4A Homework Set 9

Univeral Law of Gravity

1.
Tidal force: Two equal masses a distance L apart joined by a string are in free
fall toward a planet of mass M. The string joining them is aligned radially
towards the center of the planet. The mass closest to the planet is a distance
R from the center of the planet. So the "outer" mass is a distance
R+L from the center of the planet. Find the tension in the string.Ignore
the gravitaional attraction between the two equal
masses.

2. Dinosaurs beware: An asteroid of mass M infinitely far away has a known
non-zero initial velocity that is not initially directed toward the center of a
planet but instead in a direction such that if it proceeded in a straight line
it would miss the center of the planet by a distance *b*. This would be
called its impact parameter. From this, find the "distance of closest
approach" the asteroid will ever get to the planet; if this distance is
less than the radius of the planet a collision will occur.

3. An exploding star: A planet of mass m is in a circular orbit about a star of
mass M at an initial distance of r. The non-rotating star then explodes and
ejects half of its mass radially outward in a symmetric fashion with none of
its ejecta hitting the planet. After the star's
explosion find the new radius of the planet about the star.

4. Get out of Dodge: An object of mass m has an orbital radius R about a planet
of mass M. Find the additional speed it would need from its orbit to escape to
infinity with a zero final kinetic energy.

5. Journey to the moon: Given is the mass of the earth and moon, the radius of
the earth and moon, and the distance between their centers. Find the minimum
speed required for an object launched from the Earth to *just* make it to
the moon.

6. The Moon is leaving us: Given that we know the mass of the moon and the
earth and the distance between their centers as the moon orbits the earth, if
the earth's angular velocity about its own axis is slowing down from say some
initial given omega to a final omega (due to tidal friction in reality), find
out what happens to the orbital distance between the earth and moon as a
consequence.

7. Put your observatory here: Although a single planet of given mass m in
orbit about let's say, a star of mass M, with a known orbital radius can have
only one period of its orbit for that given radius, if a *third* object
were placed at certain special points within this system (i.e. the Lagrange
points), then this third object object (with *two*
forces of gravity acting on it from the star and then the planet) can have the
same period of orbit as the planet has about the star. Find the radius of this
third object from the star such that this could occur. One position would be in
between the star and the planet and another position would be outside of the
planet and star; both of those positions would be along a line joining the star
and planet at any given instant. Find these two points. There are other
Lagrange points but that's not this problem.

8. The universal law is not always “center
to center”. A fixed sphere of mass M and radius R has a rod of mass m and length
L oriented in a radial fashion from the center of the sphere where the end of
the rod that is closest to the sphere is a distance D from the center of the
sphere (but the rod is outside the sphere). Find the gravitational force of the
sphere on the rod.