A particle is at a distance r from the axis of rotation. A given torque `tau` produces some angular acceleration in it. If the mass of the particle is doubled and its distance from the axis is halved, the value of torque to produce the same angular acceleration is -

Options

A. `tau//2`
B. `tau`
C. `2tau`
D. `4 tau`

Answer

To solve the problem step by step, let's break it down clearly.

### Given:
- Initial distance from the axis of rotation: \( r \)
- Initial mass of the particle: \( m \)
- Initial torque: \( \tau \)
- Initial angular acceleration: \( \alpha \)

### Step 1: Write the expression for torque
The torque (\( \tau \)) acting on a particle is given by the formula:
\[ \tau = I \alpha \]
where \( I \) is the moment of inertia.

### Step 2: Calculate the initial moment of inertia
The moment of inertia (\( I \)) for a point mass is given by:
\[ I = m r^2 \]
Thus, the initial torque can be expressed as:
\[ \tau = m r^2 \alpha \]

### Step 3: Analyze the new conditions
Now, the mass of the particle is doubled, and the distance from the axis is halved:
- New mass: \( 2m \)
- New distance: \( \frac{r}{2} \)

### Step 4: Calculate the new moment of inertia
The new moment of inertia (\( I' \)) can be calculated as:
\[ I' = (2m) \left(\frac{r}{2}\right)^2 = 2m \cdot \frac{r^2}{4} = \frac{mr^2}{2} \]

### Step 5: Write the expression for the new torque
Let the new torque be \( \tau' \). The expression for the new torque is:
\[ \tau' = I' \alpha = \left(\frac{mr^2}{2}\right) \alpha \]

### Step 6: Relate the new torque to the initial torque
From the initial torque equation, we have:
\[ \tau = m r^2 \alpha \]
Now, substituting this into the new torque equation:
\[ \tau' = \frac{1}{2} \tau \]

### Conclusion
Thus, the value of the new torque \( \tau' \) required to produce the same angular acceleration is:
\[ \tau' = \frac{\tau}{2} \]

### Final Answer
The value of torque to produce the same angular acceleration is \( \frac{\tau}{2} \).

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