Problem Set 7

Physics 5200

Due 30 Mar. 1999

1. (2 points) A 6100 kg rocket is set for vertical firing from the Earth's surface. If the exhaust speed is 1200 m/s, how much gas must be ejected each second if the thrust
   1. is to equal the weight of the rocket and
   2. is to give the rocket an initial upward acceleration of 21 m/s²?
      
2. (2 points) A solid uniform sphere of radius a has a spherical cavity of radius a/2 centered at a point a/2 from the center of the sphere. Find the center of mass. (Hint: no integral required.)
   
3. (3 points) Find the moment of inertia of the sphere in the previous problem about an axis passing through the centers of the sphere and cavity.
   

4. (3 points) A star of mass M = 2×10³⁰ kg and radius R₀= 7×10⁸ m rotates with angular velocity . Suddenly the star collapses under the force of its own gravity, and its radius is reduced to R₁= 10⁴ m. For this problem, "collapse" means that the radius is reduced while the mass remains constant. Assume that the star remains spherical with a uniform mass density.

   1. What is the angular velocity of the collapsed star? You can look up the moment of inertia of a uniform sphere.

   2. Stability of the collapsed star demands that for an element of mass, dm, at the surface of the collapsed star, the gravitational acceleration exceeds the centripetal acceleration. Use this fact to derive a limit on the angular velocity after collapse, w₁.