Physics 105 - How Things Work - Fall, 2003

Midterm Examination - Solutions

Given Friday, October 10, 2003 from 1:00pm to 1:50pm

(Click on Distribution Graph to Enlarge)

PART I: MULTIPLE CHOICE QUESTIONS

Please mark the correct answer for each question on the bubble sheet. Fill in the dot completely with #2 pencil. Part I is worth 67% of the grade on the midterm examination.

Problem 1:

When a sport utility vehicle (SUV) makes a sudden left turn on the highway, it is in danger of flipping over. One valid explanation for this effect is:

(A) the SUV's angular mass increases as it turns and its angular momentum decreases. A transfer of angular momentum from its wheels to its frame then causes it to begin rotating so that it flips over. [2.8% picked]
(B) the centrifugal force on the SUV as it goes around the turn pushes outward on the top of the SUV. The wheels are held in place by friction, so only the top of the SUV accelerates outward and it flips over. [18.5% picked]
(C) the force of the SUV's momentum pushes it forward while the road pushes it to the left and these two forces twist the SUV so that it undergoes angular acceleration and flips over. [22.1% picked]
(D) the leftward frictional force that causes the SUV to accelerate leftward during the turn also exerts a torque on the SUV about its center of mass and can cause the SUV to undergo angular acceleration and flip over. [56.5% picked]

Answer: (D) the leftward frictional force that causes the SUV to accelerate leftward during the turn also exerts a torque on the SUV about its center of mass and can cause the SUV to undergo angular acceleration and flip over.[56.5% picked]

Why: For the SUV to accelerate toward the left, something must push it toward that left. That leftward push is exerted by the ground through a frictional force. This leftward frictional force is the only horizontal force on the SUV, so it accelerates toward the left. However, because that force is exerted very low on the vehicle, it exerts at torque on the SUV about its center of mass. While the SUV has enough static stability to compensate for modest torques, when the frictional torque becomes strong enough, the SUV flips over.

Problem 2:

You and your friend slap your right hands together in "high fives"--your hands make a loud noise, but neither hand is able to push the other hand forward during the slap. Overall the two of you exchange

(A) both energy and momentum. [53.8% picked]
(B) neither energy nor momentum. [6.6% picked]
(C) momentum but no energy. [20.9% picked]
(D) energy but no momentum. [18.7% picked]

Answer: (C) momentum but no energy. [20.9% picked]

Why: For a transfer of energy to occur, one hand must do work on the other. But since neither hand is able to push the other hand forward during the slap, the forces they exert on one another do no work on one another. For a transfer of momentum to occur, one hand must do an impulse on the other. Since each hand pushes on the other for a period of time, they both do impulses on one another. There is a substantial transfer of momentum during the slap, but no energy transfer.

Problem 3:

Jumping from your balcony into the swimming pool will hurt less than an identical jump onto the pool's concrete deck because the impulse (momentum transfer) you will experience when hitting the water will be

(A) the same as when hitting the deck, but will involve a smaller force for a longer time. [84.9% picked]
(B) smaller than when hitting the deck and will involve a smaller force for a longer time. [14.5% picked]
(C) smaller than when hitting the deck and will involve a smaller force for the same time. [0.4% picked]
(D) larger than when hitting the deck, but will involve a smaller force for a smaller time. [0.2% picked]

Answer: (A) the same as when hitting the deck, but will involve a smaller force for a longer time. [84.9% picked]

Why: You will come to a stop regardless of which object you hit, so the amount of momentum you transfer during your "landing" will be the same in either case. But by hitting the water, you allow that impulse to extend over a large period of time and therefore involve a smaller upward force on you.

Problem 4:

If you hit the sealed top of a glass bottle full of root beer, Pepsi, or Coke with a rubber mallet, the glass will accelerate downward while the beverage remains in place. The beverage will soon find itself in the neck of the bottle, compressing the gas there. The resulting pressure imbalance will accelerate the beverage toward the bottom of the bottle. When the beverage reaches the bottom of the bottle, it will

(A) knock the bottom out of the bottle. [66.5% picked]
(B) have already slowed to a stop and nothing dramatic will happen. [8.4% picked]
(C) emit a piercing whistling sound because of its angular momentum. [1.6% picked]
(D) blow the cap off the bottle. [23.5% picked]

Answer: (A) knock the bottom out of the bottle. [66.5% picked]

Why: This experiment was done as a demonstration in class. When the beverage hits the bottom of the bottle, the bottle acts to stop the beverage from passing through its bottom. For the beverage to accelerate backward (to stop), it must experience a higher pressure in front of it than behind it. And sure enough, the pressure in front of the moving beverage skyrockets when it encounters the bottom of the bottle. That pressure becomes so high that it can and often does push the bottom right out of the bottle.

Problem 5:

You are standing still in an elevator, with a briefcase hanging from your motionless hand. As the elevator rises toward the top of the building, you are doing

(A) zero work on the briefcase and the elevator is doing zero work on you. [3.8% picked]
(B) (positive) work on the briefcase and the elevator is zero work on you. [0.4% picked]
(C) zero work on the briefcase and the elevator is doing (positive) work on you. [74.5% picked]
(D) (positive) work on the briefcase and the elevator is doing (positive) work on you. [21.3% picked]

Answer: (D) (positive) work on the briefcase and the elevator is doing (positive) work on you. [21.3% picked]

Why: To do work on something, all you have to do is to push on that object and have that object move in the direction of your push. In the present situation, when the elevator pushes upward on you (as it does in order to support your weight and possibly to make you accelerate somewhat) and you move upward (the elevator is rising, after all), the elevator does work on you. And since you are pushing upward on your briefcase (again to support its weight and possibly to make it accelerate as well) and the briefcase moves upward, you do work on the elevator. This activity may not be very tiring, because the energy needed to lift the briefcase is flowing into you from the elevator and out of you into the briefcase -- you are merely a conduit for this energy. But you are still having work done on you by the elevator and you are still doing work on the briefcase.

Problem 6:

You roll a marble down the side of a round bowl--a bowl with a spherical bottom. The marble rolls right through the bottom of the bowl and up the far side. At this moment, the marble is exactly at the bottom of the bowl. It is rolling away from you--it has just rolled down your side of the bowl and is about to begin rolling up the far side of the bowl [added words]. It is accelerating

(A) downward. [39.2% picked]
(B) forward. [16.8% picked]
(C) upward [11.7% picked]
(D) backward. [32.3% picked]

Answer: (C) upward [11.7% picked]

Why: The fact that none of the possible answers was "zero acceleration" is a clue that the marble is not coasting as it passes through the bottom of the bowl. The marble's speed is constant, but it is actually bending its path from slightly downward just before it reaches the bottom of the bowl to slightly upward just after it reaches the bottom of the bowl. The marble's acceleration is therefore directly upward. In fact, it is an object traveling in a circle around the center of the bowl's spherical curvature. The marble's speed isn't constant through the rolling process, so it isn't an true example of uniform circular motion. However, in this case the marble travels at a pretty steady speed near the very bottom of the bowl and is pretty nearly an object in uniform circular motion. It is therefore accelerating toward the center of its circular path, which is above it. It is accelerate directly upward. [Aside: This was clearly the hardest problem on the exam, made only slightly easier by the absence of the obviously appealing but incorrect answer, "zero acceleration."]

Problem 7:

You are running in a race and are trying to catch up to the lead runner. According to the spectators, the lead runner is traveling at 10 mph (miles-per-hour) and you are traveling in same direction at 11 mph. From your perspective or "frame-of-reference", the lead runner is moving

(A) toward you at 1 mph. [81.3% picked]
(B) away from you at 1 mph. [15.3% picked]
(C) toward you at 21 mph. [2.4% picked]
(D) away from you at 21 mph. [1.0% picked]

Answer: (A) toward you at 1 mph. [81.3% picked]

Why: Since the two runners are heading in the same direction (according to the spectators), most of their overall velocities cancel when we look for the difference between those velocities. In this case, you are gradually catching up to the lead runner, so that lead runner appears to be coming toward you. And the difference between your two speeds is just 1 mph.

Problem 8:

You are standing at the top of a tall cliff above a calm lake. You have two stones in your hand, one twice as heavy as the other, and you throw them together horizontally at the same speed. Both stones soon hit the water. Neglecting any air resistance, the heavier stone reaches the water

(A) in half the time and half as far from the cliff as the lighter stone. [1.8% picked]
(B) in half the time but at the same place as the lighter stone. [0.4% picked]
(C) at the same time but half as far from the cliff as the lighter stone. [62.0% picked]
(D) at the same time and at the same place as the lighter stone. [35.7% picked]

Answer: (D) at the same time and at the same place as the lighter stone. [35.7% picked]

Why: In the absence of air resistance, the two objects fall at the same rate. These two stones were "dropped" at the same moment and they therefore fall perfectly together. They will reach the water at the same moment. As for where they hit, they both start out heading sideways (horizontally) at the same speed and they will continue to do travel at the same horizontal speed forever. So when they hit the water, they will have traveled the same horizontal distance and will hit the water in the same place. Basically, these two stones travel exactly the same path and hit simultaneously and in the same place. Had they been glued together to become a single larger stone, you would expect this combined stone to hit at one time and place. But the glue really wouldn't have mattered... the two stones would have traveled together even if the glue had broken part way through the trip, or before the two stones actually left your hand.

Problem 9:

A fast-moving softball is heading directly toward the pitcher after being hit by the batter. As it travels forward, the softball is experiencing

(A) a forward force that is equal to its momentum. [13.5% picked]
(B) a forward force that is equal to the square root of its momentum. [4.2% picked]
(C) a forward force that is equal to 1 divided by its momentum. [1.2% picked]
(D) zero forward force. [81.1% picked]

Answer: (D) zero forward force. [81.1% picked]

Why: The softball is coasting! It does not need a forward force to keep it moving forward; it does that for free. A forward force would make it accelerate forward, but it's simply moving forward at constant velocity and needs no forward acceleration. In reality, it's probably experiencing air resistance backward and is accelerating backward. Nonetheless, it will continue to move forward because of its own inertia and forward momentum.

Problem 10:

You are heading straight forward on a bicycle when you and the bicycle accidentally begin leaning toward the left. The bicycle responds automatically and prevents you from tipping over by

(A) developing a restoring force toward the right that pushes you back toward your upright stable equilibrium. [11.4% picked]
(B) using its gravitational potential energy to push you back toward your upright stable equilibrium. [1.4% picked]
(C) using its elastic potential energy to push you back toward your upright stable equilibrium. [1.6% picked]
(D) steering toward the left and driving its wheels under your overall center of gravity. [85.5% picked]

Answer: (D) steering toward the left and driving its wheels under your overall center of gravity. [85.5% picked]

Why: This question recalls the principal observation about a bicycle's marvelous stability in motion: it automatically steers so as to drive under you when you begin to fall over and return you to the upright unstable equilibrium. That unstable equilibrium occurs when your center of gravity lies directly above the line along which the wheels contact the pavement. Thus the bicycle steers so as to drive under your center of gravity and return you to your unstable equilibrium.

Problem 11:

You're playing volleyball with friends and the ball is passing you on its way toward the other team. It isn't going fast enough, so you reach out and push the ball toward the other team. The force you exert on the moving volleyball is 1 pound. The force that volleyball exerts on you in return is

(A) more than 1 pound. [0.8% picked]
(B) 1 pound. [94.8% picked]
(C) less than 1 pound, but more than zero. [3.6% picked]
(D) zero. [0.8% picked]

Answer: (B) 1 pound. [94.8% picked]

Why: All forces appear in equal but oppositely directed pairs. When you push the volleyball forward with a force of 1 pound, it pushes you backward with an equal but oppositely directed force of 1 pound. The fact that the volleyball is moving or accelerating makes absolutely no difference. You may find it relatively difficult to hit a forward-moving volleyball hard, but it will push back on you just as hard as you push on it.

Problem 12:

You are holding your friend's new television motionless in your hands while she fumbles in her purse for the keys to her apartment. Beads of sweat are dripping down your face. While you stand there like this, you are

(A) doing negative work on the television and it is doing (positive) work on you. [5.2% picked]
(B) doing (positive) work on the television. [9.5% picked]
(C) doing negative work on the television and it is doing negative work on you. [1.2% picked]
(D) doing zero work on the television. [84.1% picked]

Answer: (D) doing zero work on the television. [84.1% picked]

Why: Although you are getting tired, you are doing zero work on the television. That's because you're pushing it upward to support it, but it is not moving in the direction of your push. Your exhaustion reflects the fact that you waste your own energy while standing still. You are converting your food energy (specifically, chemical potential energy) into thermal energy, but you are not transferring any energy to the television by way of work.

Problem 13:

You are riding your bicycle on a smooth horizontal road when you drive over a bump. You are thrown briefly into the air. While your bicycle is not touching the road, you are

(A) not accelerating as you rise, but are accelerating downward as you descend back to the pavement. [6.8% picked]
(B) accelerating upward as you rise and downward as you descend back to the pavement. [7.6% picked]
(C) accelerating downward. [79.7% picked]
(D) not accelerating. [5.8% picked]

Answer: (C) accelerating downward. [79.7% picked]

Why: Once you are off the pavement, you and the bicycle are freely falling objects. The only force you are experiencing is your weight and you are accelerating straight downward.

Problem 14:

When water from a hose flows steadily and frictionlessly through a nozzle, its total energy

(A) remains unchanged, but its pressure drops and its speed increases. [77.2% picked]
(B) remains unchanged, but both its pressure and speed increase. [16.5% picked]
(C) decreases, but its pressure and speed both increase. [2.2% picked]
(D) increases as its speed increases. [4.0% picked]

Answer: (A) remains unchanged, but its pressure drops and its speed increases. [77.2% picked]

Why: As long as no energy is wasted via friction, the water's total energy per liter remains constant as it flows through plumbing. The nozzle merely causes a conversion of the water's pressure potential energy into kinetic energy. Basically, the water must speed up as it pass into and through the narrow opening of the nozzle and the resulting increased kinetic energy comes at the expense of pressure potential energy. Overall, the water's pressure drops and its speed increases.

Problem 15:

Your tiny Italian sports car collides with a city bus. Your car's velocity changes abruptly and you drive into a haystack. The bus barely accelerates at all. Fortunately, no one is hurt. The momentum your car received during the collision was

(A) zero, because no distance was traveled by your car during the collision itself. [0.8% picked]
(B) greater than the momentum the bus lost during the collision. [9.6% picked]
(C) equal to the momentum the bus lost during the collision. [86.3% picked]
(D) less than the momentum the bus lost during the collision, but more than zero. [3.2% picked]

Answer: (C) equal to the momentum the bus lost during the collision. [86.3% picked]

Why: Momentum is conserved, so if your car received a certain amount of momentum from the bus, then the bus lost that same amount of momentum. The transfer must be perfect so the momentum that one vehicle received is equal to the momentum that the other vehicle lost.

Problem 16:

You have a stiff, 1-liter bottle full of helium. You add some more helium to that bottle, but its temperature and volume remain unchanged. As a result of this addition, the bottle's pressure

(A) increases, its weight increases, but the buoyant force on it remains the same. [84.3% picked]
(B) and weight remain the same, but the buoyant force on it increases. [3.0% picked]
(C) remains the same, but its weight increases and the buoyant force on it increases. [4.2% picked]
(D) increases, its weight decreases and the buoyant force on it increases. [8.4% picked]

Answer: (A) increases, its weight increases, but the buoyant force on it remains the same. [84.3% picked]

Why: Both that helium's pressure and weight depend on how much helium (the number of helium atoms) there is in the bottle. Adding more helium increases both quantities. However, the buoyant force that the bottle experiences depends only on the bottle's volume and on the surrounding air, and those don't change when you add helium to the stiff bottle.

Problem 17:

A baby's sipping cup has a round bottom with a weight inside it. The cup always returns to upright, even after you tip it over. As you tip the cup, its center of gravity

(A) rises and its kinetic energy increases. [6.4% picked]
(B) descends and its total potential energy decreases. [9.9% picked]
(C) descends and its kinetic energy increases. [8.0% picked]
(D) rises and its total potential energy increases. [75.7% picked]

Answer: (D) rises and its total potential energy increases. [75.7% picked]

Why: For the sipping cup to be in a stable equilibrium, and therefore pop upright whenever it has been tipped, its total potential energy must increase whenever it tips. It will then accelerate back toward the upright equilibrium and recover from any tip. The only significant potential energy here is gravitational potential and the cup's gravitational potential energy is proportional to the altitude of its center of gravity -- the higher the center of gravity gets, the more gravitational potential energy it has. Therefore, the cup's center of gravity must be rising when it tips. When released, the cup accelerates back toward upright so as to lower both its center of gravity and its gravitational (and total) potential energy.

Problem 18:

You are riding a roller coaster and are upside-down at the very top of its loop-the-loop. You are traveling fast and you feel pressed into your seat. At this moment, you are accelerating

(A) backward. [3.4% picked]
(B) forward. [12.0% picked]
(C) downward. [67.3% picked]
(D) upward. [17.3% picked]

Answer: (C) downward. [67.3% picked]

Why: You are traveling in a circular motion at a relatively constant speed while at the top of the circular loop, so you are accelerating toward the center of that loop -- which is in the downward direction. Alternatively, you can note that while your speed doesn't change while you travel almost horizontally through the top of the loop, you go from heading slightly upward before reaching the exact top of the loop to slight downward after passing through the exact top. You are therefore bending your velocity in the downward direction -- a downward acceleration.

Problem 19:

You are designing a 2-foot high access ramp for a new library. One group wants a shorter ramp to save space. Another group wants to minimize the uphill force needed to push a wheelchair steadily up the ramp. If you double the ramp's length, the required uphill force will

(A) be reduced by half, but the product of that force times the ramp's length will remain unchanged. [76.7% picked]
(B) remain the same, but the product of that force times the ramp's travel time will be reduced by half. [3.2% picked]
(C) remain the same, but the product of that force times the ramp's length will be reduced by half. [9.0% picked]
(D) be reduced by half, but the product of that force times the ramp's travel time will remain unchanged. [11.0% picked]

Answer: (A) be reduced by half, but the product of that force times the ramp's length will remain unchanged. [76.7% picked]

Why: The work that needs to be done to lift the wheelchair up the 2-foot tall ramp depends only on the ramp's height, not on any other details of its shape. Therefore the product of the uphill force one must exert on the wheelchair times the distance the wheelchair moves uphill (in the direction of that force) doesn't depend on the ramp's details. If you double the length of the ramp, the uphill force must be reduced by half so as to keep the product of those two quantities unchanged.

Problem 20:

A heavy vase rests motionless on the floor. The vase is experiencing its weight downward and an equally strong force upward from the floor. We know that these two forces on the vase are equal in amount but oppositely directed because

(A) for every action, there is an equal but oppositely directed reaction. [6.4% picked]
(B) the vase has zero velocity. [2.4% picked]
(C) Newton's third law requires that forces always appear in equal but oppositely directed pairs. [30.3% picked]
(D) the vase is not accelerating so the two forces must sum to zero. [60.8% picked]

Answer: (D) the vase is not accelerating so the two forces must sum to zero. [60.8% picked]

Why: The two forces on the vase are not a Newton's third force pair. Instead they are two independent forces that happen to be equal but oppositely directed. They are the only two forces acting on the vase and so the vase accelerates in response to their sum -- which is the net force on the vase. If the two forces on the vase were not equal but oppositely directed, the vase would be accelerating. For example, if you drop the vase on the floor, the upward force that the floor will exert on the vase at impact will greatly exceed the vase's downward weight and the vase will accelerate upward. Hopefully the vase will survive that huge upward force!

Problem 21:

The air inside your friend's hot air balloon is always heated to the same temperature, regardless of the weather. On a hot day, her balloon can carry

(A) more weight than it can on a cold day. [12.7% picked]
(B) the same weight that it can on a cold day. [13.9% picked]
(C) less weight than it can on a cold day. [73.3% picked]
(D) a complete line of fashion accessories. [0.2% picked]

Answer: (C) less weight than it can on a cold day. [73.3% picked]

Why: Since the balloon's temperature is always the same, the balloon's weight never varies. However, when the outside air gets hotter and less dense, the buoyant force that the balloon experiences decreases. The balloon is still the same size as before, but it is now displacing lighter air so it experiences less buoyant force. Thus, on the hot day, the balloon's downward weight is the same as always, but the upward buoyant force acting on it is less than on a cold day, so the balloon can't carry as much as on a cold day.

Problem 22:

When you drop a bouncy ball onto the sandy beach, the normally "lively" ball barely bounces at all. The non-lively sand dominates this bouncing process because it receives most of the collision energy and then wastes that energy as thermal energy. The sand receives most of the collision energy because

(A) while the ball and sand push equally hard on one another, the ball dents more during the collision. [3.8% picked]
(B) the ball pushes harder on the sand than the sand pushes on the ball. [2.0% picked]
(C) while the ball and sand push equally hard on one another, the sand dents more during the collision. [93.6% picked]
(D) the sand pushes harder on the ball than the ball pushes on the sand. [0.6% picked]

Answer: (C) while the ball and sand push equally hard on one another, the sand dents more during the collision. [93.6% picked]

Why: The force that the ball exerts on the sand must be equal but opposite to the force that the sand exerts on the ball -- they are a Newton's third law pair of forces. However, the sand dents more than the ball and has more work done on it during the impact. The sand is therefore responsible for most of the collision energy and most of the bouncing process. Unfortunately, the sand wastes all the work done on it and it doesn't contribute anything to the bounce. The bouncy ball just flops onto the sand.

Problem 23:

You are swinging a bucket full of water around you in a big horizontal circle at a constant speed [added words]. You are at the center of its circular path. The net force on the bucket points directly

(A) along the bucket's velocity (along its direction of travel). [22.0% picked]
(B) downward. [7.3% picked]
(C) away from you. [12.1% picked]
(D) toward you. [58.6% picked]

Answer: (D) toward you. [58.6% picked]

Why: The bucket is moving in a horizontal circle at a steady pace. It is therefore accelerating toward the center of the circle. All of the forces that act on the bucket must sum to a simple centripetal force. While it is true that the bucket is experiencing its weight, it must also be experiencing an equal amount of upward force from you, otherwise it wouldn't travel in a horizontal circle. Instead, it would corkscrew into the ground.

Problem 24:

You drop a bouncy ball on a granite floor and the ball bounces back almost to its original height. During the bounce, the ball and the floor exchanged

(A) considerable energy but almost zero momentum. [11.7% picked]
(B) considerable energy and momentum. [40.2% picked]
(C) considerable momentum but almost zero energy. [33.8% picked]
(D) almost zero momentum and almost zero energy. [14.3% picked]

Answer: (C) considerable momentum but almost zero energy. [33.8% picked]

Why: The ball gave the floor a big impulse (downward force times a certain amount of time), but the ball did not do work on the floor (which did not move while the ball was pushing on it). The end result is that the ball completely reverses its momentum (a huge overall change in momentum), but retains pretty much all of its energy.

Problem 25:

In certain situations, the flow of a liquid breaks up into turbulent swirls and eddies. This transition to turbulence depends in part on the liquid's density (its mass-per-volume) and its viscosity (the measure of its "thickness" or "syrupiness.") The liquid is most likely to become turbulent when it is moving

(A) slowly, has a low density, and a low viscosity. [5.8% picked]
(B) slowly, has a high density, and a high viscosity. [2.6% picked]
(C) fast, has a low density, and a high viscosity. [14.5% picked]
(D) fast, has a high density, and a low viscosity. [77.1% picked]

Answer: (D) fast, has a high density, and a low viscosity. [77.1% picked]

Why: Turbulence occurs when inertia dominates a fluid's flow. Things that favor inertia include: high speeds, high densities, low viscosities, and big obstacles. The first two items increase momentum (velocity times mass) while the third means that adjacent portions of fluid don't "talk" to each other well (weak viscous forces). Finally big obstacles give inertia plenty of opportunity to rip a fluid apart into separate flows.

PART II: SHORT ANSWER QUESTIONS

Please give a brief answer in the space provided. Part II is worth 33% of the grade on the midterm examination.

Problem 1:

Some pranksters have "decorated" the trees in front of your neighbor's house with toilet paper. It's much easier to throw the paper into the trees than to remove it, so you are being a good friend and helping him remove it. You are using a garden hose and a nozzle to spray water at the paper hanging in the trees. Unfortunately, the paper is high up and the spray of water is having trouble reaching it.

(A) You find that the spray goes highest when you adjust the nozzle so that the amount of water it releases is relatively small. Why does letting much more water flow through the hose reduce the maximum height of the spray?

Answer: As more water flows through the hose, it moves faster and wastes a larger fraction of its energy as thermal energy through friction (or viscous effects).

Why: As water flows through the hose, it rubs against the walls and wastes some of its ordered energy as thermal energy. The faster the water travels through the hose, the harder it rubs and the more of its energy it wastes. By permitting lots of water to flow through the hose, you cause it to waste a larger fraction of its total energy and it can't rise as high anymore.

(B) The water jet rises 60 feet above the nozzle, which isn't quite high enough. You climb a 10-foot stepladder, but now the jet rises only 50 feet above the nozzle and the water peaks at the same altitude as before. Why?

Answer: The water has only enough total energy per liter to rise to the 60 foot height.

Why: How the water travels upward to its peak height simply doesn't matter... it only has enough energy to go 60 feet above the original nozzle height. By climbing on the ladder, you make the water converts some of its pressure potential energy into gravitational potential energy before it passes through the nozzle. But after the water leaves the nozzle, it travels upward just as before.

(C) You borrow an electric water pump from another neighbor and increase the pressure of water in the hose. The water jet now rises much higher and you succeed in clearing the paper from the trees. Why did this water pump have to be plugged into the electric socket, instead of using the water passing through the pump to power itself?

Answer: The pump is adding to the water's total energy, so it must obtain that energy from somewhere else (and not the water itself).

Why: A water pump adds energy to the water passing through it and therefore cannot use this same water to power itself. The water has only so much energy per liter and trying to use that energy to operate a pump is self-defeating. It would be like using your own money to pay yourself -- you don't get any richer.

(D) When the rising jet of water hits the paper, it pushes the paper away from it. In free flight, the water is at atmospheric pressure. What is the water's pressure where it hits the paper?

Answer: The water's pressure rises (above atmospheric pressure).

Why: When the water slows down upon hitting the paper, it converts some of its kinetic energy into pressure potential energy. Its pressure rises above atmospheric pressure and this increased pressure is what pushes the paper away from the stream of water.

Problem 2:

A tall flagpole sits motionless and perfectly upright on a calm summer day. Suddenly, a flying Frisbee hits the flagpole hard near its top and it begins swaying back and forth rhythmically--toward you and away from you. Let's neglect air resistance or any forms of friction, so the flagpole will sway like this forever.

(A) When during its swaying is the flagpole accelerating toward you the fastest?

Answer: It is accelerating toward you fastest when it is farthest away from you.

Why: Accelerations are caused by forces and the force experienced by the springy flagpole is strongest when it is bent the farthest. The moment when the flagpole is experiencing the strongest force toward you is when it is bent the farthest away from you. That is also the moment of maximum acceleration toward you.

(B) When during its swaying is the flagpole moving toward you the fastest? (i.e. has its maximum velocity toward you.)

Answer: It is moving toward you fastest when it is passing through equilibrium on its way toward you.

Why: When it's bend, the flagpole always accelerates toward the equilibrium point. While heading toward you from its most distant point, it accelerates forward and speeds up until it reaches equilibrium. After passing through equilibrium, it accelerates backward and slows down. Since it reaches its highest speed just as it stops accelerating forward, its velocity toward you is greatest just as it passes through the equilibrium point on its way toward you.

(C) When during its swaying does the flagpole have its lowest amount of total potential energy?

Answer: It has the lowest total potential energy when it is at equilibrium.

Why: The flagpole is moving about its stable equilibrium. Since it accelerates toward the lowest total potential energy, that equilibrium point is also the point of lowest total potential energy. In fact, this minimum of total potential energy is why that point is a stable equilibrium point.

(D) When during its swaying does the flagpole have its largest amount of momentum toward you?

Answer: It has its largest momentum toward you when it is passing through equilibrium on its its way toward you.

Why: Since its momentum is proportional to its velocity, it reaches maximum momentum toward you at the same moment it reaches maximum velocity toward you.

Problem 3:

You are trying to develop a computer-guided baseball glove. This glove should catch the ball on its own after you throw the glove into the air. Unfortunately, the conserved quantities of physics are preventing the glove from working well. [Assume the air exerts no forces or torques on the glove and that the glove remains a single, intact object.]

(A) You find that the computerized glove cannot rise any higher than a similarly thrown ordinary glove. Which conserved physical quantity prevents the glove from going higher and why can't the glove get more of it while in the air?

Answer: Energy is the conserved quantity and nothing does work on the glove to give it more.

Why: Since nothing exerts a force on the falling glove, the glove has no work done on it and its total energy remains constant in flight. It simply doesn't have enough energy to go higher than a certain altitude and nothing the computer does can change that limit.

(B) You find that the computerized glove follows the same arcing path through space as a similarly thrown ordinary glove. Which conserved physical quantity is determining the glove's path through space and, neglecting the effects of gravity, why can't the glove get more of it while in the air?

Answer: Momentum is the conserved quantity and nothing (other than gravity) does an impulse on the glove to give it more.

Why: Since nothing (besides gravity) pushes on the glove, the glove has no impulse done on it and its total momentum changes only in the manner caused by gravity. It follows the path of a freely falling object, no matter what the computer tries to do.

(C) You find that if the computerized glove leaves the player's hand without any angular velocity (i.e. not spinning in any way), that it behaves just like a similarly thrown ordinary glove: neither one can start itself spinning. Which conserved physical quantity is determining the glove's spin and why can't the glove get more of it while in the air?

Answer: Angular momentum is the conserved quantity and nothing does an angular impulse on the glove to give it more.

Why: Since nothing exerts a torque on the glove, the glove has no angular impulse done on it and its total angular momentum remains at zero. The computer can't start it spinning because there is no angular momentum to be had.

(D) Finally a small victory: if the computerized glove is thrown so that it is spinning, it can alter its angular velocity even though nothing is touch it. What does it do to change its angular velocity?

Answer: The glove changes its angular mass (or rotational mass, or moment-of-inertia).

Why: Although the glove's angular momentum can't change, its angular momentum is equal to the product of its angular mass times its angular velocity. By altering its shape and therefore its angular mass, the glove can cause its angular velocity to change in the inverse manner. For example, by reducing its angular mass, the glove can make itself spin faster (larger angular velocity).