Physics 105N - How Things Work - Fall, 1995

Problem Set 2 - Answers


Due Monday, October 23, 1995, In Class

Please Answer Each Question As Briefly As Possible

You May Work Together, But Write Up Your Answers Separately


Question 1: (Case 5 from Chapter 2)

A trampoline has an elastic surface, supported at the edges by springs or elastic bands so that it is normally flat. This surface stores energy during a bounce so that you can jump very high on it.

(A) When you get on a particular trampoline and stand in its center, its surface distorts downward 10 cm. It is behaving like a spring that is distorted away from its equilibrium shape. The distorted trampoline surface is exerting a restoring force on you in what direction?

Answer: Upward

Why: The trampoline's equilibrium shape is flat. Since you distort it downward, it experiences a restoring force upward, back toward its equilibrium shape.

(B) If someone with twice your weight climbed into the middle of the empty trampoline, how far downward would its surface distort?

Answer: 20 cm

Why: Since the trampoline behaves like a spring, the restoring force it experiences is proportional to its distortion. Since the force on it is twice as great as one that distorts it 10 cm, it must distort 20 cm.

(C) You begin bouncing up and down on the trampoline. As you land on the trampoline during one of the bounces, its surface distorts downward 30 cm. Your weight hasn't changed so how can you make it distort so far downward?

Answer: The trampoline is stopping your descent and pushing you back upward. It is accelerating you. To make you accelerate, it must give you a net upward force. Since you are also experiencing your weight downward, it must push upward on you with a force greater than your weight.

Why: There is no limit to the force that the trampoline exerts on you. If it pushes on you with a force equal to your weight, then you won't accelerate at all. But it can and does push on you with a force greater than your weight when it is acting to keep you from passing through it. It pushes upward on you very hard and you accelerate upward.

(D) While you are in the air above the trampoline, nothing is pushing on you except gravity. On both your way upward and your way downward (while in the air), do you feel weightless, your normal weight, or particularly heavy?

Answer: You are weightless both on your way up and your way down.

Why: When you are above the trampoline, the only force you experience is gravity. You accelerate downward at 9.8 m/s2. You feel weightless because all of your parts fall together without having to support one another. Another way to view this situation is that you are accelerating downward and experience an upward fictitious force that exactly balances you downward perception of weight. You feel no weight. It doesn't matter whether you are rising or fall because only your acceleration counts and you are accelerating downward in both cases.

(E) While you are touching the trampoline during a bounce, its surface is pushing upward on you. Near the bottom of the bounce, you distort the trampoline's surface downward more than 10 cm. On both your way downward and your way upward (while the surface is distorted downward more than 10 cm), do you feel weightless, your normal weight, or particularly heavy?

Answer: You feel particularly heavy both on your way down and your way up.

Why: As you near the bottom of a bounce, you are accelerating upward fast. You feel a strong downward fictitious force that adds to your weight and makes you feel very heavy.

Question 2: (Case 6 from Chapter 2)

Bumper cars are a popular ride at many amusement parks. You drive about an enclosed area in a small, electrically powered vehicle. This vehicle has a large rubber bumper wrapped all the way around its exterior.

(A) Half the fun of driving a bumper car is crashing into other people's cars. When you drive forward at high speed and slam into the car in front of you, you find yourself thrown forward in your car. Which way is your car accelerating?

Answer: Backward

Why: Your velocity is changing from a forward velocity to a backward velocity. That change requires a backward acceleration. You feel thrown forward because you experience a fictitious force in the direction opposite your backward acceleration.

(B) When you are stopped and someone else slams into the front of your car, you find yourself thrown forward in your car. Which way is your car accelerating?

Answer: Backward

Why: Your velocity is changing from a stationary one to a backward velocity. That change requires a backward acceleration, too. You again feel thrown forward because you experience a fictitious force in the direction opposite your backward acceleration.

(C) What would happen to the support forces between cars and to the accelerations those cars experienced if the soft rubber bumpers were replaced by hard, steel bumpers?

Answer: Both the support forces and the accelerations become much greater.

Why: Steel bumpers would be elastic like rubber, but much stiffer. You'd still bounce off of the other cars, but the forces involved would be enormous and they would cause very strong accelerations. It would be a very unpleasant ride.

(D) Suppose that all of the cars are traveling at 20 km/h in various directions, as viewed by the people waiting in line for a turn. You crash first into a car that is heading in about the same direction as yours and feel a gentle thud. You then crash into a car that is heading toward you and experience a tremendous jolt. Explain briefly why these two collisions are different.

Answer: When you collide with a car moving in the same direction as you are, that car's relative velocity is very small. You bounce gentle from one another, exchanging only small amounts of momentum with modest forces. When you collide with a car moving toward you, that car's relative velocity is very large. You hit violently, exchanging large amounts of momentum with large forces.

Why: When you're traveling in a bumper car, what matters is the velocity of other cars relative to you. What the spectators see doesn't matter because they aren't colliding with other cars. If a car approaches you slowly from your reference frame, then the collision will be gentle. If the approach is fast, the collision will be jarring.

(E) If you were to fill your car with metal bars, so that its mass was 10 times that of any other bumper car, how would it affect the jolts you experience during crashes, as compared to riding in a normal car?

Answer: You'd feel much weaker jolts.

Why: By increasing the mass of your car, you are making it much harder to accelerate. A collision that would have caused a large acceleration in a normal car would then cause only a small acceleration. The same forces would be acting on the car, but its increased mass would reduce its acceleration.

Question 3: (Case 4 from Chapter 3)

A scuba diver swims below the surface of the water, breathing air from the steel tanks of an aqualung.

(A) It is easiest for her to maintain a constant depth if the buoyant force on her and her equipment exactly balances their weight. The air tanks float by themselves so she wears a heavy weight belt to compensate. How should her average density compare with that of the water around her?

Answer: Her density should be exactly equal to the density of the water around her.

Why: She'll hover only when the net force on her is zero, so the buoyant force she experiences must equal her weight. She must thus displace exactly her weight in water, which she can do by having an average density (mass per volume) equal to the density of water.

(B) When she is 20 m below the surface of the water, what is the pressure pushing inward on her chest?

Answer: About 3 times atmospheric pressure (300,000 Pa).

Why: A column of water 20 m tall weighs about 2 times as much as a column of the earth's atmosphere, all the way up. When she is swimming 20 m below the surface, the diver is supporting the weight of the air column and the water column, for a total of 1 + 2 atmospheres. Each atmosphere is about 100,000 Pa of pressure (100,000 N/m2).

(C) When she breathes, she expects air to flow into her lungs. She can change the pressure in her lungs slightly, using the muscles in her chest and diaphragm. How must the pressure in her mouth compare with that in her lungs in order for air to begin flowing into her lungs?

Answer: The pressure in her mouth must be greater than the pressure in her lungs.

Why: Air accelerates toward lower pressure. For stationary air to begin flowing into the diver's lungs, the pressure in her lungs must be lower than that in her mouth.

(D) If instead of taking compressed air with her, she were to try to breathe surface air through a straw, the air entering her mouth would be at atmospheric pressure. What would happen when she tried to breathe?

Answer: Air will flow out of her lungs and into the straw.

Why: When she tries to breathe through the straw, the pressure in her mouth will be atmospheric pressure and the pressure in and around her lungs will be much greater. Air will accelerate toward her lungs and she will breathe out rather than in.

(E) The special pressure regulator of an aqualung delivers air to her mouth at exactly the same pressure as that of the surrounding water. She can breathe this air easily. However, the deeper she dives, the faster she consumes air molecules from the tanks and the sooner she must return to the surface. Explain.

Answer: As she dives deeper, the pressure of the air that's delivered to her mouth must increase. High pressure air is very dense because the air molecules must be packed tightly together to create the higher pressure. Thus each breathe she takes of the high pressure air must contain more air molecules and she will empty the tanks quickly as a result.

Why: The only way to increase air's pressure, without heating it up, is to pack its air molecules more tightly together. One breathe of high pressure air contains more molecules than one breathe of low pressure air.

Question 4: (Case 6 from Chapter 4)

You and your best friend live on the 58th floors of two adjacent high-rise apartment buildings. You have windows that face one another across an open courtyard. One day, the city turns off all the water to your friend's building. You decide to help your friend obtain water. You immediately buy about 500 meters of garden hose, enough to reach from either apartment to the ground at least two times.

(A) Your first thought is to run water from the gardener's faucet, which is at street level in the courtyard, up to your friend's apartment. Although water runs briefly into the hose, it never reaches your friend's apartment. Why not?

Answer: The water pressure in the faucet isn't high enough to support a column of water that reaches upward 58 floors.

Why: As water flows upward toward the 58th floor, it must convert its pressure potential energy into gravitational potential energy. 58 floors is a long way, and the pressure at the faucet doesn't provide enough energy to lift the water that high.

(B) You decide to obtain water from a faucet in your apartment. You run the hose out of your window, down to the courtyard, and up to your friend's apartment. To your surprise, the hose bursts in the courtyard when you turn on the water. Why does the hose burst?

Answer: The water at the bottom of the loop, in the courtyard, must support the weight of the water in the hose, all the way up to the 58th floor. The pressure required to support that much water is enormous and, when exerted on the walls of the hose, enough to make the hose burst.

Why: A 200 m tall column of water weighs quite a bit and must be supported by a very high pressure in the water at the bottom. This pressure would have been enough to lift water up the other side of the hose, had the hose been strong enough to withstand it. But the hose could contain such high pressure water.

(C) You patch the hose and decide not to let it droop down to the courtyard. With the hose running almost directly across the gap between windows, you turn on the faucet. What happens this time?

Answer: The water flows easily into the other apartment.

Why: As long as the hose is more or less level between the apartments, the water flows through the pipe without any problems. The water's gravitational potential energy doesn't change much, so there are no strange pressure problems.

(D) Water is flowing rapidly through all 500 meters of hose as your friend fills a bathtub. To stop the flow, your friend suddenly makes a kink in the end of the hose. The pressure in the hose surges upward, and the hose bursts. What produced this pressure surge?

Answer: Water hammer.

Why: The 500 meter long flow of water is like a shaft of water flowing toward the other apartment. Stopping this shaft of water takes an enormous force, exerted by pressure at the far end of the hose when it is kinked. The pressure surges upward as the hose is kinked, so as to stop the water flow as quickly as possible. The hose can't take this much pressure and bursts.

(E) As water flows out of the narrow split in the hose, it accelerates to very high speed. What happens to its pressure as its speed increases?

Answer: The water pressure drops as it accelerates to high speed.

Why: The water is converting its pressure potential energy into kinetic energy in order to flow quickly out of the split in the hose. The water in the split experiences a pressure imbalance, pushing it out of the hose and causing it to accelerate. It picks up speed as its pressure drops.

Question 5: (Case 3 from Chapter 4)

You are watching a downhill ski race. In this race, the skiers are trying to descend a mountain as quickly as possible without crashing. The veteran skier standing nearby makes a number of observations about the racers that actually have a sound scientific basis.

(A) The veteran points out that one of the skiers is skidding around corners and that such skids slow her down. During a skid, the ski slips sideways across the snow rather than sliding straight forward through it. Why will skidding always slow the skier down?

Answer: Skidding is sliding against sliding friction and the sliding friction does negative work on the skier. Since this negative work decreases the skier's energy, the skier slows down.

Why: As the skier skids, the snow exerts a huge force on the skier in the direction opposite the skier's motion. Since the force that the snow exerts is opposite the motion of the skier, the snow does negative work on the skier. The skier's two main types of energy are kinetic and gravitational potential. Skidding decrease one or both of these types of energy and ultimately slows the skier down.

(B) The veteran notes that a skier also slows down by remaining airborne too long after passing over a jump. Why should the skier regain contact with the slope as soon as possible after the jump?

Answer: The skier experiences a forward component of force (part of the downhill ramp force) only while in contact with the ground.

Why: In the air, the skier experiences only gravity and a backward drag force. With no forward forces, the skier slows down. On the ground, the skier experiences gravity, drag, a little sliding friction, and a support force. The support force is tilted forward because it's at right angles to the slope. This support force exerts a forward force on the skier, keeping that skier moving forward quickly.

(C) A skier loses control and skis off the course into a net. The veteran remarks that it is much better to hit a net than a tree. In terms of force and momentum, why is the skier less likely to be injured by crashing into a net than by crashing into a tree?

Answer: Hitting the net will slow the skier gradually, with small forces. Hitting the tree will slow the skier quickly, with large forces. Either way, the skier will transfer all their momentum to the earth. But with the net, the transfer is slow enough that it doesn't hurt much.

Why: When you come to a stop, you give up all your momentum. How you stop determines how quickly to give up your momentum and the forces involved in the transfer. The faster you stop, the larger the forces must be and the more uncomfortable the stop.

(D) The veteran is impressed by one skier's consistent compact form, a tight tucked shape. Why doesn't she want to push on the air during her descent?

Answer: Pushing the air forward and having the air move forward means that she is doing work on the air and giving up some of her energy to it. She slows down.

Why: The skier wants to retain as much energy as possible. She doesn't want to do work on anything that she doesn't have to. Thus she avoids pushing the air forward because the air will take away her energy.

(E) The veteran remarks about the high air pressure that skiers feel in front of them as they travel swiftly downhill. What causes this high pressure?

Answer: As the air approaches the skier, it slows down on the skier's chest. The air's kinetic energy decreases but its pressure potential energy increases.

Why: The proper frame of reference in which to view this problem is the skier's frame. As the skier descends the hill, the air rushes toward the skier. This air flows around the skier, but first comes almost to a stop on the skier's chest. That air experiences a large rise in pressure and the skier can feel this rise. You can feel a similar rise in pressure if you hold you hand out the window of a moving car.