Question

If I is the moment of inertia of a ring having $\alpha$ coefficient of linear expansion, then the change in $I$ corresponding to a small change in temperature $\Delta T$ is

• A. $\alpha I\Delta T$
• B. $\frac{1}{2}\alpha I\Delta T$
• C. $2\alpha I\Delta T$
• D. $3\alpha I\Delta T$

Answer 1

The correct option is C $2\alpha I\Delta T$

Moment of inertia $I=M{R}^2$
or $\frac{dI}{dR}=2MR$
or $\Delta I=2MR.\Delta R...\left(1\right)$
By the definition of linear co-efficient of expansion
$\alpha =\frac{R-R_0}{R.\Delta T}=\frac{\Delta R}{R\Delta T}$
from equation (1)
$\Delta I=2MR.\alpha R\Delta T$
or $\Delta I=2I\alpha \Delta T$.