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:
Initial Potential Energy=Kinetic Energy at A mg(R−r)=Kinetic Energy at A
Thus, the total kinetic energy of the sphere at the lowest point is:
mg(R−r)
Correct Answer: mg(R−r)