A small spherical steel ball is placed at a distance...

A small spherical steel ball is placed at a distance slightly away from the center of a concave mirror having radius of curvature $250 \mathrm{~cm}$. If the ball is released, it will now move on the curved surface. What will be the periodic time of this motion? Ignore frictional force and take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. (A) $(\pi / 4) \mathrm{s}$ (B) $\pi \mathrm{s}$ (C) $(\pi / 2) \mathrm{s}$ (D) $2 \pi \mathrm{s}$

A small spherical steel ball is placed at a distance slightly away from the center of a concave mirror having radius of curvature $250 \mathrm{~cm}$. If the ball is released, it will now move on the curved surface. What will be the periodic time of this motion? Ignore frictional force and take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$.
(A) $(\pi / 4) \mathrm{s}$
(B) $\pi \mathrm{s}$
(C) $(\pi / 2) \mathrm{s}$
(D) $2 \pi \mathrm{s}$

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Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes periodic oscillations where the restoring force is proportional to the displacement from equilibrium and directed opposite to it. In these systems, the motion can be characterized by a constant period that is independent of the amplitude for small oscillations.

Restoring Force on Curved Surfaces

When an object moves on a curved surface, gravity provides a component of force along the surface that acts as a restoring force. For small deviations, this component can often be approximated as directly proportional to the displacement, resulting in motion that closely resembles simple harmonic motion.

Small Angle Approximation

In the analysis of small oscillatory motions, the small angle approximation (sin? ? ?) simplifies the relationship between the displacement and the restoring force. This linear approximation is essential for reducing the equations of motion to those of simple harmonic oscillators.

Radius of Curvature and Geometrical Effects

The radius of curvature is a fundamental geometrical parameter that defines the shape of a curved surface. In problems involving oscillations on curved surfaces, it directly affects the effective restoring force and the period of oscillation, linking the geometry of the path to the dynamics of the motion.

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