At a temperature ' t ' the moment of inertia of a body is I....

At a temperature ' $t$ ' the moment of inertia of a body is $I$. When the temperature of the body is increased from $t+\Delta t$, its moment of inertia also increases from $I$ to $(I+\Delta l) .$ If coefficient of linear expansion of the body is $\alpha$ ', then the ratio $\frac{\Delta I}{I}$ is : (a) $\frac{\Delta t}{t}$ (b) $\frac{2 \Delta t}{t}$ (c) $\alpha \Delta t$ (d) $2 \alpha \Delta t$

At a temperature ' $t$ ' the moment of inertia of a body is $I$. When the temperature of the body is increased from $t+\Delta t$, its moment of inertia also increases from $I$ to $(I+\Delta l) .$ If coefficient of linear expansion of the body is $\alpha$ ', then the ratio $\frac{\Delta I}{I}$ is :
(a) $\frac{\Delta t}{t}$
(b) $\frac{2 \Delta t}{t}$
(c) $\alpha \Delta t$
(d) $2 \alpha \Delta t$

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Key Concepts

Moment of Inertia

This is a measure of an object's resistance to changes in its rotational motion about a specific axis. It depends on both the mass of the object and the distribution of that mass relative to the axis. Any change in the object's dimensions can lead to a change in its moment of inertia, particularly when the distances from the axis are altered.

Coefficient of Linear Thermal Expansion

This coefficient quantifies the degree to which a material expands per unit change in temperature. It represents the fractional change in length for a given temperature change, serving as a fundamental parameter in understanding how thermal effects can modify the physical dimensions of an object.

Geometric Scaling Under Thermal Expansion

When an object undergoes thermal expansion, each of its linear dimensions increases by a factor that depends on the temperature change and the coefficient of linear expansion. Since moment of inertia depends on the square of these linear dimensions, its change is governed by the square of the linear expansion factor. For small temperature changes, this results in an approximately doubled effect relative to the linear change, highlighting the significance of scaling laws in thermal physics.

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