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7. (a) Suppose a rotating spherical body such as a planet has a radius r
and a uniform density
.
, and the time required for one rotation is T. At
the surface of the planet, the apparent acceleration of a falling object is
reduced by acceleration of the ground out from under it. Derive an
equation for the apparent acceleration of gravity, g, at the equator in terms
of r,
.
, T, and G.
(b) Applying your equation from (a), by what fraction is your apparent
weight reduced at the equator compared to the poles, due to the Earth’s
rotation.
(c) Using your equation from (a), derive an equation giving the value of
T for which the apparent acceleration of gravity becomes zero, i.e. objects
can spontaneously drift off the surface of the planet. Show that T only
depends on
.
, and not on r.
(d) Applying your equation from (c), how long would a day have to be in
order to reduce the apparent weight of objects at the equator of the Earth
to zero. [Answer: 1.4 hours]
(e) Observational astronomers have recently found objects they called
pulsars, which emit bursts of radiation at regular intervals of less than a
second. If a pulsar is to be interpreted as a rotating sphere beaming out a
natural "searchlight" that sweeps past the earth with each rotation, use
your equation from (c) to show that its density would have to be much
greater than that of ordinary matter.
(f) Theoretical astronomers predicted decades ago that certain stars that
used up their sources of energy could collapse, forming a ball of neutrons
with the fantastic density of ~10
17
kg/m
3
. If this is what pulsars really are,
use your equation from (c) to explain why no pulsar has ever been ob-
served that flashes with a period of less than 1 ms or so.
8. You are considering going on a space voyage to Mars, in which your
route would be half an ellipse, tangent to the Earth’s orbit at one end and
tangent to Mars’ orbit at the other. Your spacecraft’s engines will only be
used at the beginning and end, not during the voyage. How long would
the outward leg of your trip last. (Assume the orbits of Earth and Mars are
circular.)
9. (a) If the earth was of uniform density, would your weight be in-
creased or decreased at the bottom of a mine shaft. Explain. (b) In real life,
objects weight slightly more at the bottom of a mine shaft. What does that
allow us to infer about the Earth.
10 S. Ceres, the largest asteroid in our solar system, is a spherical body
with a mass 6000 times less than the earth’s, and a radius which is 13
times smaller. If an astronaut who weighs 400 N on earth is visiting the
surface of Ceres, what is her weight.
11 S. Prove, based on Newton’s laws of motion and Newton’s law of
gravity, that all falling objects have the same acceleration if they are
dropped at the same location on the earth and if other forces such as
friction are unimportant. Do not just say, “g=9.8 m/s
2
-- it’s constant.”
You are supposed to be proving that g should be the same number for all
objects.
Chapter 10Gravity