Given your recent involvement in international espionage, you're hardly surprised when James Bond appears at your doorstep one evening. You offer him a drink, but in keeping with the politically correct nature of this problem set, you point out to him that he is going to have to do without the usual vodka martini. Nonetheless, he asks that you serve his fruit smoothie "shaken, not stirred". You're only too happy to oblige.
1. You begin to assemble the drink by mixing together strawberries and bananas in a blender. You place those fruits in the blender's plastic container and turn on the motor. A sharp blade attached to the motor spins rapidly in the middle of the plastic container and slices the fruit into tiny pieces. The container is nearly frictionless, so what holds the fruit in place as the blender's spinning blade slices through it?
Answer:
Inertia
Why: If no significant forces
are exerted on a piece of fruit, it will continue to move at constant velocity.
In the blender, that constant velocity is typically zero. The sharp blade of
the blender simply passes through the fruit while the fruit’s own inertia keeps
the fruit in place. Of course, if the blender’s blade were dull, it would exert
a substantial force on each piece of fruit it struck and the fruit might
succeed in accelerating out of the way before it was cut in half.
2. Having chopped up the fruit, it's time to add orange juice and ice. To impress Mr. Bond, you decide to drop the ice into the drink from several feet above it. You release several pieces of ice simultaneously and they fall dramatically into the liquid. Supposing that air resistance has no significant effect on their motion, which pieces of ice fall fastest: large, small, or neither? Justify your answer.
Answer:
Neither piece of ice falls faster. On the earth’s surface, all things fall
downward at the same rate.
Why: Assuming that there is no
significant air resistance, all things that are falling accelerate downward at
9.8 m/s2, regardless of mass or weight. That’s because heavier
things are also more massive, so that their increased weight (cause of
acceleration) only serves to compensate for their increased mass (resistance to
acceleration).
3. It's time for the shake. You put the top on the container and begin to shake it vigorously up and down. As you shake the container, you can feel yourself pushing upward on it as you move it upward and pushing downward on it as you move it downward. At what times are you doing (positive) work on the container: moving it upward, moving it downward, or both? Explain why that is the case.
Answer:
You are doing work both on the way up and on the way down. You are always
pushing the object in the direction it moves.
Why: Any time you push on
something and it moves in the direction of your push, you do work on it. In the
present case, you are pushing the container in the direction of its motion. The
fact that the container is moving downward some of the time is irrelevant; you
are still pushing it downward so you are doing work on it.
4. As you reverse directions and begin moving the container downward, the drink inside it often rises upward and hits the top of the container. During those moments, is there a force pushing the drink upward—toward the top of the container? If so, what is that force? If not, what is causing the drink to rise, even though the container is descending?
Answer:
There is no upward force on the rising drink. The drink’s inertia keeps it
moving upward.
Why: The drink is coasting
upward when it hits the top of the container. Even though there is nothing
pushing it upward, it continues to rise until something (namely gravity and forces
from the top of the container) accelerate it to a stop (or equivalently,
decelerate it).
You've made plans to go mountain biking on Observatory Hill and are about to ride up its side for the first time. For simplicity, we'll assume that the uphill path is straight and has a uniform slope the entire way. You are balancing precariously on your bicycle at the bottom of that path, waiting for this problem to begin.
5. While you are not moving anywhere yet, you are getting tired. Balancing the bicycle without moving is exhausting. As you balance there on the motionless bicycle, are you doing (positive) work, zero work, or negative work on the bicycle? Explain your answer.
Answer:
You are doing zero work. Since the bicycle isn’t moving, you cannot do work on
it.
Why: For you to do work on the
bicycle, (1) you have to push on it and (2) it has to move a distance in the
direction of that push. Since it’s motionless, part (2) isn’t occurring and you
do no work.
6. It's time to start climbing the hill. You push down extra hard on the pedals and you and the bicycle begin to accelerate up the hill. In what direction does the net force on you and the bicycle point during the time when you are accelerating uphill?
Answer:
The net force points uphill.
Why: Since you are
accelerating uphill, the net force on you must also be uphill. Net force and
acceleration are always in the same direction. While the individual forces on
you and the bicycle are rather complicated, the net force is simple: it’s in
the direction of your acceleration.
7. After a few seconds, you are traveling at a good pace and decide to stop accelerating. You then continue uphill at a constant velocity. While you are going uphill at a constant velocity, what is the net force on you and the bicycle? (Remember that your answer may involve an amount and a direction.)
Answer:
The net force is zero.
Why: Since you are moving at
constant velocity, you are not accelerating and the net force on you must be
zero. Whether you are moving uphill or downhill is irrelevant; all that matters
is your acceleration and at present it’s zero.
8. Your friend is also riding a bicycle up Observatory Hill. While you head straight uphill to the top, your friend weaves back and forth on the road. As a result, your friend travels exactly twice as far as you do in the process of climbing the hill. The two of you started and finished at the same place and you both weigh the same amount. Neglecting friction and air resistance, compare the amounts of work the two of you did while climbing the hill: are they the same? Are they different? If different, by what fraction do they differ?
Answer:
You both do the same amount of work.
Why: You each undergo the same change in gravitational potential energy in climbing the hill. That’s because you both have the same weight and both rise the same distance upward. The amount of work each of you must do in climbing the hill is exactly this increase in gravitational potential energy. Regardless of how you make the journey, you will have done this amount of work when you reach the top of the hill.