Physics 105 - How Things Work - Fall, 2000

Problem Set #2 – Laws of Motion, Part 2 - Solutions

Your friends have dared you to try the “Test-Your-Strength” game at the county fair. Your task is to swing a huge mallet over your head and hit one end of a small lever with it. The lever will then propel a metal weight resting on its other end up a vertical track toward a bell high overhead. If you are strong enough and you hit the lever hard enough with the mallet, the weight will travel all the way to the top of the track and ring the bell. You’ll win a huge stuffed animal and everyone’s admiration. Being such a daredevil and showoff, you can’t resist. You pay the cashier and pick up the mallet.

1. The mallet is heavier than you had expected and you find that you can't swing it well. To make it easier to swing, you "choke up" -- that is, instead of gripping the end of the mallet's handle, you grip the handle about midway between its end and the head of the mallet. You find that you can then swing the mallet more easily. The mass of the mallet didn't change, so why is it easier to swing once you have choked up?

Answer: Moving most of the mallet’s mass closer to the pivot (your arms and shoulders) reduces its moment of inertia (or, equivalently, reducing the lever arm between the mallet’s head and the pivot around which mallet head is swinging means that it exerts less torque to oppose your swing).

Why: Objects that have their mass distributed far from the pivot about which they turn are difficult angular accelerate. It’s must easier to twist a ball back and forth than a long stick with the same mass as the ball. By choking up on the mallet, you move its mass closer to the pivot and make it easier to angular accelerate. This reduction in the mallet’s resistance to angular acceleration is characterized by its reduced moment of inertia.

2. When the mallet hits the lever, it exerts an enormous downward force on the lever. This downward force is far greater than the mallet's weight, even with your hands attached to it. How is the mallet able to exert this enormous downward force?

Answer: The lever acts to stop the rapidly descending mallet from passing through its surface by making the mallet accelerate upward extremely rapidly. The lever exerts a huge upward force on the mallet, causing the mallet to accelerate upward quickly and stop descending. The mallet responds to this upward force by exerting a huge downward force on the lever.

Why: Gravity has little to do with the force the mallet exerts on the lever. The lever is simply acting to keep from being penetrated and it does this by pushing up hard on the approaching mallet. The mallet slows to a stop (it decelerates) and, as required by Newton’s third law, the mallet pushes back on the lever with an equal but oppositely directed force. It is this enormous downward force of the mallet on the lever that ultimately tosses the weight up at the bell.

3. When the mallet hits the lever, the lever begins to rotate and begins to push the weight upward. During this process, what torques are acting on the lever and why does the lever undergo angular acceleration in a clockwise manner (according to the figure) rather than counterclockwise?

Answer: The mallet exerts a clockwise torque on the lever and the weight (which the lever is pushing up the track) exerts a counterclockwise torque on the lever. The massive mallet exerts the larger torque and so the lever undergoes clockwise angular acceleration.

Why: The mallet pushes down on the right side of the lever, thus exerting a large clockwise torque about its pivot. But the weight also exerts a torque on the lever. It does this because as the lever rotates clockwise, the weight’s inertia tends to keep it in place. The lever has to “swat” the weight up and out of the way by exerting a substantial upward force on it. The weight pushes back on the lever and exerts a counterclockwise torque on it about its pivot. Ignoring gravity, the weight only exerts this counterclockwise torque on the lever if the lever continues to undergo clockwise angular acceleration—the lever has to keep chasing the weight up the track. The presence of gravity complicates things by allowing the weight to exert a counterclockwise torque on the lever even when there is no angular acceleration, but since the question assumes the there is clockwise angular acceleration, the main source of counterclockwise torque is the weight’s inertia rather than the weight’s weight.

4. The lever always pushes on the weight for a certain distance as the lever turns. After that distance, the weight loses contact with the lever and must coast up the track on its own, against the force of gravity. The main determinant of how high the weight will go is the angular acceleration of the lever: the faster the lever undergoes angular acceleration, the higher the weight will go. That's because more rapid angular acceleration leads to a larger transfer of energy to the weight, which in turn lets it travel farther up the track before all of its kinetic energy has become gravitational potential energy. In terms of forces, distances, and work, explain briefly (two good sentences should be enough) why a more rapid angular acceleration of the lever leads to a larger transfer of energy to the weight.

Answer: The faster the lever angular accelerates, the faster the weight must accelerate up the track to get out of its way and the larger the upward force the lever must thus be exerting on the weight. When the lever exerts a larger upward force on the weight while traveling a fixed distance in the direction of that force, it does more work on the weight and gives it more energy.

Why: As the lever undergoes angular acceleration, it begins to push the weight upward and out of the way. The force that the lever exerts on the weight must support its weight and also cause enough upward acceleration to get the weight out of the way. A faster angular acceleration for the lever means a larger upward acceleration for the weight and therefore a larger upward force of the lever on the weight. Since the work done on the weight by the lever is proportional to the force that the lever exerts on the weight (work = force x distance) and the distance that this work is done over is fixed, then increasing the lever’s angular acceleration leads to an increase in the work that the lever does on the weight and thus to an increase in the weight’s final energy.

You have become bored with sedate sports like street luge, sky surfing, and competitive snowboarding, so you decide to take up something truly thrilling: lawn bowling. In this adrenaline junkie's delight, you and your opponents take turns rolling heavy balls along the grass, trying to see who can leave his/her balls closest to a smaller ball known as the jack. You are allowed to use your balls to knock opponents' balls away from the jack.

5. It's your turn to throw, so you pick up a ball and throw it underhanded across the lawn. The ball arcs briefly through the air before landing on the grass and continuing on toward the jack. At the moment the ball first touches the grass, the ball is not spinning. During the ball's first few moments of contact with the ground, what happens to its total energy, its momentum, and its angular momentum? (report all three)

Answer: The ball’s total energy decreases, its momentum decreases, but its angular momentum increases.

Why: The ball is skidding across the grass during this period in its travels. Skidding involves sliding friction, so the ball is converting some of its kinetic energy into thermal energy. While some of that thermal energy stays in the ball and contributes to the ball’s total energy, some leaves the ball and ends up heating the grass. The ball is also losing velocity as it skids, so its momentum decreases. But since friction exerts its slowing force at the bottom of the ball, well below the ball’s center of mass, it exerts a torque on the ball and causes it to undergo angular acceleration. As its angular velocity increases, so does its angular momentum.

6. After a few moments, the ball begins to roll smoothly along the grass without sliding at all. During this period of its travels, does the ball's total energy increase, decrease, or stay constant?

Answer: The ball’s total energy stays the same.

Why: The rolling ball is experiencing only static friction—its turning surface touches the grass and then lifts off, all without sliding. Since static friction wastes no energy, the ball continues on with constant total energy.

7. Exactly as planned, your ball slams head on into your opponent's ball and knocks it far away from the jack. Your ball comes abruptly to a stop following the collision. It has transferred most of its energy and momentum to your opponent's ball. In one sentence each, use words like force, time, and distance to show that (a) energy and (b) momentum were transferred to the opponent's ball.

Answer: Your ball pushes your opponent’s ball forward and your opponent’s ball moves forward, so your ball does work on your opponent’s ball and transfers energy to it. Your ball pushes your opponent’s ball forward for a certain amount of time, so your ball applies an impulse to your opponent’s ball and transfers momentum to it.

Why: Energy is transferred by doing work. Since your ball pushes your opponent’s ball in the same direction that your opponent’s ball moves, your ball does work on your opponent’s ball. Momentum is transfer by an impulse. Since your ball pushes your opponent’s ball forward for an amount of time, your ball applies and impulse to your opponent’s ball.

8. Lawn bowling is normally done on a level lawn. If the lawn had some bowl-shaped depressions in it, then any ball that passed through one of these depressions would tend to accelerate toward the bottom of the depression (and would not go straight). Use the concept of potential energy to show why a ball passing through one of these depressions would accelerate directly toward the bottom of the depression.

Answer: Your ball accelerates in the direction that reduces its overall potential energy as quickly as possible, which in this case is toward the bottom of the depression.

Why: Forces and potential energy are intimately intertwined. Recall that potential energy is the energy stored in forces. The direction in which an object’s potential energy diminishes as quickly as possible is the direction in which the net force acts on that object. In the case of the ball rolling through the depression, the net force on the ball is toward the bottom of the depression—the direction in which the ball’s potential energy diminishes most quickly.