How do the forces on a slowly pulled ball change as it moves over a hemisphere?

01:00 Solution starts | 02:33 Hint given | 04:17 Answer

Question N1L2: Concepts used in solution

Equilibrium | Vector addition | Properties of triangles

Problem statement

A small metal ball is being pulled slowly from the bottom to the top of a fixed frictionless hemisphere, as shown in the figure. Assuming that the radii of the ball and the pulley are much smaller than that of the hemisphere, do the magnitudes of the pulling force F and the contact force N between the ball and the hemisphere increase, decrease or remain unchanged?

How to replace a pulley and two masses with an effective mass

00:40 Hint given | 01:46 Solution starts | 04:34 M from limits | 11:13 M from F=ma

Question N2L2: Concepts used in solution

Limits |  F=ma | Simple two mass pulley system

Problem statement

A ball of mass m is connected by rope to a pulley (P) on the other side as shown in figure (a). Balls of mass m1 and m2 are tied to the ends of a rope that passes over this pulley (P). Find the acceleration of mass m, assuming the pulleys and ropes to be massless and frictionless. Find the equivalent mass M of a new ball that could replace the pulley (P) and masses m1 and m2, as shown in figure (b), but provide the correct acceleration for mass m.