A disc is rotating with angular velocity `omega`. A force F acts at a point whose position vector with respect to the axis of rotation is r. The power associated with torque due to the force is given by

Options

A. `(r xxF). omega`
B. `(r xx F) xx omega`
C. `r xx (F. omega)`
D. `r. (F xx omega)`

Explanation

To find the power associated with the torque due to the force acting on a rotating disc, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concepts**:
- In rotational motion, the analogous quantity to linear force is torque (τ).
- The power (P) in linear motion is given by the dot product of force (F) and velocity (v). For rotational motion, we will use torque and angular velocity (ω).

2. **Define Torque**:
- Torque (τ) is defined as the cross product of the position vector (r) and the force (F):
\[ \tau = \mathbf{r} \times \mathbf{F} \]

3. **Relate Power to Torque and Angular Velocity**:
- The power associated with the torque can be expressed as:
\[ P = \tau \cdot \omega \]
- Here, τ is the torque and ω is the angular velocity vector.

4. **Substitute for Torque**:
- Substitute the expression for torque into the power equation:
\[ P = (\mathbf{r} \times \mathbf{F}) \cdot \mathbf{\omega} \]

5. **Final Expression**:
- Thus, the power associated with the torque due to the force acting on the disc is given by:
\[ P = (\mathbf{r} \times \mathbf{F}) \cdot \mathbf{\omega} \]

### Conclusion:
The power associated with the torque due to the force acting on the rotating disc is given by the equation:
\[ P = (\mathbf{r} \times \mathbf{F}) \cdot \mathbf{\omega} \]