Question

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centers. The moment of inertia and angular velocity of the first disc are $0.1kg/m^2$ and $10rad/s$ respectively while those for the second one are $0.2kg/m^2$ and $5rad/s$ respectively. At some instant, they get stuck together and start rotating as a single system about their common axis with some angular speed. The Kinetic energy of the combined system is:

• A. $\frac{2}{3}J$
• B. $\frac{10}{3}J$
• C. $\frac{5}{3}J$
• D. $\frac{20}{3}J$

Answer 1

The correct option is D

$\frac{20}{3}J$

Step 1: Given

Moment of inertia of first disc: $I_1=0.1kg/m^2$

Moment of inertia of second disc: $I_2=0.2kg/m^2$

Angular velocity of first disc: ${\omega }_1=10rad/s$

Angular velocity of second disc: ${\omega }_2=5rad/s$

Step 2: Formula Used

The angular velocity of the system will be $\omega =\frac{I_1{\omega }_1+I_2{\omega }_2}{I_1+I_2}$

Kinetic Energy of a rotating system: $KE=\frac{1}{2}I{\omega }^2$

Step 3: Calculate the angular velocity of the entire system using the formula.

$$\begin{gathered} \omega =\frac{0.1\times 10+0.2\times 5}{0.1+0.2} \\ =\frac{1+1}{0.3} \\ =\frac{2}{0.3} \\ =\frac{20}{3}rad/s \end{gathered}$$

Step 4: Calculate the kinetic energy of the rotation using the formula. The moment of inertia of the entire system will be equal to the sum of moment of inertias.

$\begin{array}{rcl}KE& =& \frac{1}{2}I{\omega }^2\\ & =& \frac{1}{2}\left(I_1+I_2\right){\omega }^2\\ & =& \frac{1}{2}\left(0.1+0.2\right){\left(\frac{20}{3}\right)}^2\\ & =& \frac{1}{2}\times 0.3\times \frac{400}{9}\\ & =& \frac{20}{3}J\end{array}$

Hence, the kinetic energy is $\begin{array}{l}\frac{20}{3}J\end{array}$.