Question

A particle slides from the top of a smooth hemispherical surface of radius R which is fixed on a horizontal surface. If it separates from the hemisphere at a height h from the horizontal surface, then find the speed of the particle.
A. \[\sqrt {2g\left( {R - h} \right)} \]
B. \[\sqrt {2g\left( {R + h} \right)} \]
C. \[\sqrt {2gR} \]
D. \[\sqrt {2gh} \]

Answer 1

Hint:Before we start addressing the problem, we need to know about the conservation of energy. It states that the total energy of the system should be conserved and the energy neither be created nor be destroyed.

Formula Used:
From the conservation of energy, we have,
\[\Delta P.E = \Delta K.E\]
Where, P.E is potential energy and K.E is kinetic energy.

Complete step by step solution:
Consider a particle that is sliding from the top of a smooth hemispherical surface of radius R which is fixed on a horizontal surface. If it separates from the hemisphere at a height h from the horizontal surface, then we need to find the speed of the particle. By using the conservation of energy,
\[\Delta P.E = \Delta K.E\]
\[\Rightarrow mg\left( {R - h} \right) = \dfrac{1}{2}m{v^2}\]
\[\Rightarrow {v^2} = 2g\left( {R - h} \right)\]
\[\therefore v = \sqrt {2g\left( {R - h} \right)} \]
Therefore, the speed of the particle is \[\sqrt {2g\left( {R - h} \right)} \].

Hence, option A is the correct answer.

Additional information: There are three laws of conservation of energy, that is, conservation of energy, conservation of momentum, and the conservation of angular momentum. Some of the examples of conservation of energy are, the swinging of a pendulum, a coin tossed up in the air, when a ball rolls down an inclined plane, etc.

Note:In order to solve this problem it is important to remember what the principle of conservation of energy is, and the formula for the potential energy and kinetic energy.