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$\mathbf{P}_{17 \text { - } 19}$ : Paragraph for Questions Nos. 17 to 19 Two discs $A$ and $B$ are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc $A$ is imparted an initial angular velocity $2 \omega$ using the entire potential energy of a spring compressed by a distance $x_1$. Disc $B$ is imparted an angular velocity $\omega$ by a spring having the same spring constant and compressed by a distance $x_2$. Both the discs rotate in the clockwise direction.Question:
The loss of kinetic energy during the above process is
Options:
$\mathbf{P}_{17 \text { - } 19}$ : Paragraph for Questions Nos. 17 to 19 Two discs $A$ and $B$ are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc $A$ is imparted an initial angular velocity $2 \omega$ using the entire potential energy of a spring compressed by a distance $x_1$. Disc $B$ is imparted an angular velocity $\omega$ by a spring having the same spring constant and compressed by a distance $x_2$. Both the discs rotate in the clockwise direction.Question:
The loss of kinetic energy during the above process is
Solution:
2317 Upvotes
Verified Answer
The correct answer is:
$\frac{i \omega^2}{3}$
$\frac{i \omega^2}{3}$
Loss of kinetic energy $=K_i-K_f$
$$
\begin{aligned}
& =\left\{\frac{1}{2} I(2 \omega)^2+\frac{1}{2}(2 I)(\omega)^2\right\}-\frac{1}{2}(3 I)\left(\frac{4}{3} \omega\right)^2 \\
& =\frac{1}{3} I \omega^2
\end{aligned}
$$
$\therefore$ Option (b) is correct.
$$
\begin{aligned}
& =\left\{\frac{1}{2} I(2 \omega)^2+\frac{1}{2}(2 I)(\omega)^2\right\}-\frac{1}{2}(3 I)\left(\frac{4}{3} \omega\right)^2 \\
& =\frac{1}{3} I \omega^2
\end{aligned}
$$
$\therefore$ Option (b) is correct.
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