The torque required to produce a constant angular acceleration in a body connected to a weightless string of length r, will be proportional to?

Options

A. (a)`1/r^2`
B. (b)r
C. (c)1/r
D. (d)`r^2`

Answer

To solve the problem of determining the relationship between torque and the parameters involved in producing a constant angular acceleration in a body connected to a weightless string of length \( r \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a body of mass \( m \) connected to a weightless string of length \( r \). The body is rotating in a horizontal plane, and we need to find the torque required to maintain a constant angular acceleration \( \alpha \).

2. **Torque Definition**:
- Torque (\( \tau \)) is defined as the product of the radius (or length of the string, \( r \)) and the force (\( F \)) applied perpendicular to the radius. Mathematically, this is expressed as:
\[ \tau = r \cdot F \]

3. **Relating Torque to Angular Acceleration**:
- According to Newton's second law for rotation, the torque is also related to the moment of inertia (\( I \)) and angular acceleration (\( \alpha \)):
\[ \tau = I \cdot \alpha \]

4. **Calculating the Moment of Inertia**:
- For a point mass \( m \) at a distance \( r \) from the axis of rotation, the moment of inertia is given by:
\[ I = m \cdot r^2 \]

5. **Substituting Moment of Inertia into Torque Equation**:
- By substituting the expression for moment of inertia into the torque equation, we have:
\[ \tau = (m \cdot r^2) \cdot \alpha \]

6. **Identifying Proportional Relationships**:
- From the equation \( \tau = m \cdot r^2 \cdot \alpha \), we can see that if \( m \) and \( \alpha \) are constant, the torque \( \tau \) is directly proportional to \( r^2 \):
\[ \tau \propto r^2 \]

7. **Conclusion**:
- Therefore, the torque required to produce a constant angular acceleration in a body connected to a weightless string of length \( r \) is proportional to the square of the length of the string.

### Final Answer:
The torque required is proportional to \( r^2 \).