Question

A uniform circular disc of mass $1\text{kg}$ and radius $0.5\text{m}$ is rotating about its axis with a constant angular velocity $5\text{rad/sec}$. Two birds, each of mass $0.25\text{kg}$ sit gently at the opposite ends of the diameter of the disc. The final angular velocity of the disc will be

• A. $5\text{rad/sec}$
• B. $7.5\text{rad/sec}$
• C. $2.5\text{rad/sec}$
• D. $1\text{rad/sec}$

Answer 1

The correct option is C $2.5\text{rad/sec}$

Let the final angular velocity of disc is ${\omega }_2$
Given, mass of disc, $M=1\text{kg}$
radius of disc, $r=0.5\text{m}$
Initial angular velocity of disc, ${\omega }_1=5\text{rad/sec}$.
Mass of bird, $m=0.25\text{kg}$

Since, net external torque on the system is zero. Therefore, angular momentum will remain conserved.
Thus,
$L_i=L_f$
$I_1{\omega }_1=I_2{\omega }_2$
$\Rightarrow \left(\frac{1}{2}M{r}^2\right){\omega }_1=(\frac{1}{2}M{r}^2+2m{r}^2){\omega }_2$
$\Rightarrow (\frac{1}{2}\times 1\times {0.5}^2)\times 5=(\frac{1}{2}\times 1\times {0.5}^2+2\times 0.25\times {0.5}^2)\times w_2$
$\Rightarrow {\omega }_2=2.5\text{rad/sec}$
So, the final angular velocity of the disc will be $2.5\text{rad/sec}$.