What is the acceleration of these pure rolling rings?

00:46 Hint given | 01:51 Constraint equation | 04:32 Force solution | 08:51 Energy solution | 11:28 Answer

Question GM3L3: Concepts used in solution

F=ma | Constraint equation | Energy conservation | Torque

Problem statement

Two identical rings of mass M and radius R stand on a rough horizontal surface. The rings are in contact at point P. The radius vector to P makes an angle theta with the horizontal as shown in the figure.  A small cylindrical object of mass m is placed symmetrically on the rings at point P and released. It slides on the rings without friction. Find the acceleration of the center of the rings immediately after the release. The rings do not slip on the surface.

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?

What is the force between hemispheres of a charged solid sphere?

00:34 Hint given | 01:17 Solution starts | 08:10 Using COM | 11:34 Answer

Question ES1L3: Concepts used in solution

Electric field inside sphere | Spherical coordinates | Center of mass

Elemental volume in spherical coordinate system

Problem statement

What is the net force on the northern hemisphere due to the southern hemisphere of a uniformly charged solid sphere of radius R and the total charge Q? Could this problem be solved using the position of the center of mass of the hemisphere?