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, we need to analyze the relationship between torque, moment of inertia, and angular acceleration. Let's break it down step by step.

### Step 1: Understand the relationship between torque and angular acceleration
Torque (\( \tau \)) is related to moment of inertia (\( I \)) and angular acceleration (\( \alpha \)) by the formula:
\[ \tau = I \cdot \alpha \]
We will use this relationship for two different scenarios.

### Step 2: Define the initial conditions
Let:
- The initial mass of the particle be \( m \).
- The initial distance from the axis of rotation be \( r \).
- The initial torque be \( \tau \).
- The initial moment of inertia \( I_1 \) can be expressed as:
\[ I_1 = m \cdot r^2 \]
Thus, the initial torque can be expressed as:
\[ \tau = I_1 \cdot \alpha = m \cdot r^2 \cdot \alpha \]

### Step 3: Define the new conditions after changes
Now, according to the problem:
- The mass of the particle is doubled: \( m' = 2m \).
- The distance from the axis is halved: \( r' = \frac{r}{2} \).

### Step 4: Calculate the new moment of inertia
The new moment of inertia \( I_2 \) can be calculated as:
\[ I_2 = m' \cdot (r')^2 = (2m) \cdot \left(\frac{r}{2}\right)^2 = (2m) \cdot \left(\frac{r^2}{4}\right) = \frac{2m \cdot r^2}{4} = \frac{m \cdot r^2}{2} \]

### Step 5: Relate the new torque to the angular acceleration
We want to find the new torque \( \tau_2 \) that produces the same angular acceleration \( \alpha \):
\[ \tau_2 = I_2 \cdot \alpha = \left(\frac{m \cdot r^2}{2}\right) \cdot \alpha \]

### Step 6: Compare the two torques
Now we can compare the initial torque \( \tau \) and the new torque \( \tau_2 \):
From the initial torque:
\[ \tau = m \cdot r^2 \cdot \alpha \]
From the new torque:
\[ \tau_2 = \left(\frac{m \cdot r^2}{2}\right) \cdot \alpha \]

### Step 7: Express \( \tau_2 \) in terms of \( \tau \)
Now, we can express \( \tau_2 \) in terms of \( \tau \):
\[ \tau_2 = \frac{1}{2} \cdot (m \cdot r^2 \cdot \alpha) = \frac{\tau}{2} \]

### Conclusion
Thus, the value of the torque required to produce the same angular acceleration after doubling the mass and halving the distance from the axis is:
\[ \tau_2 = \frac{\tau}{2} \]

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

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