Question

A small sphere of mass m and radius R slides down the smooth surface of a large hemispherical bowl of radius R. If the sphere starts sliding from rest, the total kinetic energy of the sphere at the lowest point A of the bowl will be:

• A. mg(R − r)
• B. \(\frac{7}{10}mg(R - r)\)
• C. \(\frac{2}{7}mg(R - r)\)
• D. \(\frac{7}{10}mg(R - r)\)

Hint:

For rolling motion, the total kinetic energy is the sum of translational and rotational kinetic energies. Use energy conservation to determine the total kinetic energy at the lowest point.

Answer 1

The Correct Option is A

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:
\[ \text{Initial Potential Energy} = \text{Kinetic Energy at A} \] \[ m g (R - r) = \text{Kinetic Energy at A} \]

Thus, the total kinetic energy of the sphere at the lowest point is: 
\[ \boxed{m g (R - r)} \]

Correct Answer: \( \mathbf{mg(R - r)} \)