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. `(vecr xx vecF).vecomega`
B. `(vecr xx vecF) xx vecomega`
C. `vecr . (vecF xx vecomega)`
D. `vecr xx (vecF. vecomega)`

Explanation

To solve the problem, we need to find the power associated with the torque due to the force acting on a rotating disc. Let's break this down step by step.

### Step 1: Understand the relationship between torque and force
Torque (τ) is defined as the cross product of the position vector (r) and the force vector (F). Mathematically, this is expressed as:
\[ \tau = r \times F \]

### Step 2: Relate power to torque
Power (P) associated with torque can be expressed as:
\[ P = \tau \cdot \omega \]
where \( \omega \) is the angular velocity of the disc.

### Step 3: Substitute the expression for torque
Substituting the expression for torque into the power equation, we have:
\[ P = (r \times F) \cdot \omega \]

### Step 4: Simplify the expression
The expression \( (r \times F) \cdot \omega \) represents the power associated with the torque due to the force acting on the disc. This is the final expression for power in terms of torque and angular velocity.

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