Two particles revolve concentrically in a horizontal plane in the same direction. The time required to complete one revolution for particle `A` is `3min` , while for particle `B` is `1min` . The time required for `A` to complete one revolution relative to `B` is

Options

A. `2` min
B. `1` min
C. `1.5` min
D. `1.25` min

Explanation

To solve the problem of finding the time required for particle A to complete one revolution relative to particle B, we can follow these steps:

### Step 1: Identify the time periods of the particles
- Particle A completes one revolution in \( T_A = 3 \) minutes.
- Particle B completes one revolution in \( T_B = 1 \) minute.

### Step 2: Calculate the angular velocities of the particles
The angular velocity \( \omega \) is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
For particle A:
\[ \omega_A = \frac{2\pi}{T_A} = \frac{2\pi}{3} \text{ radians/min} \]
For particle B:
\[ \omega_B = \frac{2\pi}{T_B} = \frac{2\pi}{1} = 2\pi \text{ radians/min} \]

### Step 3: Determine the relative angular velocity
The relative angular velocity of particle A with respect to particle B is:
\[ \omega_{AB} = \omega_B - \omega_A \]
Substituting the values:
\[ \omega_{AB} = 2\pi - \frac{2\pi}{3} = 2\pi \left(1 - \frac{1}{3}\right) = 2\pi \cdot \frac{2}{3} = \frac{4\pi}{3} \text{ radians/min} \]

### Step 4: Calculate the time for A to complete one revolution relative to B
The time \( t \) required for A to complete one revolution relative to B can be calculated using the formula:
\[ t = \frac{2\pi}{\omega_{AB}} \]
Substituting the value of \( \omega_{AB} \):
\[ t = \frac{2\pi}{\frac{4\pi}{3}} = 2\pi \cdot \frac{3}{4\pi} = \frac{3}{2} \text{ minutes} \]

### Solution Summary
The time required for particle A to complete one revolution relative to particle B is \( \frac{3}{2} \) minutes.

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