a particle of mass m slides inside a hemispherical bowl of radius r show THAT for small displacement from equilibrium the particle exhibits simple harmonic motion with an angular frequency equal to that of a simple pendulum of length r.

To understand how a particle of mass m sliding inside a hemispherical bowl of radius r exhibits simple harmonic motion (SHM) for small displacements from equilibrium, we can break this down step by step. The key is to analyze the forces acting on the particle and how they relate to its oscillatory motion.

The Setup

Imagine a particle placed at the center of a hemispherical bowl. When the particle is at rest at the lowest point of the bowl, it is in a state of equilibrium. If we gently displace the particle slightly from this position, the gravitational force acting on it will cause it to move back towards the equilibrium position. This scenario is similar to a simple pendulum.

Forces Acting on the Particle

When the particle is displaced by a small angle θ from the vertical, two main forces are acting on it:

• Gravitational Force (mg): This acts downward and can be decomposed into two components: one along the surface of the bowl and one perpendicular to it.
• Normal Force (N): This acts perpendicular to the surface of the bowl and counteracts the component of gravitational force acting perpendicular to the bowl.

Analyzing the Motion

For small angles, we can use the small angle approximation, where sin(θ) ≈ θ (in radians). The restoring force that brings the particle back to equilibrium is the component of the gravitational force acting along the surface of the bowl.

The component of the gravitational force acting along the surface can be expressed as:

F = mg sin(θ) ≈ mg θ

This force is directed towards the equilibrium position, indicating that it is a restoring force.

Relating to Simple Harmonic Motion

According to the principles of SHM, the restoring force is proportional to the displacement from equilibrium:

F = -k x

where k is the force constant and x is the displacement from equilibrium. From our earlier expression, we can see that:

mg θ = -k θ

Thus, we can identify that:

k = mg

Finding the Angular Frequency

For SHM, the angular frequency ω is given by the formula:

ω = √(k/m)

Substituting our expression for k, we get:

ω = √(mg/m) = √(g/r)

This is the same angular frequency as that of a simple pendulum with length r, which is known to be:

ω_pendulum = √(g/L) where L is the length of the pendulum. In this case, L = r.

Conclusion on Motion

Therefore, for small displacements from the equilibrium position, the particle indeed exhibits simple harmonic motion with an angular frequency equal to that of a simple pendulum of length r. This relationship highlights the underlying simplicity and symmetry between different physical systems, illustrating how similar principles govern their behavior.