Question

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are $0.1{\text{kg.m}}^2\text{and}10{\text{rad s}}^{–1}$ respectively while those for the second one are $0.2{\text{kg m}}^2\text{and}5{\text{rad s}}^{–1}$ 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. $\frac{10}{3}\text{J}$
• B. $\frac{20}{3}\text{J}$
• C. $\frac{5}{3}\text{J}$
• D. $\frac{2}{3}\text{J}$

Answer 1

The correct option is B $\frac{20}{3}\text{J}$

Initial angular momentum of discs be $I_1{\omega }_1\&I_2{\omega }_2$ respectively
Let $\omega$ be angular speed of the combined system.
Final angular momentum of discs be $I_1\omega \&I_2\omega$ respectively
According to conservation of angular momentum
$(I_1+I_2)\omega =I_1{\omega }_1+I_2{\omega }_2$
$\Rightarrow \omega =\frac{I_1{\omega }_1+I_2{\omega }_2}{I_1+I_2}=\frac{0.1\times 10+0.2\times 5}{0.1+0.2}=\frac{20}{3}$
Final rotational kinetic energy
$K_f=\frac{1}{2}{I}_1{\omega }^2+\frac{1}{2}{I}_2{\omega }^2=\frac{1}{2}(0.1+0.2)\times {\left(\frac{20}{3}\right)}^2$
$\Rightarrow K_f=\frac{20}{3}\text{J}$