Extra Credit Homework #1

1. A plastic ball of mass m filled with air with pressure p hits the wall. The ball is deformed by x, which is much smaller than its radius R. The pressure p is greater than the atmospheric pressure and its change due to deformation is negligible. Estimate the time the ball interact with the wall. (Hint: treat the system as a linear harmonic oscillator.)

2. A tall cyllindrical bucket of radius R is so supported that it can rotate around its vertical axis. The bucket is half-filled with water and is rotating with a constant angular velocity ω. After everything is settled down, the bottom of the bucket is still completely covered with water (and water is not overflowing over the edge of the bucket). What is the equation describing the surface of water?
3. A thread is connected to a wall on one end, and to a bob on the other, and is ran through a roller mounted on a block of mass m₀, as shown on the Figure below.  Both the block and the roller on it can move without friction. At the initial moment, the bob is raised to angle alpha and released. Using Newton's laws, determine the acceleration of the block if the angle between the thread and the vertical axis is constant. What is the mass of the bob?
   

4. Solve the previous problem using Lagrange's equation. Use the length of the thread from the roller to the bob (let's call it r) as one generalized coordinate, and the angle between the thread and the roller as the other (and call it θ). Note that the requirement θ = const. = α is an integral of the equations of motion, and not a constraint! Only compute the acceleration, don't bother with the mass of the bob. (Make sure you get the same answer as in the previous problem :)