Problem Set 6

Physics 5200

Due 23 Mar. 1999

1. (3 points) Show that the total torque on a system of particles can be written as a torque acting on the CM of the system, and a torque about the CM, N_tot = N_CM + N_rel. Then show that
   N_CM = dL_CM/dt and N_rel = dL_rel/dt.
   This result should be somewhat familiar from introductory physics. It is the basis of statics problems such as the one below.
   
2. (2 points) A window washer stands on a uniform plank of mass M and length l supported by 2 ropes attached at the ends of the plank. The window washer has a mass m. What is the tension in each of the ropes as a function of the window washer's displacement from the center of the plank. (Call the distance from the center of the plank x.)
   
   
3. (3 points) A uniform rod of length l and mass m is allowed to pivot freely about a point a distance l/3 from one end. Determine
   1. the angular momentum of the CM of the rod about the pivot, L_CM,
   2. the angular momentum about the CM of the rod, L_rel, and
   3. the total angular momentum of the rod about the pivot, L_tot. Calculate L_tot by direct integration and make the comparison with L_tot = L_CM + L_rel.
      

4. (2 points) Consider the "pedagogical machine" drawn below. All surfaces are frictionless; the pulley is frictionless; and the pulley and rope are massless. Find the force F which must be applied such that the relative positions of all the masses remain constant. Hint: If all the masses equal M, then F = 3Mg.