Is it possible to make a pendulum bob undergo uniform circular motion in a vertical plane?

00:48 Hint given | 01:50 Solution starts | 06:23 Answer | 06:28 Animation

Watch the inner end of string move along the plotted curve. Change parameters and see the effect on the path.

Animation of solution on Desmos

Question N1L3: Concepts used in solution

Centripetal acceleration | F=ma |

Problem statement

A small bob of mass m is attached to one end of a light string of length l. How should the other end of the string be moved such that the bob undergoes uniform circular motion along a circle of radius l, in a vertical plane under gravity?

Solving the pulley problem without writing the force equations

00:56 Hint given | 01:59 Solution starts | 06:35 Tension | 08:53 Acceleration of COM

Question N1L1: Concepts used in solution

Acceleration due to gravity | Center of mass | Vectors

Solving the pulley problem by writing the force equations by MIT OCW

Problem statement

Without writing the force equations, find the acceleration (a) of mass m in the pulley system shown in the figure. Reason the dependence of acceleration on the given masses m, M and acceleration due to gravity g by varying their values and comparing with known limits. Also, use the symmetries in the problem. Assume the pulley and rope to be massless and surfaces frictionless. Similarly, find tension T in rope and A the acceleration of the center of mass of m and M.

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.