The radius of the circular path of a particle is doubled but its frecuency of rotation remains unchanged . If the initial centripetal force be F, then the final centripetal force will be

Options

A. (a) F
B. (b)`(F)/(2)`
C. (c)4F
D. (d)2F

Answer

To solve the problem step by step, we will analyze the relationship between the centripetal force, radius, and frequency of rotation.

### Step-by-Step Solution:

1. **Understanding Centripetal Force**:
The centripetal force \( F \) acting on a particle moving in a circular path is given by the formula:
\[ F = m \cdot a_c \]
where \( a_c \) is the centripetal acceleration. The centripetal acceleration can be expressed in terms of the radius \( r \) and angular velocity \( \omega \):
\[ a_c = \frac{v^2}{r} = r \cdot \omega^2 \]
Thus, the centripetal force can also be expressed as:
\[ F = m \cdot r \cdot \omega^2 \]

2. **Initial Conditions**:
Let the initial radius be \( r_1 \) and the initial centripetal force be \( F \). Therefore, we can write:
\[ F = m \cdot r_1 \cdot \omega^2 \]

3. **Doubling the Radius**:
According to the problem, the radius is doubled, so the new radius \( r_2 \) is:
\[ r_2 = 2 \cdot r_1 \]

4. **Frequency Remains Unchanged**:
Since the frequency of rotation remains unchanged, the angular velocity \( \omega \) also remains constant. Thus, we have:
\[ \omega_2 = \omega_1 = \omega \]

5. **Calculating the New Centripetal Force**:
The new centripetal force \( F_2 \) when the radius is doubled can be expressed as:
\[ F_2 = m \cdot r_2 \cdot \omega^2 \]
Substituting \( r_2 \):
\[ F_2 = m \cdot (2 \cdot r_1) \cdot \omega^2 \]
This simplifies to:
\[ F_2 = 2 \cdot (m \cdot r_1 \cdot \omega^2) \]

6. **Relating to Initial Centripetal Force**:
From our initial condition, we know that:
\[ F = m \cdot r_1 \cdot \omega^2 \]
Therefore, substituting this into our expression for \( F_2 \):
\[ F_2 = 2 \cdot F \]

7. **Final Result**:
The final centripetal force when the radius is doubled while keeping the frequency constant is:
\[ F_2 = 2F \]

### Conclusion:
The final centripetal force will be \( 2F \).

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