A simple pendulum suspended from the ceilling of a stationary trolley has a length l. its period of oscillation is `2pisqrt(l//g)`. What will be its period of oscillation if the trolley moves forward with an acceleration f ?

A. `2pisqrt(l/(f-g))`
B. `2pisqrt(l/(f+g))`
C. `2pisqrt(l/((f^(2)+g^(2))^(1//2)))`
D. `2pisqrt(l/(f^(2)-g^(2)))`

Options

A. `2pisqrt(l/(f-g))`
B. `2pisqrt(l/(f+g))`
C. `2pisqrt(l/((f^(2)+g^(2))^(1//2)))`
D. `2pisqrt(l/(f^(2)-g^(2)))`

Explanation

To solve the problem of finding the period of oscillation of a simple pendulum suspended from a trolley that is accelerating, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Condition**:
- The period of a simple pendulum when the trolley is stationary is given by the formula:
\[ T_0 = 2\pi \sqrt{\frac{l}{g}} \]
where \( l \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

2. **Analyze the New Condition**:
- When the trolley accelerates forward with an acceleration \( f \), a pseudo force acts on the pendulum bob in the opposite direction of the trolley's acceleration.

3. **Determine the Effective Acceleration**:
- The effective acceleration acting on the pendulum bob is a combination of the gravitational acceleration \( g \) acting downward and the pseudo force due to the trolley's acceleration \( f \) acting horizontally.
- Since these two accelerations are perpendicular to each other, we can find the resultant acceleration using the Pythagorean theorem:
\[ a_{\text{net}} = \sqrt{f^2 + g^2} \]

4. **Calculate the New Period of Oscillation**:
- The period of oscillation for the pendulum in the accelerating frame can be expressed as:
\[ T = 2\pi \sqrt{\frac{l}{a_{\text{net}}}} \]
- Substituting the expression for \( a_{\text{net}} \):
\[ T = 2\pi \sqrt{\frac{l}{\sqrt{f^2 + g^2}}} \]
- This can be rewritten as:
\[ T = 2\pi \sqrt{\frac{l}{(f^2 + g^2)^{1/2}}} \]

5. **Select the Correct Option**:
- From the options provided, the correct expression for the period of oscillation is:
\[ T = 2\pi \sqrt{\frac{l}{(f^2 + g^2)^{1/2}}} \]
- Thus, the correct answer is option C.

### Final Answer:
**C.** \( 2\pi \sqrt{\frac{l}{(f^2 + g^2)^{1/2}}} \)

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