Download Solutions to Quiz 7

Solutions to Quiz 7
April 28, 2010
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A Party Trick
You decide to impress your friends at your next party with the following
demonstration: you want to levitate a coin by firing a concentrated laser
beam at it.
Take the coin to be a penny which weighs 2.5 g. Assuming that the coin
is balanced and doesn’t topple, find the power rating of the laser needed to
perform this trick. Compare this power rating to that of a 75 W light bulb.
1.1
Solution
The force imparted by all of the photons is given by
F =
P
c
(1)
where P is the power rating of the source. Since all the photons are being
absorbed by the coin, we can balance the forces
mg = P c
(2)
and solve for a power rating of 75 · 105 W or 105 the power output of a 75W
light bulb.
2
Solar Sail
You are back in your regular day job on the Starship Enterprise. The ship’s
engine is not working, and the ship is now in danger of being pulled into a
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nearby star. Inspired by your stunt the night before, you come up with the
idea of using a giant sail to propel the ship with the radiation pressure of
the star. Remember that unlike the laser, the starlight is spread out in all
directions.
Derive a formula for the minimum area of the sail needed to overcome
the gravitational pull on the ship. Assume that the sail is black and absorbs
all light falling on it. You should express your answer in terms of the mass
of the ship, the mass of the star, the power output of the star and any other
constants you may need.
2.1
Solution
The star is exerting two forces on the spaceship. The first is the gravitational
attraction, and the second is the force being exerted on the ship by photons
that are hitting it. To escape a fiery demise, we want the force from the
radiation to exceed the gravitational force.
Fradiation ≥
GM m
r2
(3)
Next, we derive a formula for the radiation pressure of the star. We know
that
I
Pradiation =
(4)
c
where I is the intensity of the light source. The light is spread in all direction,
and is thus distributed over the surface of a sphere. So the intensity
I=
Pstar
4πr2
(5)
The force on the sail due to this radiation pressure is then given by
Fradiation = Asail Pradiation
(6)
Combining all the above equations, we get
Asail Pstar
GM m
≥
2
c 4πr
r2
(7)
which simplifies to
Asail ≥
4πcGM m
Pstar
2
(8)
Note that the distance to the star cancels out, which means that once
you have a sail of sufficient size, it will always be big enough.
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