Two particles of equal masses are revolving in circular paths of radii 2 m and 8 m respectively with the same period .Then ratio of their centripetal forces is ,

Options

A. `8:1`
B. `2:1`
C. `1:4`
D. `1:1`

Explanation

To solve the problem of finding the ratio of centripetal forces for two particles revolving in circular paths of different radii but with the same period, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Data:**
- Let the radius of the first particle (r₁) = 2 m
- Let the radius of the second particle (r₂) = 8 m
- Both particles have equal mass (m) and the same period (T).

2. **Centripetal Force Formula:**
The centripetal force (F) acting on a particle moving in a circular path is given by the formula:
\[ F = \frac{mv^2}{r} \]
where:
- \( m \) = mass of the particle
- \( v \) = linear velocity of the particle
- \( r \) = radius of the circular path

3. **Relate Velocity to Period:**
The linear velocity (v) of a particle in circular motion can be expressed in terms of the radius (r) and the period (T):
\[ v = \frac{2\pi r}{T} \]

4. **Calculate Velocities for Both Particles:**
For particle 1 (radius r₁):
\[ v_1 = \frac{2\pi r_1}{T} = \frac{2\pi \cdot 2}{T} = \frac{4\pi}{T} \]
For particle 2 (radius r₂):
\[ v_2 = \frac{2\pi r_2}{T} = \frac{2\pi \cdot 8}{T} = \frac{16\pi}{T} \]

5. **Substitute Velocities into Centripetal Force Formula:**
Now, we can find the centripetal forces for both particles:
For particle 1:
\[ F_1 = \frac{m v_1^2}{r_1} = \frac{m \left(\frac{4\pi}{T}\right)^2}{2} = \frac{m \cdot \frac{16\pi^2}{T^2}}{2} = \frac{8m\pi^2}{T^2} \]
For particle 2:
\[ F_2 = \frac{m v_2^2}{r_2} = \frac{m \left(\frac{16\pi}{T}\right)^2}{8} = \frac{m \cdot \frac{256\pi^2}{T^2}}{8} = \frac{32m\pi^2}{T^2} \]

6. **Find the Ratio of the Centripetal Forces:**
Now, we can find the ratio of the centripetal forces:
\[ \frac{F_1}{F_2} = \frac{\frac{8m\pi^2}{T^2}}{\frac{32m\pi^2}{T^2}} = \frac{8}{32} = \frac{1}{4} \]

### Final Answer:
The ratio of the centripetal forces \( F_1 : F_2 \) is \( 1 : 4 \).

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