Let g be the acceleration due to gravity at earth's surface and k be the rotational K . E. of ea

step 1 The question it is given that let G be the acceleration due to the gravity at Earth's surface an

Let g be the acceleration due to gravity at earth's surface...

Let $g$ be the acceleration due to gravity at earth's surface and $\mathrm{k}$ be the rotational $\mathrm{K} . \mathrm{E}$. of earth suppose the earth's radius decreases by $2 \%$ keeping alt other quantities same then (A) $g$ decreases by $2 \%$ and $\mathrm{K}$ decreases by $4 \%$ (B) $g$ decreases by $4 \%$ and $K$ increases by $2 \%$ (C) $\mathrm{g}$ increases by $4 \%$ and $\mathrm{K}$ increases by $4 \%$ (D) g decreases by $4 \%$ and $\mathrm{K}$ increases by $4 \%$

Let $g$ be the acceleration due to gravity at earth's surface and $\mathrm{k}$ be the rotational $\mathrm{K} . \mathrm{E}$. of earth suppose the earth's radius decreases by $2 \%$ keeping alt other quantities same then
(A) $g$ decreases by $2 \%$ and $\mathrm{K}$ decreases by $4 \%$
(B) $g$ decreases by $4 \%$ and $K$ increases by $2 \%$
(C) $\mathrm{g}$ increases by $4 \%$ and $\mathrm{K}$ increases by $4 \%$
(D) g decreases by $4 \%$ and $\mathrm{K}$ increases by $4 \%$

Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy is given by the formula K = ½ I ?², where I is the moment of inertia and ? is the angular velocity. When the moment of inertia decreases and the angular velocity increases (as required by the conservation of angular momentum), the net effect on the kinetic energy depends on the precise changes in these quantities, typically resulting in an increase in kinetic energy if the increase in angular velocity more than compensates for the decrease in moment of inertia.

Gravitational Acceleration

Gravitational acceleration at a body’s surface is determined by the formula g = GM/R², where G is the gravitational constant, M is the mass of the body, and R is its radius. This inverse square relationship means that if the radius decreases while mass remains constant, g will increase proportionally by the square of the factor by which R is reduced.

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to changes in its rotation and depends on how its mass is distributed relative to its axis of rotation, typically scaling with the square of the radius (I ? R²) for simple symmetric objects. A reduction in radius results in a decrease in the moment of inertia if the mass distribution remains similar.

Conservation of Angular Momentum

In the absence of external torques, the angular momentum of a rotating system is conserved. Thus, if a rotating body’s moment of inertia decreases (for instance, due to a reduction in radius), its angular velocity must increase to maintain constant angular momentum, influencing other rotational characteristics such as kinetic energy.

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