1
Phy 3221
Due: March 2, 2011
Homework set # 7
Problem 1: Imagine that you are at rest relative to the surface of a planet with the same
mass and radius as the Earth. What would be the length of a day on that planet if it were
rotating so fast that the acceleration of gravity on the equator (at rest relative to the surface
of the planet) were zero? hint: The surface is rotating around the axis of the planet, so if
you are at rest on the surface then you are subject to a “centrifugal force” which arises from
the acceleration of your frame of reference.
Problem 2: A commuting scheme involves boring a straight tunnel through the crust of
the Earth between New York City and San Francisco. If you were to drop a ball down into
the tunnel in NYC then how long does it take for the ball to pop up in SF? Assume that
there is no friction or air resistance in the tunnel. Assume that the earth is round and has
constant density, and that at the surface of the Earth g = 10 m/s2 . Finally assume that
the triangle connecting NYC, SF and the center of the Earth is an equilateral triangle (60◦
for each angle). Assume that the radius of the Earth is 6,400 km. Hint: Ignore the Earth’s
rotation and centrifugal force affects in this problem!
Problem 3: Two objects, each of mass m are attached to the ends of a spring of spring
constant k whose unstretched length is `. The masses are at rest on a perfectly horizontal,
frictionless table. Assume that the Earth is perfectly round and has mass M and radius
R. The gravitational force on each mass points directly toward the center of the Earth,
and these two forces are not quite parallel because the masses are a distance ` away from
each other. As a result the forces of gravity are not quite perpendicular to the surface of
the table and this causes the spring to be stretched or compressed. What is the change
in the length of the spring caused by this “tidal force?” Hint: In solving this problem,
take advantage of the fact that the length of the spring is much less than the radius of
the Earth. In this case the angle θ subtended by the spring at the center of the earth has
tan θ ≈ sin θ ≈ `/R.
Problem 4: Lagrange Points: The Earth and the moon orbit their common center
of mass. If a small satellite where placed in just the right spot and with just the right
velocity, then the satellite would also circle the center of mass of the Earth and Moon with
the same orbital period as the Moon. In this case, the relative positions of the Earth, Moon
and satellite would not change. There are five of these special places and they are called
Lagrange points. Look up Lagrangian point on Wikpedia. The Lagrange points labeled
L4 and L5 are particularly interesting and have a rather easy description. Mathematically
describe where L4 is and then demonstrate that a satellite at L4 would, in fact, orbit the
center of mass of the Earth-Moon system with the same orbital frequency as the Moon orbits
the Earth.