Work Power Energy

To determine the total kinetic energy of a small sphere of mass m and radius r sliding down the smooth surface of a hemispherical bowl of radius R, follow the steps below:

• The sphere starts sliding from rest, so its initial potential energy is entirely due to its height above the lowest point A.
• The height difference between the center of the bowl and the sphere at point A is given by R - r, where r is the radius of the sphere.
• At the lowest point A, the total mechanical energy is converted into kinetic energy, assuming no energy loss due to friction or other forces.

Using the conservation of mechanical energy:

Thus, the total kinetic energy of the sphere at the lowest point is:

Correct Answer:

1. Step 1: Equilibrium points occur where the derivative of the potential energy with respect to x is zero, i.e., where the force is zero.
2. Step 2: From the graph, there are three points where the potential energy is at a minimum or maximum. The point at the minimum corresponds to a stable equilibrium, while the points at the maxima correspond to unstable equilibria.
3. Step 3: Therefore, the particle has three equilibrium points, with one stable and two unstable.