Three forces of magnitude F, F and `sqrt(2F)` are acting on the periphery of a disc of mass m and radius R as shown in the figure. The net torque about the centre of the disc is

Options

A. `FRhatK`
B. `-FRhatK`
C. `3FRhatK`
D. `-3FRhatK`

Answer

To solve the problem of calculating the net torque about the center of the disc due to the three forces acting on it, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces and Their Directions**:
- We have three forces: \( F \), \( F' \) (which we can denote as \( F \) for simplicity), and \( \sqrt{2F} \).
- The force \( \sqrt{2F} \) is acting at an angle of \( 45^\circ \) with respect to the radius of the disc.

2. **Calculate the Torque Due to Each Force**:
- Torque (\( \tau \)) is calculated using the formula:
\[ \tau = r \times F \times \sin(\theta) \]
- Where \( r \) is the radius of the disc, \( F \) is the force applied, and \( \theta \) is the angle between the force and the radius.

3. **Torque from the Force \( F \)**:
- The torque due to the force \( F \) acting tangentially (assuming it acts perpendicular to the radius):
\[ \tau_1 = R \cdot F \cdot \sin(90^\circ) = R \cdot F \]

4. **Torque from the Force \( F' \)**:
- The torque due to the second force \( F' \) (acting in the opposite direction):
\[ \tau_2 = R \cdot F' \cdot \sin(90^\circ) = R \cdot F \]
- This torque will act in the opposite direction, so we consider it as negative.

5. **Torque from the Force \( \sqrt{2F} \)**:
- The torque due to the force \( \sqrt{2F} \) acting at an angle of \( 45^\circ \):
\[ \tau_3 = R \cdot \sqrt{2F} \cdot \sin(45^\circ) = R \cdot \sqrt{2F} \cdot \frac{\sqrt{2}}{2} = R \cdot F \]
- This torque will also act in the clockwise direction.

6. **Calculate the Net Torque**:
- Now, we can sum up the torques considering their directions:
\[ \text{Net Torque} = \tau_1 - \tau_2 + \tau_3 = R \cdot F - R \cdot F + R \cdot F = R \cdot F \]
- Since \( \tau_2 \) is negative, it cancels out \( \tau_1 \), leaving us with the torque from \( \sqrt{2F} \).

7. **Final Result**:
- The net torque about the center of the disc is:
\[ \tau_{\text{net}} = -R \cdot F \hat{k} \]
- The negative sign indicates the direction of the torque.