Two particles of mass M and m are moving in a circle of radii R and r. if their time period are the same, what will be the ratio of their linear velocities?

Options

A. `MR:mr`
B. `M:m`
C. `R:r`
D. `1:1`

Explanation

To solve the problem, we need to find the ratio of the linear velocities of two particles moving in circular paths with the same time period. Let's denote the two particles as Particle 1 (mass M, radius R) and Particle 2 (mass m, radius r).

### Step 1: Understanding Linear Velocity
The linear velocity \( v \) of an object moving in a circular path is given by the formula:
\[ v = \frac{2\pi R}{T} \]
where \( R \) is the radius of the circular path and \( T \) is the time period of the motion.

### Step 2: Write the Linear Velocities for Both Particles
For Particle 1 (mass M, radius R):
\[ v_1 = \frac{2\pi R}{T} \]

For Particle 2 (mass m, radius r):
\[ v_2 = \frac{2\pi r}{T} \]

### Step 3: Find the Ratio of Linear Velocities
To find the ratio of the linear velocities \( v_1 \) and \( v_2 \), we will divide the two equations:
\[ \frac{v_1}{v_2} = \frac{\frac{2\pi R}{T}}{\frac{2\pi r}{T}} \]

### Step 4: Simplifying the Ratio
The \( 2\pi \) and \( T \) terms cancel out:
\[ \frac{v_1}{v_2} = \frac{R}{r} \]

### Conclusion
Thus, the ratio of the linear velocities of the two particles is:
\[ \frac{v_1}{v_2} = \frac{R}{r} \]