newton2ndlaw - Efrain Lopez shows that when the forces...

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Unformatted text preview: Efrain Lopez shows that when the forces balance to zero, no acceleration occurs. lilifi“ Newton’s Second Law Newton’s Second Law of Motion n Chapter 2, we discussed the concept of mechanical equilibrium, 2F: 0, where Forces are balanced. In this chapter, we consider what happens when Forces aren’t balanced—when net Forces do not equal zero. The net Force on a kicked soccer ball, For example, is greater than zero, and the motion of the ball changes abruptly. Its path through the air is not a straight line but curves downward due to gravityiagain, a change in motion. Most of the motion we see undergoes change. This chapter covers changes in motion—accelerated motion. In Chapter 3, we learned that acceleration describes how quickly motion changes. Specifically, it is the change in velocity during a certain time interval. Recall the defini- tion of acceleration: change in velocity acceleration I . . time Interval We will now Focus on the cause or” acceleration: force. Force Causes FIGURE 4.1 Kick the ball and it accelerates. 58 Acceleration Consider a hockey puck at rest on ice. Apply a force, and it starts to move— it accelerates. When the hockey stick is no longer pushing it, the puck moves at constant velocity. Apply another force by striking the puck again, and again the motion changes. Applied force produces acceleration. Most often, the applied force is not the only force acting on an object. Other forces may act as well. Recall, from Chapter 2, that the combination of forces acting on an object is the net force. Acceleration depends on the net force. To increase the acceleration of an object, you must increase the net force acts ing on it. If you double the net force on an object, its acceleration doubles; if you triple the net force, its acceleration triples; and so on. This makes good sense. We say an object’s acceleration is directly proportional to the net force acting on it. We write acceleration ~ net force Chapter 4 Newton ’5 Second Law of Motion 59 The symbol ~ stands for “is directly proportional to.” That means any change (:3 in one is the same amount of change in the other. Force Causes Acceleration 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 slides on a very rough floor, how will the acceleration compare with pushing the crate on a smooth Floor? (Think before you read the answer below!) I ' Friction When surfaces slide or tend to slide over one another, a force of friction aets. When you apply a force to an object, friction usually reduces the net force and the resulting acceleration. Friction is caused by the irregularities in the surfaces in mutual contacr, and it 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. When one object slides against another, it must either rise over the irregular bumps or else scrape atoms off. Either way requires force. Force 0f hand The direction of the friction force is always in a direction opposing OCCEleres motion. An object sliding down an incline experiences friction directed up lh_e_ bl.le the incline; an object that slides to the right experiences friction toward the m 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 Twice as much force each other. The zero net force then results in zero acceleration and constant produces twice as velocity. much acceleration FIGURE 4.3 Friction results from the mutual contact of irregularities in the surfaces of sliding objects. Even surfaces that appear to be smooth have irregular surfaces when viewed at the microscopic level. Twice the force on twrce the mass gives the same acceleration FIGURE 4.2 Acceleration is directly proportional to force. _—________.__——_————-—— CHECK YOUR ANSWERS glecs 1. It will have four times as much acceleration. Friction 2. it will have less acceleration because friction will reduce the net force. W 60 Part One Mechanics FIGURE 4.4 The direction ofthe force of friction always opposes the direction ofmotion. (Left) Push the crate to the right, and friction acts toward the left. (Right) The sack falls downward, and air Friction (air resistance) acts upward. (What is the acceleration of the sack when air resistance equals the sack’s weight?) Air Resistance No friction exists on a crate that sits at rest on a level floor. But, if you push the crate horizontally, you’ll disturb the contact surfaces and friction is produced. How much? If the crate is still at test, then the friction that opposes motion is just enough to cancel your push. If you push horizontally with, say, 70 newtons, friction builds up to become 70 newtons. If you push harder— say, 100 newtonsiand 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.‘ 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. For given surfaces, static friction is somewhat greater than sliding friction. If you push on a crate, it takes more force to get it going than it takes to keep it sliding. Before the time of antilock brake systems, slamming on the brakes of a car was quite prob— lematic. When tires lock, they slide, providing less friction than if they are made to roll to a stop. A rolling tire does not slide along the road surface, and friction is static friction, with more grab than sliding friction. But once the tires start to slide, the frictional force is reduced—not a good thing. An antilock brake system keeps the tires below the threshold of breaking loose into a slide. It’s also interesting that the force of fricrion does not depend on speed. A car skidding at low speed has approximately the same friction as the same car skidding 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. It may be more when the crate is at rest and on the verge of sliding, but, once the crate is sliding, the friction force remains approximately the same. More interesting still, friction does not depend on the area of contact. If you slide the crate on its smallest surface, 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 LEven though 1{ may not seem so yet. most of the COHCEFN in physics are not really complicated. But friction is different. Unlike most concepts in physics. it is a very complicated phenomenon. ‘l‘he findings are empirical [gained from a wide range of experiments) and the predictions are approxnnatc (also based on experimenti. FIGURE 4.5 Friction between the tire and the ground is nearly the same whether the tire is wide or narrow. The purpose of the greater contact area is to reduce heating and wear. Chapter 4 Newton’s Second Law ofMotion 61 area to reduce heating and wear. Similarly, the friction between a truck and the ground is the same whether the truck has four tires or eighteen! More tires spread 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. Friction is not restricted to solids sliding over one another. Friction occurs also in liquids and gases, both of which are called fluids (because they flow). Fluid friction occurs as an object pushes aside the fluid it is moving through. Have you ever attempted a IOU—m dash through waist—deep water? The friction of fluids is appreciable, even at low speeds. A very common form of fluid fricrion for something moving through air is air resistance, also called air drag. You usually aren‘t aware of air resistance when walking or jogging, but you notice it at the higher speeds when riding a bicycle or when skiing downhill. Air resistance increases with increasing speed. The falling sack shown in Figure 4.4 wiil reach a conStant velocity when air resistance balances the sack’s weight. CHECK YOURSELF 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? Mass and Weight FIGURE 4.6 An anvil in outer space— between the Earth and the Moon, for example—may be weightless, but it is not massless. The acceleration imparted to an object depends not only on applied forces and friction forces, but on the inertia of the object. How much inertia an object possesses depends on the amount of matter in the object—the more matter, the more inertia. 1n speaking of how much matter something has, we use the term mass. The greater the mass of an object, the greater its inertia. Mass is a mea- sure of the inertia of a material object. Mass corresponds to our intuitive notion of weight. We casually say that something has a lot of matter if it weighs a lot. But there is a difference between mass and weight. We can define each as follows: Mass: The quantity of matter in an object. It is also the measure of the inertia or sluggisbness that an object exhibits in response to any effort made to start it, stop it, or change its state of motion in any way. Weight: The force upon an object due to gravity. In the absence of acceleration, mass and weight are directly proportional to each other.2 If the mass of an object is doubled, its weight is also doubled; CHECK YOUR ANSWER 10 N in the direction ofyour push (110 N - 100 N). ___________.__—_.————-—-— 2Weight and mass are directly proportional; weight = mg. where g is the constant of proportionality and has the value 9.8 leg, Fquii'alently, g is the acceleration due to gravity. 9.8 I'ni's2 [the units kag are equivalent to Ii‘lfszl. In Chapter 9. we'!l extend the definition of weight as the force that an object exerts on a supporting surface. 62 Part One Mechanics FIGURE 4.7 The astronaut in space finds that it isjust as difficult to shake the “weightless” anvil as it would be on Earth. if the anvil were more massive than the astronaut, which would shake morerthe anvil or the astronaut? FIGURE 4.8 Why will a slow, continuous increase in downward Force break the string above the massive ball, while a sudden increase would break the lower string? if the mass is halved, the weight is halved. Because of this, mass and weight are often interchanged. Also, mass and weight are sometimes confused because it is customary to measure the quantity of matter in things (mass) by their gravitational attraction to the Earth (weight). But mass is more fun— damental than weight; it is a fundamental quantity that completely escapes the notice of most people. There are times when weight corresponds to our unconscious notion of inertia. For example, if you are trying to determine which of two small objects is the heav- ier one, you might shake them back and forth in your hands or move them in some way instead of lifting them. In doing so, you are judging which of the two is more difficult to get moving, feeling which of the two is most resistant to a change in motion. You are really comparing the inertias of the objects. In the United States, the quantity of matter in an object is commonly described by the gravitational pull between it and the Earth, or its weight, usu— ally expressed in pounds. In most of the world, however, the measure of matter is commonly expressed in a mass unit, the kilogram. At the surface of the Earth, a brick with a mass of 1 kilogram weighs 2.2 pounds. In metric units, the unit of force is the newton, which is equal to a little less than a quarterrpound (like the weight of a quarter-pound hamburger after it is cooked). A 1-kilogram brick weighs about 10 newtons (more precisely, 9.8 NI).3 Away from the Earth’s sur- face, where the influence of gravity is less, a 1-kilogram brick weighs less. It would also weigh less on the surface of planets with less gravity than Earth. 0n the Moon’s surface, for example, where the gravitational force on things is only one- sixth as strong as on Earth, a 1-kilogram brick weighs about 1.6 newtons (or 0.36 pounds). On planets with stronger gravity, it would weigh more, but the mass of the brick is the same everywhere. The brick offers the same resistance to speeding up or slowing down regardless of whether it’s on Earth, on the Moon, or on any other body attracting it. In a drifting spaceship, where a scale with a brick on it reads zero, the brick still has mass. Even though it doesn’t press down on the scale, the brick has the same resistance to a change in motion as it has on Earth. just as much force would have to be exerted by an astronaut in the spaceship to shake it back and forth as would be required to shake it back and forth while on Earth. You’d .have to provide the same amount of push to accel- erate a huge truck to a given speed on a level surface on the Moon as on Earth. The difficulty of liftng it against gravity (weight), however, is something else. Mass and weight are different from each other (Figures 4.6 and 4.7). A nice demonstration that distinguishes mass and weight is the massive ball suspended on the string, shown in Figure 4.8. The top string breaks when the lower string is pulled with a gradual increase in force, but the bottom string breaks when the lower string is jerked. Which of these cases illustrates the weight of the ball, and which illustrates the mass of the ball? Note that only the top string bears the weight of the ball. 50, when the lower string is grad- ually pulled, the tension supplied by the pull is transmitted to the top string. The total tension in the top string is caused by the pull plus the weight of the ball. The top string breaks when the breaking point is reached. But, when the ASo 2.2 lb equals 9.8 N, or 'i N is approximately equal to 0.22 Ibrabour the weight of an apple. In the metric system, it is customary to specify quantities of matter in units of mass [in grams or kilograms) and rarely in units of weight (in newtons). In the United States and countries that use the British system of units, however, quantities of matter are customarily specified in units of weight (in pounds). (The British unit of mass, the slug, is not well known.) See Appendix I for more about systems of measurement. When two things are directly proportional to each other, as one increases, the other increases also. How- ever, when two things are inversely propor- tional to each other, as one increases, the other decreases. Chapter 4 Newton’s Second Law of Motion 63 bottom string is jerked, the mass of the ball—its tendency to remain at rest— is responsible for the bottom string breaking. It is also easy to confuse mass and volume. When we think of a massive object, we often think of a big object. An object’s size (volume), however, is not necessarily a good way to judge its mass. Which is easier to get moving: a car battery or an empty cardboard box of the same size? So, we find that mass is neither weight nor volume. CHECK YOURSELF 1. Does a 2-kg iron brick have twice as much inertia as a 'l-kg iron brick? Twice as much mass? Twice as much volume? Twice as much weight? 2. Would it be easier to lift a cement truck on the Earth’s surface or to lift it on the Moon’s surface? 3. Ask a friend to drive a small nail into a piece of wood placed on top ofa pile of books on your head. Why doesn’t this hurt you? Acceleration Push your friend on a skateboard and your friend accelerates. Now push equally hard on an elephant on a skateboard and the 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 acceler- ation; 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, 1 mass Acceleration ~ CHECK YOUR ANSWERS 1. The answers to all parts areyes. A 2—kg iron brick has twice as many iron atoms and therefore twice the amount of matter and mass. In the same location, it also has twice the weight. And, since both bricks have the same density (the same ratio of mass to volume), the 2-kg brick also has twice the volume. 2. A cement truck would be easier to lift on the Moon because the gravitational force is less on the Moon. When you lift an object, you are contending with the force of gravity (its weight). Although its mass is the same on the Earth, on the Moon, or anywhere, its weight is only 1/6 as much on the Moon, so only U6 as much effort is required to lift it there. To move it horizontally, however, you are not pushing against gravity. When mass is the only factor, equal forces will produce equal accelerations, whether the object is on the Earth or the Moon. 3. The relatively large mass of the books and block atop your head resists being moved. The force that is successful in driving the nail will not be as successful in accelerating the massive books and block, which don’t move very much when the nail is struck. Can you see the similarity between this and the suspended massive ball demonstration, where the supporting string doesn’t break when the bottom string isjerkecl? —___—_—————-——— 64 Part One Mechanics FIGURE 4.1 0 An enormous Force is required to accelerate this three-story—high earth mover when it carries a typical BSD-ton loads FIGURE 4.9 Interactive Figure * The greater the mass, the greater the Force must be For a given acceleration, By inversely We mean that the two values change in opposite directions. As the denominator increases, the whole quantity decreases. For example, the quantity 1/100 is less than 1/10. Newton’s Second Law of Motion conceptsfacceleration, force, and mass. He proposed one of the most impor— tant rules of nature, his second law of motion. Newton’s second law states sics Newton was the first to discover the relationship among three basic physical CB Newton’s Second 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. In summarized form, this is , net force Force of hand Acceleration ~ ——— OfiCEIcrates mass 1' I . - I cs' ' 3‘ e brlci We use the Wiggly line m as a symbol meaning is proportional to. We say i that acceleration a is directly proportional to the overall net force F and inversely proportional to the mass in. By this we mean that, if F increases, The same force a increases by the same factor (it P doubles, a doubles); but if in increases, accelerates Z bricks it decreases by the same factor (if m doubles, a is cut in half). V2 ‘35 mUCh By using consistent units, such as newtons (N) for force, kilograms (kg) for mass, and meters per second squared (m/szi for acceleration, the proportional- ity may be expressed as an exact equation: .' . net force 3 bricks, 1/3 as acceleration = 1 much acceleration mass In its bricfest form, where a is acceleration, PM is net force, and m is mass, it becomes 4- FUE[ a _ in FIGURE 4.11 Acceleration is inversely An object is accelerated in the direction or the force acting on it. Applied in proportional to mass. the direction of the object’s motion, a force will increase the object’s speed. Chapter 4 Newton’s Second Law ofMation 55 Applied in the opposite direction, it will decrease the speed of the object. Applied at right angles, it will deflect the object. Any other direction of application will result in a combination of speed change and deflection. The acceleration ofari object is always in the direction of the net force. CHECK YOURSELF 1. in the previous chapter, acceleration was defined to be the time rate of change in velocity; that is, a : (change in vly-‘time. in this chapter, are we saying that acceleration is instead the ratio of force to mass—that is, a : F/m? Which is it? 2. Ajumbo jet cruises at a constant velocity of l000 km/h when the thrusting force ofits engines is a constant 100,000 N. What is the acceleration of the jet? What is the force of air resistance on the jet? When Acceleration ls g—Free Fall i ll ‘5 = g» ..2_F :: m m 9 FIGURE 4.12 Interactive Figure The ratio ofweight (F) to mass (m) is the same for all objects in the same locality; hence, their accelerations are the same in the absence ofair resistance. Only a single Force acts on something in free fall—the force of gravity. Although Galileo founded both the concept of inertia and the concept of accel- eration, and although he was the first to measure the acceleration of falling objects, he 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 Earth because of the gravitational force of attraction between the object and 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. The greater the mass of an object, the greater is the gravitational force of attraction between it and the Earth. The double brick in Figure 4.12, for exam- ple, has twice the gravitational attraction of the single brick. Why, then, as Aris— totle 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 Ii'llEL“ Free- Fall Acceleration Explained CHECK YOUR ANSWERS 1. Acceleration is defined as the time rate of change of velocity and is produced by a Force. How much force/mass (the cause) determines the rate change in v/time (the effect). So, whereas we defined acceleration in Chapter 3, in this chapter we define the terms that produce acceleration. 2. The acceleration is zero because the velocity is constant. Since the acceleration is zero, it follows From Newton’s second law that the net Force is zero, which means that the force of air drag must just equal the thrusting Force oF100,000 N and must act in the opposite direction. So the air drag on the jet is 100,000 N. (Note that we don’t need to know the velocity of the jet to answer this question. We need only to know that it is constant, our clue that acceleration—and there- Fore net force—is zero.) ,_,_—_—.—._.————-——— 66 Part One Mechanics =‘lT é Ulfl D FIGURE 4.13 The ratio ofweight (F) to mass (m) is the same For the large rock and the small Feather; similarly, the ratio of circumference (C) to diame ter (D) is the same For the large and the small circle. weight_but also 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 a, to denote that acceleration is due to gravity alone. The ratio of weight to mass for freely falling objects equals a constant! g. This is similar to the constant ratio of circumference to diameter for cir- cles, which equals the constant 17. The ratio of weight to mass is the same both for heavy objects and for light objects, just as the ratio of circumfer- ence to diameter is the same borh for large circles and for small circles (Figure 4.13). 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 with the same acceleration as the pebble because, although the force on the boulder (its weight) is 100 times greater than the force on the pebble, its resistance to a change in motion (its mass) is 100 times that of the pebble. The greater force offsets the equally greater mass. CHECK YOURSELF In a vacuum, a coin and a Feather fall at the same rate, side by side. Would it be correct to say that equal forces of gravity act on both the coin and the feather when in a vacuum? When Acceleration Is Less Than g—Nonfree Fall ,_: When Galileo tried to explain why all objects fall with equal accel- erations, wouldn’t he have loved to know the rule a : F/m? Iill§£°s Falling and Air Resistance Objects falling in a vacuum are one thing, but 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, how— ever, 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 in which air resistance can be neglected. the net force is the weight because it is the only force. In the presence ______________————————— CHECK YOUR ANSWER No, no, no, a thousand times no! These objects accelerate equally not because the Forces of gravity on them are equal, but because the ratios of their weights to their masses are equal. Although air resistance is net present in a vacuum, gravity is. (You’d know this if you stuck your hand into a vacuum chamber and the truck shown in Figure 4.10 rolled over it!) If you answered yes to this question, let this be a warn- ing to be more careful when you think physics! _____,_______—————— FIGURE 4.14 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. Air Resistance Air Resistance T Weight FIGURE 4.15 Interactive Figure k The heavier parachutist must fall Faster than the lighter parachutist for air resistance to cancel his greater weight. Chapter 4 Newton’s Second Law of Motion 67 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.4 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 mol- ecules 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. In some cases, air drag greatly affects falling; in other cases, it doesn’t. Air drag is important for a falling feather. Because a feather has so much area compared with its small weight, it doesn’t have to fall very fast before the upward-acting air resistance cancels the downward-acting weight. The net force on the feather is then zero and acceleration terminates. When acceleration terminates, we say that 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. 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 acceler— ates; she has reached her terminal velocity. For a feather, terminal velocity is a few centimeters per second, whereas, for a skydivet, 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 veloc- ity. A smaller terminal velocity is attained by spreading oneself out like a flying squirrel. Minimum terminal velocity is attained when the parachute is opened. Consider a man and woman parachuting together from the same altitude (Figure 4.15). Suppose that the man is twice as heavy as the woman and that their samensized parachutes are initially opened. Having parachutes of the same size means that, at equal speeds, the air resistance is the same on both of them. Who reaches the ground first—the heavy man or the lighter woman? The answer is that 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 para- chutes are the same, the terminal speeds for each would be the same and, there- fore, that both would reach the ground at the same time. This doesn’t happen, however, because air drag depends on speed. Greater speed means greater force of air impact. The woman will reach her terminal speed when the air drag against her parachute equals her weight. When this occurs, the air drag against the para- chute of the man will not yet equal his weight. He must fall faster than she does for the 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. 4In mathematical flotation, Fm mg - R u' i 7” H1 where mg 15 the weight and R is the air resistance. Note that when R = mg, a = 0-, then, with no acceleration, the ohiect falls at constant velocity. \Xlith elementary algebra. we can go another step and get we r R R g=—= 7 .8 m m m W’s: see that the acceleration a will always be less than 3 if air resistance R impedes falling. Only when R = 0 does .1 : 3‘ 3Terminal speed for the twice-as-heavy man will be about 41 percent greater than the woman‘s terminal speed, . . . . . . . 7 . . 7 because the retarding force of air resistance is proportional to speed squared. (21,",“7emu“; = 1.41“ = 2.) 68 Part One Mechanics JUL c When the Force 0F gravity and air drag act on a Falling object, it is not in Free Fall. FIGURE 4.16 A stroboscopic study oFa golF ball (leFt) and a StyroFoam ball (right) Falling in air. The air resistance is negligible For the heavier golFball, and its acceleration is nearly equal to g. Air resistance is not negligie ble For the lighter StyroFoam ball, which reaches its termi- nal velocity sooner. CHECK YOURSELF A skydiver jumps From a highuFlying helicopter. As she Falls Faster and Faster through the air, does her acceleration increase, decrease, or remain the same? one a regular hollow ball and the other filled the iron-filled ball is consider— ably heavier than the regular ball. If you hold them above your head and drop them simultaneously, you’ll see that they strike the ground at about the same time. But if you drop them From a greater heightfisay, from the top of a building— Consider a pair of tennis balls, with iron pellets. Although they are the same size, you’ll note the heavier ball strikes the ground First. Why? In the first case, the balls do net gain much speed in their short fall. The air drag they encounter is small compared with their weights, even {or the regular ball. The tiny difference in their arrival time is not noticed. But, when they are dropped from a greater height, the greater speeds of fall are met with greater air resistance. At any given speed, each ball encounters the same air resistance because each has the same size. This same air resistance may be a lot compared with the weight of the lighter ball, but only a little compared with the weight of the heavier ball (like the parachutists in Figure 4.15). For example, 1 N of air drag acting on a 2-N object will reduce its accel- eration by half, but 1 N of air drag on a ZOO-N object will only slightly diminish its acceleration. So, even with equal air resistances, the accelerations of each are different. There is a moral to be learned here. Whenever you consider the acceler— ation of something, use the equation of Newton’s second law to guide your think- ing: The acceleration is equal to the ratio of net force to the mass. For the falling tennis balls, the net force on the hollow ball is appreciably reduced as air drag builds up, while the net Force on the iron—filled ball is comparably only slightly reduced. Acceleration decreases as net force decreases, which, in turn, decreases as air drag increases. If and when the air drag builds up to equal the weight of the falling object, then the net force becomes zero and acceleration terminates. _________________—————-— CHECK YOUR ANSWER Acceleration decreases because the net Force on her decreases. Net Force is equal to her weight minus her air resistance, and, because air resistance increases with increas- ing speed, net Force and hence acceleration decrease. By Newton’s second law, Fner _ M? R a : m m where mg is her weight and R is the air resistance she encounters. As R increases, a 0; then, with no decreases. Note that, iF she Falls Fast enough so that R = mg, a — acceleration, she Falls at a constant speed. ...
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