Assignment 6

Problem 1:

A star of mass M and radius R is moving with constant velocity v through a cloud of particles of density ρ.
If all the particles which collide with the star are trapped by it, show that the mass of the star will increase at a rate
dM/dt = πρv(R2 + 2GMR/v2).

imageProblem 2:

A satellite is in a circular orbit of radius r around an airless spherical planet of radius R.  An asteroid of equal mass falls radially towards the planet, starting at zero velocity from a very large distance. The satellite and the asteroid collide inelastically and stick together, moving in a new orbit that just misses the planet's surface.  What was the radius r of the satellite's original circular orbit in terms of R?

Problem 3:

Consider a particle of mass m moving in the xy plane.  The potential energy function is
U(x,y) = (k/2)/(x2 + y2), with k a positive constant.
(a)  Find the equations of motion.
(b)  Are there circular orbits?  If yes, do they all have the same period?
(c)  Is the total energy conserved?

Problem 4:

(a)  Derive the relationship between the impact parameter b and the scattering angle θ for Rutherford scattering of a projectile of mass m by a fixed target particle of mass M. 
(b)  If 4 MeV alpha particles are incident on a gold foil (Z = 79), calculate the impact parameter that would give a deflection of 10 degrees.
(c)  Explain what modifications of this calculation must be made if the gold atom is allowed to recoil during the collision.

Problem 5:

Three stars with masses m1, m2, and m3 are forming a peculiar triple-star system, where each of the stars is situated in the corners of an equilateral triangle with a side length d.  The stars are attracting each other with gravitational forces.  Determine the direction and magnitude of the rotational velocity ω which will leave the relative position of the three stars unchanged.