Problem 1. Warm-Up (15 points)
Each of the following problems is worth 5 points. Be sure to justify your reasoning for full credit!
- Pipe Betting You release a hollow pipe of radius a and mass m, a solid sphere of radius 2a and mass m, and a solid pipe of radius a and mass 2m from rest at the same time from the top of an incline. Recall that the moment of inertia of a hollow cylinder is MR2; the moment of inertia of a solid sphere is 2/5 MR2; and the moment of inertia of a solid cylinder is 1/2 MR2. Which arrives at the bottom first?
- Gravity × 2017 Consider a ring of 2018 evenly spaced point masses. The gravitational field at the center of the ring is 0; that is, a mass placed at point P in the center of the ring will experience no net force. Now suppose you remove one of the masses. What is the magnitude and direction of the resultant gravitational field at point P?
Fig. 1 shows the scenario with 8 point masses.
- Stable or Unstable? Consider the potential energy given by U(x) = sin xex in the interval [0; 2π]: What are the points of equilibrium? Identify each as stable or unstable.