Extra Credit Homework #2

1. ``A toy model of the expansion of the universe:'' At the initial moment (the ``Big Bang''), a spherical cloud of galaxies has an uniform distribution of matter, ρ, and a distribution of velocities which is the linear function of radius from the center, v = Hr (the ``Hubble's law''). What is the maximal initial density, ρ = ρ_max, at which the cloud will still expand forever? (Hint: only consider the dynamics of the galaxies on the surface of the cloud, and rely on problem #1 from this week's regular homework.)

2. A spherical asteroid of radius R and mass M has a spherical hole of radius r. The distance from the center of the hole to the center of the asteroid is d, with d < R-r. Calculate the strength of the gravitational field outside the asteroid. (Hint: calculate the field of an asteroid without the hole, and of the material which would fill the hole; then use the principle of the superposition.)

3. A metal sphere of radius R is suspended in vacuum and fixed. Electrons are being shot, one at a time, from the distance of d >> R towards the sphere. The direction of the electrons is displaced from the axis of the sphere by 0 < b < R. The electrons are shot with the initial velocity v. The sphere is thus slowly being charged, and the charge spreads uniformly over the surface of the sphere. What is the maximal charge Q = Ne of the sphere? (Neglect the image charge of the incoming electron; it is assumed that v is large enough so that N >> 1.)

4. A half-cylinder of mass M and radius R is laying on the table. It can roll without slipping on the table. What is the period of the small oscillations of the half-cylinder when it's rocking on the surface of the table?

5. A slice of toast with jam is balanced at the edge of the kitchen table of height h. The toast is an a × a square, thickness d_toast, and mass m_toast. Jam is spread so that its thickness is negligible, covers the whole one side of the toast, and has mass m_jam. The toast starts tipping over and falls down. Under which condition it will fall with the jam facing the kitchen floor? Measure these parameters in your kitchen and predict whether the toast will fall jam-side down on your own kitchen floor.