In a case of virtual reality run amok, your ebook version of the classic children’s story “Peter Pan” has somehow taken over your life. Even more troubling is the fact that Captain Hook and his pirates have captured you and will soon make you “walk the plank.” Projecting horizontally from the side of Hook’s ship is a flexible wooden board several meters long and it is this “plank” that you will have to walk out on before stepping off into space. After noticing a huge pile of mattress lying on the dock just under the board, you realize that the pirates aren’t so bad after all and that they’re just trying to have a little fun. Since no one is really getting hurt, you begin to enjoy the adventure.
1. There is a line of other victims ahead of you, so you can relax for the moment and watch the other people walk onto the board, one by one. You notice that as a victim walks out onto the board, the board gradually bends downward. Since the victim is losing altitude, she is also losing gravitational potential energy. Where is her gravitational potential energy going?
Answer: It
is being transformed into elastic potential energy (in the board)
Why: As she walks out onto the
board, it bends downward to support her weight. As the board bends, she does
work on it and it extracts gravitational potential energy from her. This energy
becomes stored in the elastic forces within the board.
2. As the victim stands motionless at the end of the board, you observe that the board is bent downward by an amount that is exactly proportional to her weight; the previous victim was twice her weight and the board bent downward twice as far. Explain why the board’s downward deflection is proportional to the victim’s weight.
Answer: At
equilibrium, the board's upward support force on the victim exactly balances
her weight. Since the elastic board acts as a spring, this upward support force
is proportional to its downward distortion. The downward distortion is
therefore proportional to the victim's weight.
Why: There are two separate
relationships here: (1) at equilibrium, the upward force from the board on the
victim is equal to the victim's weight and (2) the upward force that the board
exerts on the victim is proportional to the downward deflection of the board.
(1) is true because at equilibrium there is no net force on the victim. (2) is
true because the board is acting as a spring and exerting a force on its end
that is proportional to the deflection of that end. Combining (1) and (2) gives
us the result that the weight of the victim is proportional to the deflection
of the board.
3. It’s now your turn to go. You walk bravely off the board and drop onto the pile of mattresses below. To your relief, you plunge into the soft heap, come to a stop deep in the mattress pile, and then rebound upward. After a couple more bounces, you come to rest. You then climb off the mattresses and walk off in search of Peter Pan himself. What was the direction of your acceleration (a) just before you reached the deepest point in the pile and (b) just after you reached the deepest point in the pile?
Answer: (a)
acceleration is upward and (b) acceleration is upward.
Why: Near the deepest point in
the mattresses, the restoring force the mattresses exert on you is much greater
than your weight. We know this because the mattresses are in the process of
stopping your downward descent and causing you to begin rising upward. They are
pushing upward on you hard enough to accelerate you upward, despite your
weight. At the moment of deepest descent, your velocity reaches zero but you
are accelerating upward. In fact, you are accelerating upward faster than at
any other time in the story. Just before and after that moment of deepest
descent, you are still accelerating upward, but slightly less quickly. In all
three cases, the net force on you is upward and you accelerate upward. In
contrast, your velocity shifts from downward before the deepest descent, to
zero at the deepest descent, to upward after the deepest descent.
4. What forms did your energy take as you bounced up and down in the pile of mattress and at what moment was each form of energy at its maximum?
Answer: (a)
Gravitational potential energy, (b) kinetic energy, and (c) elastic potential
energy. The maximum of (a) occurred before you left the board, (b) as you
passed through the equilibrium position part way into the pile of mattresses,
and (c) at your deepest descent into the pile of mattresses.
Why: At your highest point
during the event, you have a maximal amount of energy stored in the
gravitational forces. As you are passing through the equilibrium point within
the pile of mattresses, the net force on you switches from downward to upward
and you stop accelerating downward--thus you stop gaining kinetic energy and
start losing it. And as you reach the deepest descent into the pile of
mattresses, you have done a maximal amount of work on the elastic mattresses
and begin to extract energy from them. Of these three, the kinetic energy part
is the most surprising. Just touching the mattresses doesn't cause your kinetic
energy to stop rising. You still accelerate downward when you first begin to
touch the mattresses so your kinetic energy continues to rise. It isn't until
you have dented the mattresses enough that their restoring force equals your
weight that your kinetic energy stops rising. That point is the equilibrium point,
the point at which you would come to rest if you simply sat on the mattress and
let the bouncing stop. Whenever you are above that point, you accelerate
downward and whenever you are below that point, you accelerate upward. It is a
stable equilibrium, after all.
You’re playing racquetball with a friend. Each of you has a metal racket with nylon strings and you’re taking turns smacking the ball against the front wall of a hard-surfaced room. The ball ricochets back from the front wall after every hit and the two of you are working up quite a sweat trying to keep it bouncing off that front wall.
5. You swing your racket forward and hit the ball forward. You notice that your racket slows down slightly as a result of the collision. Use the concept of momentum to explain why the racket must slow down during this sort of collision.
Answer: Pushing the ball forward for a period of time gives the ball a
forward impulse and thus extracts forward momentum from the racket. The
racket's forward velocity must decrease as its forward momentum decreases.
Why: Any forward push on the
ball that lasts longer than an instant gives that ball a forward impulse. Since
an impulse is a transfer of momentum, the ball receives forward momentum from
this push. That momentum must come from somewhere. In the present situation, it
comes from your racket. Since the racket's forward momentum is associated with
its forward velocity and its mass, the loss of forward momentum must involve a
loss of forward velocity. Your racket accelerates backward and slows down.
6. Instead of swinging forward as the ball approaches him from the front wall, your opponent swings his racket gently backward (away from the front wall). The ball hits the racket and bounces remarkably weakly. In terms of energy and work, why did allowing the ball to hit the racket as the racket moved away from the front wall cause the weak bounce?
Answer: The racket
pushes the ball forward as the ball moves backward, so the racket does a
negative amount of work on the ball. The ball loses energy and barely moves
after the bounce.
Why: The situation here
resembles the task of pushing a child on a swing. If you push the child forward
as she heads forward, you do work on her and her energy increases. She swings
more and more vigorously. But if you push the child forward as she heads
backward, you do negative work on her and extract energy from her and her
energy decreases. She swings less and less vigorously. Returning to the ball
and racket, we see that pushing the ball forward as it moves backward also
reduces the ball's energy so that it doesn't bounce much.
7. You let the racquetball drop from rest onto the rigid floor and observe that it rebounds to 72% of its original height. What fraction of the ball’s kinetic energy at the moment before it hits the floor has been returned to the ball by the moment after it rebounds from the floor? What became of the missing energy?
Answer:
72% is returned. The rest became thermal energy.
Why: The ball's initial energy
is all gravitational energy. As it drops, this energy becomes kinetic energy
and the kinetic energy reaches a maximum the moment before it hits the ground.
The bounce then occurs, with some of the energy becoming elastic potential
energy in the ball and then being return to kinetic form during the rebound.
Whatever energy is left the moment after the ball rebounds from the floor is in
kinetic form. This remaining kinetic energy then becomes gravitational
potential energy as the ball rises to its peak height. Since gravitational
potential energy is proportional to altitude, the ball's altitude rise after
the bounce is proportional to its kinetic energy after the rebound. The
percentage of height lost between drop and rebound is equal to the percentage
of energy lost to thermal energy during the bounce.
8. You hold the racket face horizontal and drop the ball onto its surface. To your surprise, the ball bounces higher from the racket face than it does when dropped directly onto the rigid floor. Why does the racket’s nylon-mesh surface allow the ball to bounce better than it would from a solid surface?
Answer: The racket
surface is livelier than the ball itself and so it returns a larger fraction of
its share of the collision energy to the ball.
Why: When a ball bounces from
a soft surface, the collision energy is shared between the ball and the
surface. The fraction of collision energy going into each object is
proportional to how far each dents during the collision. The two objects are
then jointly responsible for the rebound. If the soft surface is livelier than
the ball, it will return a larger fraction of its share of the collision energy
and will cause the ball to rebound more strongly than it would have if it
didn't share the collision energy with the surface.