Question

When the temperature of a body increases form t to $t+\Delta t$, its moment of inertia increases from I to $I+\Delta I$. The coefficient of linear expansion of the body is $\alpha$ . The ratio $\Delta I/I$ is :

• A. $\Delta t/t$
• B. $2\Delta t/t$
• C. $\alpha \Delta t$
• D. $2\alpha \Delta t$

Answer 1

The correct option is D $2\alpha \Delta t$

Moment of inertia is given by $m{r}^2$
where $r=$ perpendicular distance of the centre of mass of the body from the rotational axis.
If $I_1$ and$I_2$ are initial and final moment of inertia for a change in temperature $dt$, we can write,
$\frac{I_2-I_1}{I_1}=\frac{{r_2}^2-{r_1}^2}{{r_1}^2}=\left(\right(1+\alpha dt{)}^2-1)=2\alpha dt+{\alpha }^2{dt}^2$
Neglecting value of ${\alpha }^2$, we get
$\frac{I_2-I_1}{I_1}=2\alpha dt$