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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:
When disc $B$ is brought in contact with disc $A$, they acquire a common angular velocity in time $t$. The average frictional torque on one disc by the other during this period is
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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:
When disc $B$ is brought in contact with disc $A$, they acquire a common angular velocity in time $t$. The average frictional torque on one disc by the other during this period is
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2127 Upvotes
Verified Answer
The correct answer is:
$\frac{2 I \omega}{3 t}$
$\frac{2 I \omega}{3 t}$
Let $\omega^{\prime}$ be the common velocity. Then from conservation of angular momentum, we have
$$
\begin{aligned}
(i+2 I) \omega^{\prime} & =I(2 \omega)+2 I(\omega) \\
\omega^{\prime} & =\frac{4}{3} \omega
\end{aligned}
$$
From the equation,
Angular impulse = change in angular momentum for any of the disc, we have
$$
\begin{aligned}
\tau t & =I(2 \omega)-I\left(\frac{4}{3} \omega\right)=\frac{2 I \omega}{3} \\
\tau & =\frac{2 I \omega}{3 t}
\end{aligned}
$$
$\therefore$ Option (a) is correct.
$$
\begin{aligned}
(i+2 I) \omega^{\prime} & =I(2 \omega)+2 I(\omega) \\
\omega^{\prime} & =\frac{4}{3} \omega
\end{aligned}
$$
From the equation,
Angular impulse = change in angular momentum for any of the disc, we have
$$
\begin{aligned}
\tau t & =I(2 \omega)-I\left(\frac{4}{3} \omega\right)=\frac{2 I \omega}{3} \\
\tau & =\frac{2 I \omega}{3 t}
\end{aligned}
$$
$\therefore$ Option (a) is correct.
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