Question

A rigid spherical body is spinning around an axis without any external torque. Due to rise of temperature its volume increases by $3\%$. Then percentage change in its angular speed is:

• A. $-2\%$
• B. $-1\%$
• C. $-3\%$
• D. $1\%$

Answer 1

Answer: (A)

$-2\%$

Since no external torque is applied,

$I_1{\omega }_1=I_2{\omega }_2$

or, $\frac{2}{5}m{R}_1^2\times {\omega }_1=\frac{2}{5}m{R}_2^2{\omega }_2$ ............................................. (i)

Now, the increase in volume may be accounted for as:

$\frac{\frac{4}{3}\pi R_2^3-\frac{4}{3}\pi R_1^3}{\frac{4}{3}\pi R_1^3}=\frac{3}{100}$

or, $(\frac{R_2}{R_1}{)}^3=(\frac{103}{100})$

So, using eqn (i), we get:

$\left(\frac{{\omega }_1}{{\omega }_2}\right)=(\frac{r_2}{r_1}{)}^2$$=(\frac{103}{100}{)}^{\frac{2}{3}}$

Requaired % change $=\frac{{\omega }_2-{\omega }_1}{{\omega }_1}\times 100$

$=(\frac{{\omega }_2}{{\omega }_1}-1)\times 100$

$=\left(\right(\frac{100}{103}{)}^{\frac{2}{3}}-1)\times 100$

$=-1.9512\%$