Due Monday, February 5, In Class
Please Answer Each Question As Briefly As Possible
You May Work Together, But Write Up Your Answers Separately
Question 1: (Case 14 from Chapter 1)
Revolving doors are popular in northern hotels and office buildings as a way to prevent cold outside air from blowing directly into the lobby. Most revolving doors have 4 panels arranged in a cross, as viewed from above. You step in between two panels of the door and push on the panel in front of you. The revolving door begins to rotate and once you reach the inside of the building, you step out into the lobby.
a. It is much easier to make the revolving door rotate by pushing on the panel far from the central pivot than it is by pushing near the pivot. Why?
Answer: Pushing the door far from the pivot produces a larger torque on the door than pushing it near the pivot.
Why: Increasing the lever arm between the pivot and the point at which you push the door increases the torque on the door.
b. As you push on the door, it begins to turn more and more quickly. What is your pushing doing to the door?
Answer: You are causing the door to undergo angular acceleration (or your pushing is exerting a torque on the door).
Why: The door is turning faster and faster because you are exerting a torque on it and its undergoing angular acceleration.
c. One of the dangers of revolving doors is being hit by the panel behind you as you step out of the door. The door tends to keep on turning after you stop pushing on it and it can bump you if you're not careful. Why does it keep on turning after you stop pushing?
Answer: Its rotational inertia (or angular momentum) keeps it turning.
Why: Objects that are spinning tend to continue spinning unless acted on by a torque.
d. What eventually stops the revolving door when no one uses it for a minute of two?
Answer: Sliding friction (probably in the door's pivot) and/or air resistance exert torques on the door and gradually bring it to a stop.
Why: Just as friction and air resistance stop objects that are undergoing translational motion, so they stop object that are undergoing rotation motion.
e. The people who built the revolving door want it to be easy to start and stop. They had to be most careful to minimize the mass of which edge of each door panel: the upper edge, lower edge, inside edge (nearest the pivot), or the outside edge (farthest from the pivot)?
Answer: They should minimize the mass at the outside edge of the door.
Why: The door's moment of inertia is the measure of its rotational inertia. Since the moment of inertia is most sensitive to mass located far from the pivot, the door should have as little mass on its outside edge as possible. That way its moment of inertia will remain small and it will be easy to start and stop its rotation.
Question 2: (Case 16 from Chapter 1)
You're a pilot for the Navy. For your airplane to be able to lift itself off the ground, it must be traveling forward with a speed of 130 miles-per-hour. At this take-off speed your airplane will have about 50,000,000 newton-meters (or 50,000,000 J) of kinetic energy.
a. During takeoff, your airplane's jet engine exerts a force of 250,000 N in the backward direction on the air leaving the engine. What force does that same air exert on the airplane (specify the amount and the direction of force)?
Answer: 250,000 newtons forward
Why: For every force, there is an equal but oppositely-directed reaction force. As the airplane engine pushes backward on the air, the air pushes forward on the engine. Since the engine exerts 250,000 newtons of force on the air, the air exerts 250,000 newtons of force on the engine.
b. The force exerted by the air on the airplane causes it to accelerate down the runway. How long must the runway be for the airplane to reach its take-off speed?
Answer: 200 meters
Why: The airplane will be able to take off when it is reaches 130 miles-per-hour, at which time it will have 50,000,000 joules of kinetic energy. To give the airplane 50,000,000 joules of kinetic energy, something must do that much work on it (and the airplane had better not do much work on anything else). 50,000,000 joules is 50,000,000 newton-meters, so that it can be obtained by exerting 1 newton from 50,000,000 meters or 50,000,000 newtons for 1 meter or something (more reasonable) in between. In this case, the force available from the engine is 250,000 newtons so 200 meters is required (250,000 newtons times 200 meters is 50,000,000 newton-meters). The airplane will reach take off energy and speed after 200 meters of accelerating forward in response to the forces from its own engines.
c. An aircraft carrier runway is only about 100 m long. As your answer to b indicates, this distance is not enough for the airplane to reach take-off speed on its own. The aircraft carrier must assist the airplane by exerting an extra force on it. The aircraft carrier uses a steam-powered catapult to help push the airplane forward. How much additional force should the catapult exert on the airplane to bring the airplane to take-off speed at the end of the 100 m runway?
Answer: 250,000 additional newtons in the forward direction.
Why: To reach take off energy and speed after only 100 meters will require a force of 500,000 newtons (500,000 newtons times 100 meters is 50,000,000 newton-meters). Since 250,000 newtons is available from the engines themselves, the catapult must exert an additional 250,000 newton in the forward direction. If the catapult were required to bring the airplane to take off speed in even less distance, still more force would be required. Below a certain length, the force would have to be so high that it would break the airplane. That explains why airplanes cannot be shot off the carrier deck with explosives, although sometimes rocket packs are used.
d. During an aircraft carrier take-off, the airplane and the catapult exert forces on one another. Which of these two objects does work on the other and which object transfers some of its energy to the other?
Answer: (1) The catapult does work on the airplane and (2) the catapult transfers energy to the airplane.
Why: The catapult exerts a forward force on the airplane and the airplane moves forward. Therefore, the catapult does work on the airplane. In the process, the catapult transfers energy to the airplane.
e. During an aircraft carrier landing, the airplane hooks onto a cable that slows the airplane to a stop. The airplane and cable exert forces on one another. Which of these two object does work on which and which object transfers some of its energy to the other?
Answer: (1) The airplane does work on the cable and (2) the airplane transfers energy to the cable.
Why: The airplane exerts a forward force on the cable and the cable moves forward. Therefore, the airplane does work on the cable. In the process, the airplane transfers energy to the cable. The cable's movement allows the airplane to transfer its energy gracefully, over a reasonable length of time. If the cable were too stiff and the forces involved were too large, the transfer would occur very, very quickly and the pilot would be thrown though the windscreen of the airplane.
Question 3: (Case 19 from Chapter 1)
You have recently taken up track and field as a way to keep in shape. You soon begin to notice how simple physical laws appear in many of the events.
a. You notice that great sprinters have extremely strong legs. Why is it so important that a sprinter be able to push back very hard on the starting blocks at the beginning of a race?
Answer: When a sprinter pushes back hard on the blocks, the blocks push forward hard on the sprinter. The sprinter then accelerates forward rapidly and wins the race.
Why: To accelerate quickly in a race, the sprinter needs as large a forward force as possible. The sprinter obtains this force by pushing back hard on the starting blocks. The blocks respond with an equal and opposite force and push the sprinter forward.
b. You find that throwing a heavy metal shot is far more difficult than throwing a baseball. Weight isn't the whole problem. Even if you try to throw the shot horizontally or downward, so that weight is not an issue, you have great difficulty getting the shot moving quickly. Why?
Answer: The shot has a large mass.
Why: The larger an object's mass, the harder it is to accelerate. The shot has a large mass, so it's hard to accelerate. Even when gravity is completely out of the picture, you just can't get the shot moving quickly.
c. As you land on the soft, foam pad beneath the pole vault, you realize that its job is to bring you to rest by accelerating you upward gradually with only modest support forces. If there were no pad there, only concrete, what would the acceleration and support forces be like during your landing?
Answer: The acceleration would be much more rapid and the support forces that cause the acceleration would be much larger.
Why: The pad's purpose is to slow the acceleration that occurs when the pole vaulter reaches the ground. The pole vaulter is definitely going to come to a stop, but it's more comfortable to stop slowly with modest forces than to stop quickly with large forces.
d. You cross the finish line at the end of a race. The net force on your body points in what direction as you slow down?
Answer: Backward (away from your direction of motion).
Why: When you are stopping, you are decelerating. That's just a special word for accelerating in the direction opposite your velocity. You are heading forward so to slow down you accelerate backward. The net force on you must thus be backward.
e. In the long jump, you run rapidly down a path and then leap into the air. You find that the best distance comes from pushing yourself upward rather than forward during the leap. Why is it so important to have a large upward component of velocity at the start of the leap?
Answer: You already have a large forward component of velocity but you need to stay aloft as long as possible. By jumped upward, you maximize your air time and allow yourself to drift a long distance forward before you return to the ground.
Why: In a standing broad jump, you need to jump forward as well as up. But when you are running forward before you even begin to jump, you're main task is to propel yourself upward as hard as possible. That way you'll stay in the air a long time while the forward component of velocity that you already had will carry you a long way forward.
Question 4: (Case 22 from Chapter 1)
Local fairs and amusement parks usually offer games in which you can win a large prize by performing a seemingly easy task. In many cases, these tasks are made surprisingly difficult by simple physical principles and few people receive prizes. Here are several of those games.
a. A pitching game requires that you knock over three milk bottles with a baseball. The bottles are filled with sand. Why does filling the bottles with sand make them so hard to knock over with the baseball?
Answer: Adding sand increases their masses so that they undergo much smaller accelerations when the ball hits them.
Why: By increasing the masses of the bottles, the operator has made it hard to accelerate the bottles. The ball rebounds from the bottles as it would from a wall. The bottles may wobble back and forth as they get rid of the momentum and energy transferred to them by the ball. But they're unlikely to rock far enough to tip over.
b. A tossing game requires that you throw a coin forward and have it come to a stop on a smooth glass plate. Why doesn't the coin stop when it hits the smooth glass plate?
Answer: The force of sliding friction slowing the coins is too small and it acts for too short a time to stop the coin before it slides off the plate.
Why: For the coin to stop, something must remove its forward momentum. The only horizontal force on the coin is sliding friction from the smooth plate. This force is so small and acts for such a short time, that the coin still has forward momentum when it reaches the end of the plate. It slides off the plate and onto the floor.
c. Another game requires that you knock over a wooden peg with a ball hanging from a string. The string is suspended from a point directly above the peg. To win, you must swing the ball past the peg and have the ball knock over the peg on its return swing. This feat can't be done. The ball keeps circling around and around the peg. The ball's suspension prevents it from experiencing any torques about an axis that includes the peg and the ball keeps swinging around and around the peg. What law of motion keeps it swinging around that peg?
Answer: Either Newton's first law of rotational motion or angular momentum.
Why: The ball is rotating around the peg so it has angular momentum. It can't reach the peg because that would require it to pass essentially through the axis of rotation. It would then have zero angular momentum. So it just keeps rotating steadily about the peg and never touches it.
d. Still another tossing game requires that you throw a basketball into a shallow basket that is tipped toward you. Whenever you throw the ball into the basket, it bounces back, rolls out of the basket, and falls onto the floor. What conserved physical quantity is the basketball unable to get rid of in time to remain in the basket?
Answer: Energy
Why: The ball can't get rid of its energy so it bounces right back out of the basket. It does manage to exchange lots of momentum with the basket (it even reverses directions). But its energy stays with it and it leaps back out.
e. A pitching game measures the speed at which you can throw a baseball. While many people can reach 80 km/h (50 mph) with a pitch, almost no one can throw a ball twice that fast. Use the concept of kinetic energy to explain why this is not surprising.
Answer: Doubling the speed of the ball is equivalent to quadrupoling the ball's kinetic energy.
Why: Professional pitchers are much stronger than the average person because they must give the baseball as much as four times as much energy as an average person can provide. Moreover, the pitcher has less time to do the work needed to reach the higher speed, so it takes even more strength.
Question 5: (Case 2 from Chapter 11)
A Van der Graaf generator is an electrostatic device that uses a moving non-conductive belt to carry electric charge into a hollow metal sphere. This sphere is insulated from the ground and can accumulate charge until enormous voltages are reached. Small Van der Graaf generators are exciting novelties while large ones are used in research and industry.
a. A typical Van der Graaf generator uses a rubber belt to carry negatively charged electrons from its base to the sphere. As the belt passes through the sphere, it is touched by a metal brush. Electrons leave the belt and flow through a wire to the surrounding sphere. Why do the electrons flow onto the sphere rather than staying on the belt?
Answer: They repel one another and maximize their separations by flowing outward to the surface of the ball.
Why: Electrons will always flow so as to maximize their separations. The charges inside the sphere on the belt are relatively close to one another and to the charges on the sphere around them. By flowing to the surface of the sphere, they get farther from the other charges, on average. The charges on the sphere rearrange to make room for them.
b. As negative charge builds up on the sphere, the belt motor begins to strain. Why does it become harder and harder for the motor to move the belt?
Answer: The charge on the belt is negative and the charge on the sphere is negative, so the motor must push the two together (the motor does work on the charges as it pushes them up into the sphere.
Why: As negative charge begins to accumulate on the sphere, it repels the charge that's approaching it on the belt. The motor must work harder and harder to add each additional charge.
c. The amount of charge that can build up on the sphere depends on many things. For one, the sphere must be very smooth. Why is it important that there be no sharp points on the sphere?
Answer: If there are sharp points on the sphere, charge will rush onto those points and will then leave the sphere by going into the air (or a corona discharge will occur).
Why: When there is a sharp point on the sphere, charge that goes onto that point can get particularly far away from the rest of the charge on the sphere. But the charge that does go onto the point is tightly packed and experiences such strong repulsions that some of it leaps off the tip and into the air. This leaving charge reduces the total charge on the sphere.
d. As you move your hand close to the negatively charged sphere, your hand becomes positively charged and a spark may leap from the sphere to your hand. How does you hand acquire a positive charge?
Answer: The negative charge on the sphere attracts positive charge onto your nearby hand and repels the negative charge off of your hand (and perhaps into the ground).
Why: The large negative charge on the sphere tends to redistribute charges on nearby objects. The negative and positive charges in your body rearrange, with positive charge shifting toward the sphere and negative charge shifting away from the sphere.
e. If you insulate yourself from the ground and put your hand on the sphere, negative charge will accumulate on you as well as the sphere. Explain why your hair stands up.
Answer: The negative charge migrates onto all of your hairs, which then repel one another.
Why: Negatively charged objects, such as hairs with negative charge on them, repel one another. When the forces become larger than the weights of the hairs, they stick up in order to get as far apart as possible.