Can gravitational potential energy ever be negative? Note that
the question refers to PE, not ΔPE, so that you must think about
how the choice of a reference level comes into play.
[Based on a
problem by Serway and Faughn.]
A ball rolls up a ramp, turns around, and comes back down.
When does it have the greatest gravitational potential energy? The
greatest kinetic energy?
[Based on a problem by Serway and Faughn.]
(a) You release a magnet on a tabletop near a big piece of
iron, and the magnet leaps across the table to the iron. Does the
magnetic potential energy increase or decrease? Explain.
(b) Suppose instead that you have two repelling magnets. You give
them an initial push towards each other, so they decelerate while approaching
each other. Does the magnetic potential energy increase
or decrease? Explain.
Let Eb be the energy required to boil one kg of water.
an equation for the minimum height from which a bucket of water
must be dropped if the energy released on impact is to vaporize it.
Assume that all the heat goes into the water, not into the dirt it
strikes, and ignore the relatively small amount of energy required to
heat the water from room temperature to 100 °C. [Numerical check,
not for credit: Plugging in Eb = 2.3 MJ/kg should give a result of
(b) Show that the units of your answer in part a come out right
based on the units given for Eb.
A grasshopper with a mass of 110 mg falls from rest from a
height of 310 cm. On the way down, it dissipates 1.1 mJ of heat due
to air resistance. At what speed, in m/s, does it hit the ground?
Solution, p. 159
A person on a bicycle is to coast down a ramp of height h and
then pass through a circular loop of radius r. What is the smallest
value of h for which the cyclist will complete the loop without
falling? (Ignore the kinetic energy of the spinning wheels.)
A skateboarder starts at nearly rest at the top of a giant cylinder,
and begins rolling down its side. (If she started exactly at rest
and exactly at the top, she would never get going!) Show that her
board loses contact with the pipe after she has dropped by a height
equal to one third the radius of the pipe.
Solution, p. 159*
(a) A circular hoop of mass m and radius r spins like a wheel
while its center remains at rest. Its period (time required for one
revolution) is T. Show that its kinetic energy equals 2π2mr2/T2.
(b) If such a hoop rolls with its center moving at velocity v, its
kinetic energy equals (1/2)mv2, plus the amount of kinetic energy
found in the first part of this problem. Show that a hoop rolls down
an inclined plane with half the acceleration that a frictionless sliding
block would have.
Students are often tempted to think of potential energy and
kinetic energy as if they were always related to each other, like
yin and yang. To show this is incorrect, give examples of physical
situations in which (a) PE is converted to another form of PE, and
(b) KE is converted to another form of KE.
Solution, p. 160
Lord Kelvin, a physicist, told the story of how he encountered
James Joule when Joule was on his honeymoon. As he traveled,
Joule would stop with his wife at various waterfalls, and measure
the difference in temperature between the top of the waterfall and
the still water at the bottom.
(a) It would surprise most people
to learn that the temperature increased. Why should there be any
such effect, and why would Joule care? How would this relate to the
energy concept, of which he was the principal inventor?
much of a gain in temperature should there be between the top
and bottom of a 50-meter waterfall?
(c) What assumptions did you
have to make in order to calculate your answer to part b? In reality,
would the temperature change be more than or less than what you
calculated? [Based on a problem by Arnold Arons.]
Make an order-of-magnitude estimate of the power represented
by the loss of gravitational energy of the water going over
Niagara Falls. If the hydroelectric plant at the bottom of the falls
could convert 100% of this to electrical power, roughly how many
households could be powered?
Solution, p. 160
When you buy a helium-filled balloon, the seller has to inflate
it from a large metal cylinder of the compressed gas. The helium
inside the cylinder has energy, as can be demonstrated for example
by releasing a little of it into the air: you hear a hissing sound,
and that sound energy must have come from somewhere. The total
amount of energy in the cylinder is very large, and if the valve is
inadvertently damaged or broken off, the cylinder can behave like
bomb or a rocket.
Suppose the company that puts the gas in the cylinders prepares
cylinder A with half the normal amount of pure helium, and cylinder
B with the normal amount. Cylinder B has twice as much energy,
and yet the temperatures of both cylinders are the same. Explain, at
the atomic level, what form of energy is involved, and why cylinder
B has twice as much.
At a given temperature, the average kinetic energy per molecule
is a fixed value, so for instance in air, the more massive oxygen
molecules are moving more slowly on the average than the nitrogen
molecules. The ratio of the masses of oxygen and nitrogen molecules
is 16 to 14. Now suppose a vessel containing some air is surrounded
by a vacuum, and the vessel has a tiny hole in it, which allows the air
to slowly leak out. The molecules are bouncing around randomly,
so a given molecule will have to try many times before it gets
lucky enough to head out through the hole. How many times more
rapidly does the nitrogen escape?
Explain in terms of conservation of energy why sweating cools
your body, even though the sweat is at the same temperature as
your body. Describe the forms of energy involved in this energy
transformation. Why don't you get the same cooling effect if you
wipe the sweat off with a towel? Hint: The sweat is evaporating.