## Problem Set 1 - Answers

• Due Monday, September 16, 1996, in class
• You may discuss the problems with one another but you must write them up separately and in your own words.
• There are 5 problems containing a total of 25 parts. Each part will be worth 4 points.

Problem 1: Chapter 1, Case 6a-d (Pg. 73)

Two cars are driving at 88 km/h (55 mph) toward an intersection. A 1500 kg sedan is heading east and a 700 kg subcompact is heading north. The subcompact has a green light and drives into the intersection. The tired driver of the sedan runs the red light and crashes directly into the driver's side of the subcompact. The subcompact experiences a sudden, strong force to its right (toward the east).

a. Which car experiences the largest force due to the collision?
Answer: The forces on the two cars have equal amounts (equal magnitudes).
Why: In accordance with Newton's third law, the two cars exert equal but oppositely directed forces on one another.

b. Which car experiences the largest acceleration due to the collision?
Answer: The subcompact accelerates more than than the sedan.
Why: Since the cars experience forces of equal magnitudes and the subcompact has the smaller mass, the subcompact experiences the larger acceleration.

c. Which car is most likely to be thrown off the roadway because of the collision?
Why: The subcompact accelerates more than the sedan and experiences the greater change in its velocity. It is thus more likely to go in a direction it hadn't intended and drive off the roadway.

d. If the force on the subcompact is directly toward that car's right (directly toward the east), what happens to the north/south component of the subcompact's velocity?
Answer: The north/south component of the subcompact's velocity doesn't change.
Why: The eastward force causes only an eastward acceleration. Since there is no north/south component of acceleration, that portion of the car's velocity doesn't change. It continues to coast northward at its original rate after the collision, but it acquires an eastward component of velocity as well.

Problem 2: Chapter 1, Case 11a-f (Pg. 74)

An escalator is essentially a moving staircase. The individual steps are supported by metal tracks that run on either side of the escalator. These steps follow one another in a complete loop, driven by an electric motor. When you step onto an escalator at the ground floor, it soon begins to carry you upward and forward at a constant velocity toward the second floor.

a. While you're moving toward the second floor at a constant velocity, what is the net force exerted on you by all outside forces (specify the amount and the direction of the net force)?
Why: If you are at constant velocity, and thus not accelerating, the net force on you must be zero.

b. You know that gravity gives you a weight in the downward direction. What force does the escalator exert on you as you move toward the second floor at a constant velocity (specify the amount and the direction of the force)?
Answer: A force equal in amount (magnitude) to your weight, but in the upward direction.
Why: Since the net force on you is zero and the only two forces you're experiencing are (1) your weight downward and (2) a support force from the escalator upward, those two forces must exactly cancel. Thus the support force must have exactly the same magnitude as your weight, but they must point in opposite directions.

c. Is the escalator doing work on you as you move toward the second floor?
Why: The escalator is pushing upward on you and you are moving (at least partially) upward. Whenever something exerts a force on you and you move at least partially in the direction of that force, that something is doing work on you.

d. As you first step onto the escalator, you begin to accelerate toward the second floor. Is the net force exerted on you by all outside forces the same as in a?
Why: Since you are accelerating, the net force on you must not be zero.

e. If the rapidly moving escalator suddenly stopped moving, you would be thrown forward and might even fall over. What causes you to be thrown forward?
Why: If the escalator were to stop suddenly, it would be accelerating backward at an enormous rate. Without some external force to stop you, you'll continue forward at constant velocity. As a result, you will appear to be "thrown" forward, when in fact you are simply obeying the laws of motion and the concept of inertia. No force "throws" you forward--just inertia.

f. You have more energy when you reach the second floor than you had on the first floor. Why aren't you moving faster as a result?
Why: The escalator does work on you as you rise upward and this energy becomes stored in the forces between you and the earth as gravitational potential energy. This energy isn't visible, but it can be released by descending back down to the first floor.

Problem 3: Chapter 1, Case 14a-e (Pg. 74)

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: The farther from the pivot you push, the more torque you produce with your force.
Why: It takes a torque to cause the door to begin rotating. The force you exert on the door produces such a torque, but the amount of that torque depends on where you push. By pushing far from the pivot, you select a large lever arm and produce a large torque. As a result, the door undergoes relatively rapid angular acceleration.

b. As you push on the door, it begins to turn more and more quickly. What is your pushing doing to the door?
Why: When a door (or any rotating object) changes its rate of rotation, it is undergoing angular acceleration. That is what you are causing by pushing on the door.

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, and it can bump you if you're not careful. Why does it keep on turning after you stop pushing?
Answer: Its angular momentum (or equivalently rotational inertia, or even simply inertia).
Why: When nothing exerts a torque on the door (or at least the torques are very small), the door will coast around and around because of rotational inertia. It has angular momentum and is unable to transfer that angular momentum elsewhere, so around it goes.

d. What eventually stops the revolving door when no one uses it for a minute or two?
Why: The door isn't really free of all torques. As it spins, it rubs against things and also pushes on the air. Friction and air resistance twist the door in the direction opposite its rotation and it gradually slows to a stop.

e. The people who built the revolving door wanted 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 outside edge (farthest from the pivot)?
Why: The ease with which you can start or stop the door from turning depends on its moment of inertia--the smaller this moment of inertia, the easier it is to start or stop the door. An object's moment of inertia depends both on its mass and on the location of that mass. The farther the mass is from the axis of rotation, the more it contributes to the moment of inertia. That's why minimizing the mass far from the door's pivot (from its outer edge) is the best way to minimize its moment of inertia.

Problem 4: Chapter 1, Case 17a-e (Pg. 75)

You're seated in a crowded bus that has stopped at a bus stop. Two people board the bus, one wearing rubber-soled shoes and one wearing inline skates (roller skates - i.e., no friction). They both have to stand in the middle of the aisle and neither one holds onto anything.

a. The bus driver is new and the bus lurches forward from the bus stop. Which way does the person wearing inline skates move in relationship to the bus?
Answer: The skater moves backward relative to the bus (but stays in one place relative to the bus stop).
Why: As the bus accelerates away from the bus stop, the ground pushes it forward. To remain with the bus, the skater also needs a forward force but the skates prevent the bus from exerting such a forward force on the skater. The wheels of the skates just spin. So instead of accelerating forward, the skater remains in place in front of the bus stop and the bus drives out from under the skater. To the passengers on the bus, the skater rolls backward through the isle and hits the back of the bus.

b. The bus is accelerating forward as it pulls away from the bus stop. Why doesn't the person wearing inline skates accelerate with the bus?
Answer: The bus can't exert the forward force on the skater that the skater needs to accelerate forward with the bus.
Why: The skate wheels turn as the bus accelerates forward, so that the skater doesn't experience the forward frictional force that a person in normal shoes would need to accelerate with the bus.

c. The person wearing rubber-soled shoes remains in place as the bus starts moving. What provides the force that causes that person to accelerate with the bus?
Why: Static friction between the bus floor and the person's rubber-soled shoes pushes the person forward and the bus floor backward. As a result, the person accelerates with the bus and the bus accelerates somewhat more slowly than it would have had the person not been in the bus.

d. Once the bus has reached a constant velocity (it's traveling along a straight road at a steady pace), the person wearing inline skates is able to stand comfortably in the aisle without rolling anywhere. What is the net force on the person wearing skates?
Why: Since the bus and the skater are both moving at constant velocity, neither one is accelerating. The skater (like the bus) is experiencing zero net force.

e. The driver accidentally runs into a curb and the bus stops so abruptly that everything slides forward, including the person wearing rubber-soled shoes. In stopping quickly, the bus experiences an enormous backward acceleration. Why doesn't the person wearing rubber-soled shoes accelerate backward with the bus?
Answer: The force of static friction can't provide enough backward force to slow the person with the bus.
Why: When the bus stops gradually, static friction between the shoes and the bus floor exert enough backward force on the person wearing the shoes to make that person accelerate backward with the bus. However, when the bus stops suddenly, the force required to accelerate the person backward with the bus is greater than the maximum force that static friction can supply. The shoes begin to slide across the floor and the person skids forward in the bus.

Problem 5: Chapter 2, Case 5a-e (Pg. 119)

A trampoline has an elastic surface, supported at the edges by springs or elastic bands so that it's 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's behaving like a spring that's distorted away from its equilibrium shape. The distorted trampoline surface is exerting a restoring force on you in which direction?
Why: Like any spring that has been distorted downward from its equilibrium shape, the trampoline exerts a force upward 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?