**Problem 1:**

The source of the first gravitational
wave event observed by the LIGO collaboration in 2015 has been interpreted as
the merger of two black holes in a binary system, each with a mass of roughly 35
solar masses (implying a radius for the event horizon of about 100 km for each,
if assumed spherical), where a solar mass is 1*.*989*10^{30
}kg. A full understanding requires general relativity, but assume
Newtonian mechanics and Newtonian gravity as a first approximation for the
orbital motion. At the peak amplitude of the detected gravitational wave, its
measured frequency indicated that the two black holes were revolving around the
center of mass about 75 times per second.

What was the approximate separation of the centers for the two black holes at
this point in the merger event?

**Problem 2:**

A particle of mass m is released a
distance b from a fixed origin of force that attracts the particle according to
the inverse square law F(x) = – k/x^{2}. Find the time required for
the particle to reach the origin. Use this result to show that, if the Earth
were suddenly stopped in its orbit, it would take approximately 65 days for it
to collide with the Sun. Assume that the Sun is as a fixed point mass and
Earth’s orbit is circular.

**Problem 3:**

Find the maximum time a comet (C) of mass m following a parabolic trajectory around the Sun (S) can spend within the orbit of the Earth (E). Assume that the Earth's orbit is circular and in the same plane as that of the comet.

**Problem 4:**

A particle of mass m moves under the action of a central force whose
potential energy function is U(r) = kr^{4},
k > 0.

(a) For what energy and angular momentum will the orbit be a circle of radius a
about the origin? What is the period of this circular motion?

(b) If the particle is slightly disturbed from this circular motion, what will
be the period of small radial oscillations about r = a?

**Problem 5:**