The torque acting on a body is the rotational, analogue of

Options

A. mass of the body
B. linear kinetic energy of the body
C. linear velocity of the body
D. force in linear motion

Answer

To solve the question, we need to understand the relationship between torque, power, and their analogs in rotational and translational motion.

### Step-by-Step Solution:

1. **Understanding Torque**:
Torque (τ) is a measure of the rotational force that causes an object to rotate about an axis. It is defined as τ = r × F, where r is the distance from the axis of rotation to the point where the force is applied, and F is the force applied.

**Hint**: Recall that torque depends on both the magnitude of the force and the distance from the pivot point.

2. **Power in Rotational Motion**:
Power (P) in rotational motion is defined as the rate at which work is done. The work done by torque can be expressed as the integral of torque over angular displacement (θ). Mathematically, this is given by:
\[ P = \frac{dW}{dt} = \frac{d}{dt}(\tau \cdot \theta) \]

**Hint**: Remember that power is the derivative of work with respect to time.

3. **Relating Work Done to Angular Displacement**:
The work done by torque can be expressed as:
\[ W = \int \tau \cdot d\theta \]
Therefore, differentiating this with respect to time gives us the power:
\[ P = \tau \cdot \frac{d\theta}{dt} \]

**Hint**: Recognize that \( \frac{d\theta}{dt} \) is the angular velocity (ω).

4. **Final Expression for Power**:
Substituting \( \frac{d\theta}{dt} \) with angular velocity (ω), we get:
\[ P = \tau \cdot \omega \]
This shows that the power associated with torque is indeed the product of torque and angular speed.

**Hint**: This relationship mirrors the translational power equation \( P = F \cdot v \).

5. **Translational Analog**:
In translational motion, power is given by the product of force (F) and velocity (v):
\[ P = F \cdot v \]
Here, force is the translational analog of torque, and velocity is the translational analog of angular velocity.

**Hint**: Compare the forms of the equations for rotational and translational power to see the analogs.

### Conclusion:
The torque acting on a body is the rotational analog of force in translational motion, and the power associated with torque is given by the product of torque and angular speed, paralleling the relationship in linear motion.