Download Exercises – Chapter 3

Exercises – Chapter 3
1. In what way does the string of a bow and arrow behave like a spring?
E.1
As you draw the string away from its equilibrium shape, it experiences a restoring force proportional to its
displacement.
2. As you wind the mainspring of a mechanical watch or clock, why does the knob get harder and harder to turn?
E.2
As the mainspring winds farther and farther from its equilibrium shape, the restoring force it exerts on its
end gets stronger. You must overcome this increasing force as you twist the knob.
3. Curly hair behaves like a weak spring that can stretch under its own weight. Why is a hanging curl straighter at the
top than at the bottom?
E.3
The top of the curl supports more weight than the bottom.
4. When you lie on a spring mattress, it pushes most strongly on the parts of you that stick into it the farthest. Why
doesn’t it push up evenly on your entire body?
E.4
The deeper you dent the mattress's surface at any given point, the stronger the local restoring force
becomes.
5. If you pull down on the basket of a hanging grocery store scale so that it reads 15 N, how much downward force are
you exerting on the basket?
E.5
15 N.
6. While you’re weighing yourself on a bathroom scale, you reach out and push downward on a nearby table. Is the
weight reported by the scale high, low, or correct?
E.6
The scale reads low; it reports less than your actual weight.
E.6
The upward force that the table exerts on your hand contributes to the force supporting you against gravity
and allows the scale to exert a smaller upward force on you.
7. There’s a bathroom scale on your kitchen table and your friend climbs up to weigh himself on it. One of the table’s
legs is weak and you’re afraid that he’ll break it, so you hold up that corner of the table. The table remains level as you
push upward on the corner with a force of 100 N. Is the weight reported by the scale high, low, or correct?
E.7
Correct.
8. If you put your bathroom scale on a ramp and stand on it, will the weight it reports be high, low, or correct?
E.8
The scale will read less than your actual weight.
E.8
The scale only reports how hard it pushes perpendicular to its surface. When you weigh yourself on a
ramp, your weight doesn't push directly toward the scale's surface and so the scale doesn't push back
directly perpendicular to its surface. The perpendicular component of its force on you is less than you full
weight.
9. When you step on a scale, it reads your weight plus the weight of your clothes. Only your shoes are touching the
scale, so how does the weight of the rest of your clothes contribute to the weight reported by the scale?
E.9
You support your clothes and the scale supports you.
10. To weigh an infant you can step on a scale once with the infant and then again without the infant. Why is the
difference between the scale’s two readings equal to the weight of the infant?
E.10 When you weigh yourself with the child, the scale reports the total force needed to support you both. When
you weigh yourself alone, the scale reports only the force needed to support you. The difference between
these two reported values is the force needed to support only the child.
11. An elastic ball that wastes 30% of the collision energy as heat when it bounces on a hard floor will rebound to 70%
of the height from which it was dropped. Explain the 30% loss in height.
E.11 A 30% loss of rebound height is a 30% loss of gravitational potential energy—equal to the energy that
became thermal energy.
12. The best running tracks have firm but elastic rubber surfaces. How does a lively surface assist a runner?
E.12 An elastic track stores energy as it dents when the runner's foot presses against it and then returns that
energy to the runner as it undents.
13. Why is it so exhausting to run on soft sand?
E.13 You do work on the sand as you step on it, but the sand doesn’t return this energy to you as you lift your
foot back up again.
14. Steep mountain roads often have emergency ramps for trucks with failed brakes. Why are these ramps most
effective when they are covered with deep, soft sand?
E.14 Sand dents easily as the truck plows through it and extracts energy from the truck. The truck does work in
pushing the sand out of the way. The sand converts that work into safe thermal energy.
15. There have been baseball seasons in which so many home runs were hit that people began to suspect that
something was wrong with the baseballs. What change in the baseballs would account for them traveling farther than
normal?
E.15 An increase in the balls’ coefficients of restitution.
16. During rehabilitation after hand surgery, patients are often asked to squeeze and knead putty to strengthen their
muscles. How does the energy transfer in squeezing putty differ from that in squeezing a rubber ball?
E.16 Energy transferred while deforming putty is converted into thermal energy and never returns to the
person's hand. However, energy transferred while deforming a rubber ball becomes elastic potential energy
in the ball and returns to the person's hand when the rubber ball returns to its spherical shape.
17. Your car is on a crowded highway with everyone heading south at about 100 km/h (62 mph). The car ahead of you
slows down slightly and your car bumps into it gently. Why is the impact so gentle?
E.17 Your relative velocity is small—in your frame of reference the car in front of you is barely moving, so the
impact is very gentle.
18. Bumper cars are an amusement park ride in which people drive small electric vehicles around a rink and
intentionally bump them into one another. All of the cars travel at about the same speed. Why are head-on collisions
more jarring than other types of collisions?
E.18 During a head-on collision, the relative velocity is enormous--the sum of the two individual velocities--and
the forces, accelerations, and momentum transfers are enormous as well.
19. When two trains are traveling side by side at breakneck speed, it’s still possible for people to jump from one train
to the other. Explain why this can be done safely.
E.19 Because the trains’ relative velocity is zero, a person jumping between them views them both as essentially
stationary.
20. If you drop a steel marble on a wooden floor, why does the floor receive most of the collision energy and
contribute most of the rebound energy?
E.20 The marble is stiffer than the floor, so the floor dents more than the marble. Since the forces are equal in
magnitude, the floor also has the most work done on it during the collision and is thus responsible for most
of the rebound energy.
21. A RIF (reduced injury factor) baseball has the same coefficient of restitution as a normal baseball except that it
deforms more severely during a collision. Why does this increased deformability lessen the forces exerted by the ball
during a bounce and reduce the chances of its causing injury?
E.21 During a bounce, the work done on a RIF ball—to store energy in it—involves a smaller force exerted for
a longer distance.
22. Padded soles in running shoes soften the blow of hitting the pavement. Why does padding reduce the forces
involved in bringing your foot to rest?
E.22 With padding in your shoes, you feet stop over a longer period of time when they encounter the pavement.
The momentum transfer or impulse is the same, but the force is smaller while the time is longer.
23. Some athletic shoes have inflatable air pockets inside them. These air pockets act like springs that become stiffer as
you pump up the air pressure. High pressure also makes you bounce back up off the floor sooner. Why does high
pressure shorten the bounce time?
E.23 At high pressure, the shoes are stiffer and exert larger forces when distorted. They accelerate more rapidly
and bounce faster.
24. Why does it hurt less to land on a soft foam pad than on bare concrete after completing a high jump?
E.24 The foam pad brings you to rest over a longer period of time and thus with a small upward force. The
smaller upward force is less uncomfortable than the huge upward force that the concrete would exert on
you while stopping you quickly.
25. Why must the surface of a hammer be very hard and stiff for it to drive a nail into wood?
E.25 The force that the hammer exerts on the nail during their collision would diminish if the hammer’s surface
distorted easily.
26. Some amusement park rides move you back and forth in a horizontal direction. Why is this motion so much more
disturbing to your body than cruising at a high speed in a jet airplane?
E.26 You only feel accelerations, so rapid side-to-side motion is easily felt. In contrast, rapid travel at constant
velocity involves no acceleration and produces no sensations at all.
27. You are traveling in a subway along a straight, level track at a constant velocity. If you close your eyes, you can’t
tell which way you’re heading. Why not?
E.27 While you can feel accelerations, you can’t feel velocity.
28. Moving a can of spray paint rapidly in one direction will not mix it nearly as well as shaking it back and forth. Why
is it so important to change directions as you mix the paint?
E.28 To make the paint move relative to the can and mix as a result, you must make the can accelerate. The
paint will then coast around inside the can and slam into the walls. If you just move the can quickly in one
direction, the paint and can will coast along together and there will be little mixing.
29. Why does a baby’s rattle only make noise when the baby moves it back and forth and not when the baby moves it
steadily in one direction?
E.29 When the rattle accelerates, the beads inside it continue on and hit the walls of the rattle. The rattle then
makes noise.
30. In some roller coasters, the cars travel through a smooth tube that bends left and right in a series of complicated
turns. Why does the car always roll up the right-hand wall of the tube during a sharp left-hand turn?
E.30 When the track makes a sharp left-hand turn, the car needs a strong leftward force to follow it. The car
obtains that leftward force by riding up on the right-hand wall of the tube.
31. Railroad tracks must make only gradual curves to prevent trains from derailing at high speeds. Why is a train likely
to derail if it encounters a sharp turn while it’s traveling fast?
E.31 The sharper the curve, the more centripetal force the train needs to accelerate around the curve. If the track
can’t supply it . . . disaster.
32. Police sometimes use metal battering rams to knock down doors. They hold the ram in their hands and swing it into
a door from about 1 m away. How does the battering ram increase the amount of force the police can exert on the door?
E.32 The police can use a long period of time and a modest forward force to give the battering ram forward
momentum. The battering ram can then transfer this momentum to a door with a huge force exerted for a
short period of time.
33. When a moving hammer hits a nail, it exerts the enormous force needed to push the nail into wood. This force is far
greater than the hammer’s weight. How is it produced?
E.33 The nail exerts an enormous force on the hammer to slow it down. The hammer pushes back, driving the
nail into the wood.
34. A hammer’s weight is downward, so how can a hammer push a nail upward into the ceiling?
E.34 When the moving hammer encounters the nail, the nail pushes back extremely hard on the hammer to stop
the hammer from penetrating into the nail. The hammer pushes back on the nail, propelling it into the
ceiling. While gravity also pushes on the hammer and slows its upward motion, the weight forces in this
situation are trivial compared to the forces between hammer and nail.
35. As you swing back and forth on a playground swing, your apparent weight changes. At what point do you feel the
heaviest?
E.35 At the bottom of each swing.
36. Some stores have coin-operated toy cars that jiggle back and forth on a fixed base. Why can’t these cars give you
the feeling of actually driving in a drag race?
E.36 A jiggling car can only accelerate you forward for a brief period of time, while a true drag racer will
accelerate you forward for a long period of time. You can feel the forward acceleration as a backward
gravity-like sensation.
37. A salad spinner is a rotating basket that dries salad after washing. How does the spinner extract the water?
E.37 As the salad undergoes rapid centripetal acceleration, the water travels in straight lines and runs off the
salad.
38. People falling from a high diving board feel weightless. Has gravity stopped exerting a force on them? If not, why
don’t they feel it?
E.38 A falling person's internal organs all fall together without having to support one another. The absence of
internal forces within a person's body is what leads them to believe that they are weightless.
39. When your car travels rapidly over a bump in the road, you suddenly feel weightless. Explain.
E.39 After going over a bump, the car begins to accelerate downward and your apparent weight is briefly less
than your real weight.
40. Astronauts learn to tolerate weightlessness by riding in an airplane (nicknamed the “vomit comet”) that follows an
unusual trajectory. How does the pilot direct the plane in order to make its occupants feel weightless?
E.40 The pilot steers the "vomit comet" in the path of a falling object: a parabolic arc through the air. The net
force on the plane is then exactly equal to its weight and it is truly falling. The objects inside it fall as well
and everyone inside feels weightless.
41. You board an elevator with a large briefcase in your hand. Why does that briefcase suddenly feel particularly heavy
when the elevator begins to move upward?
E.41 As the car accelerates upward, you must pull upward on the briefcase extra hard to make it accelerate
upward too.
42. As your car reaches the top in a smoothly turning Ferris wheel, which way are you accelerating?
E.42 You are accelerating downward at the top of a smoothly turning Ferris wheel.
E.42 In uniform circular motion, you always accelerate toward the center of the circle.
Problems – Chapter 3
1. Your new designer chair has an S-shaped tubular metal frame that behaves just like a spring. When your friend,
who weighs 600 N, sits on the chair, it bends downward 4 cm. What is the spring constant for this chair?
P.1
15,000 N/m.
2. You have another friend who weighs 1000 N. When this friend sits on the chair from Problem 1, how far does it
bend?
P.2
The chair bends downward 6.7 cm.
3. You’re squeezing a springy rubber ball in your hand. If you push inward on it with a force of 1 N, it dents inward 2
mm. How far must you dent it before it pushes outward with a force of 5 N?
P.3
10 mm.
4. When you stand on a particular trampoline, its springy surface shifts downward 0.12 m. If you bounce on it so that
its surface shifts downward 0.30 m, how hard is it pushing up on you?
P.4
The trampoline is pushing upward on you with a force that is 2.5 times your weight.
5. Engineers are trying to create artificial “gravity” in a ring-shaped space station by spinning it like a centrifuge. The
ring is 100 m in radius. How quickly must the space station turn in order to give the astronauts inside it apparent
weights equal to their real weights at the earth’s surface?
P.5
It must turn at about 31 m/s.
6. A satellite is orbiting the earth just above its surface. The centripetal force making the satellite follow a circular
trajectory is just its weight, so its centripetal acceleration is about 9.8 m/s 2 (the acceleration due to gravity near the
earth’s surface). If the earth’s radius is about 6375 km, how fast must the satellite be moving? How long will it take for
the satellite to complete one trip around the earth?
P.6
The satellite must be moving horizontally at about 7,900 m/s. It will complete its trip in about 5100
seconds or 84 minutes.
7. When you put water in a kitchen blender, it begins to travel in a 5-cm-radius circle at a speed of 1 m/s. How quickly
is the water accelerating?
P.7
20 m/s2.
8. In Problem 7, how hard must the sides of the blender push inward on 0.001 kg of the spinning water?
P.8
The blender must push inward on the water with a force of 0.02 newtons.