force offsets the equally greater mass.

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

24. The ratio of weight (F) to mass (m) is the same for the large rock and the small feather; similarly, the ratio of circumference (C) to diameter (D) is the same for the large and the small circle.

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

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

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

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

29. The heavier parachutist must fall faster than the lighter parachutist for air resistance to cancel his greater weight.

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