1. (20 points) Derive the equations of motion for a point mass in spherical coordinates.
2. (30 points) A uniform sphere with mass m1 and radius R and a mass m2 hang by massless strings from the same point (see Fig. 1). Ignore frictional forces, use D’Alembert’s principle to find at what angle θ they are in balance?
3. (25 points) In special relativity, the Lagrangian of a point mass is
(m0 is its mass, v is its velocity, c is the speed of light, and V is the potential). Derive the Lagrange’s equations for it.
4. (25 points) Let a uniform rod of length 2l slide down freely from rest on a frictionless plane (see Fig. 2), write down the equations of motion. What are the conserved quantities?
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