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. `1/r^2`
B. r
C. 1/r
D. `r^2`

Answer

To determine the relationship between the torque required to produce 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 Torque**:
Torque (\( \tau \)) is defined as the product of the force (\( F \)) applied at a distance (\( r \)) from the pivot point. Mathematically, it can be expressed as:
\[ \tau = r \times F \]

2. **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 (\( \alpha \)):
\[ \tau = I \alpha \]

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

4. **Substituting Moment of Inertia**:
We can substitute the expression for moment of inertia into the torque equation:
\[ \tau = (M r^2) \alpha \]

5. **Analyzing the Relationship**:
If we assume that the mass \( M \) and angular acceleration \( \alpha \) are constant, we can analyze the relationship between torque and the length of the string \( r \):
\[ \tau = M \alpha r^2 \]
From this equation, we can see that torque (\( \tau \)) is directly proportional to \( r^2 \):
\[ \tau \propto r^2 \]

6. **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 \( r^2 \).

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

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