Question

If the torque remains constant while the angle changes, the work done is equal to:

• A. ratio of torque and angular displacement
• B. ratio of the angular displacement and square root of torque
• C. product of torque and angular displacement
• D. product of torque and square root of angular displacement

Hint:

Many concepts in rotational mechanics have direct analogues in linear mechanics. - Position \(x\) \(\leftrightarrow\) Angle \(\theta\) - Velocity \(v\) \(\leftrightarrow\) Angular Velocity \(\omega\) - Mass \(m\) \(\leftrightarrow\) Moment of Inertia \(I\) - Force \(F\) \(\leftrightarrow\) Torque \(\tau\) - Momentum \(p=mv\) \(\leftrightarrow\) Angular Momentum \(L=I\omega\) - Work \(W=Fd\) \(\leftrightarrow\) Work \(W=\tau\theta\)

Answer 1

The Correct Option is C

Step 1: Recall the definition of work done by a constant force in linear motion. Work \(W\) is defined as the product of the force \(F\) and the displacement \(d\) in the direction of the force: \(W = F \cdot d\).
Step 2: Use the rotational analogues for force and displacement. In rotational motion, the analogue of force is torque (\(\tau\)), and the analogue of linear displacement is angular displacement (\(\theta\)).
Step 3: Formulate the expression for rotational work. By analogy, the work done by a constant torque is the product of the torque and the angular displacement. \[ W = \tau \theta \] This corresponds to option (3). If the torque is not constant, the work done is found by integrating the torque over the angular displacement: \(W = \int \tau \, d\theta\).