A small mirror of area A and mass m is suspended by means of...

A small mirror of area $A$ and mass $m$ is suspended by means of a weightless thread, in vertical plane. On the mirror a beam of light of intensity $I$, is made to incident perpendicular to mirror, due to which mirror is displaced and thread makes a small angle $\alpha$ with the vertical. Assume mirror to be totally reflecting, calculate $\alpha$ : (a) $\alpha=\frac{2 I A}{c m g}$ (b) $\alpha=\frac{I A}{\mathrm{cmg}}$ (c) $\alpha=\frac{3 I A}{c m g}$ (d) $\alpha:=\frac{L A}{2 c m g}$

A small mirror of area $A$ and mass $m$ is suspended by means of a weightless thread, in vertical plane. On the mirror a beam of light of intensity $I$, is made to incident perpendicular to mirror, due to which mirror is displaced and thread makes a small angle $\alpha$ with the vertical. Assume mirror to be totally reflecting, calculate $\alpha$ :
(a) $\alpha=\frac{2 I A}{c m g}$
(b) $\alpha=\frac{I A}{\mathrm{cmg}}$
(c) $\alpha=\frac{3 I A}{c m g}$
(d) $\alpha:=\frac{L A}{2 c m g}$

Key Concepts

Force Equilibrium

Force equilibrium involves balancing the forces acting on an object to determine its state of rest or motion. In contexts where a small external force, such as that due to radiation pressure, acts on a suspended object, it is balanced by gravitational and tension forces. This principle allows us to relate the radiation force to the gravitational force and determine resulting displacements.

Small Angle Approximation

The small angle approximation is a simplifying assumption used when angles are sufficiently small such that their trigonometric functions can be approximated by their angle in radians. For example, for small angles, the tangent of the angle is approximately equal to the angle itself. This approximation is useful in solving problems involving slight displacements or deviations in equilibrium conditions.

Radiation Pressure

Radiation pressure is the pressure exerted by electromagnetic radiation on surfaces. When light is absorbed or reflected by an object, the momentum carried by the photons is transferred to the object, resulting in a force. For a perfectly reflecting surface, the radiation pressure is effectively doubled compared to an absorbing surface, making it a crucial factor in problems involving light’s mechanical effects.

Momentum Transfer from Light

The concept of momentum transfer refers to the exchange of momentum when photons interact with a surface. Specifically, during reflection, the change in momentum is twice that of the incident momentum, particularly when the incidence is perpendicular. This doubled effect is essential in calculating the force on objects due to light.

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