Assignment 6

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*1030 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/x2.   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) = kr4,  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:

A steel disk A with radius R moves with speed v = 10 m/s when it collides with a second identical disk B at rest.  The collision is elastic and has an impact parameter “b”.  After the collision, the speed of disk A is equal to 5 m/s.  What is the value of the impact parameter “b”?  Neglect friction.