Write the formula of work done by torque in a rotational rigid body about a fixed axis.

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(N/A) For a rigid body rotating about a fixed axis,the work done $W$ by a torque $\tau$ through an angular displacement $d\theta$ is given by the integral of the torque over the angular displacement.
The infinitesimal work done is $dW = \tau \cdot d\theta$.
For a finite angular displacement from $\theta_1$ to $\theta_2$,the total work done is:
$W = \int_{\theta_1}^{\theta_2} \tau \, d\theta$.
If the torque $\tau$ is constant,the formula simplifies to:
$W = \tau (\theta_2 - \theta_1) = \tau \Delta \theta$.

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Match the linear motion formulas in Column-$I$ with their corresponding rotational motion formulas in Column-$II$.
Column-$I$ Column-$II$
$(1)$ $W = F \Delta x$ $(a)$ $P = \tau \omega$
$(2)$ $P = Fv$ $(b)$ $W = \tau \Delta \theta$
$(c)$ $L = I \omega$

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