A second pendulum is attached to roof of a car that is sliding down along a smooth inclined plane of inclination `60^(@)`. Its perios of oscillation is

Options

A. `2s`
B. `sqrt(2)s`
C. `2sqrt(2)s`
D. `1s`

Explanation

To solve the problem of finding the period of oscillation of a second pendulum attached to the roof of a car sliding down a smooth inclined plane at an angle of 60 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Pendulum**: A second pendulum is defined as one that has a period of 2 seconds when in a gravitational field. The formula for the period \(T\) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity.

2. **Identifying the Effective Gravity**: When the car is sliding down the incline, the effective acceleration due to gravity acting on the pendulum is not just \(g\) but is modified by the inclination of the plane. The effective gravitational acceleration \(g'\) acting on the pendulum can be expressed as:
\[ g' = g \cos(\theta) \]
where \(\theta\) is the angle of inclination (60 degrees in this case).

3. **Substituting into the Period Formula**: We substitute \(g'\) into the period formula:
\[ T = 2\pi \sqrt{\frac{L}{g \cos(\theta)}} \]

4. **Simplifying the Expression**: Since we know that for a second pendulum, \(L = \frac{g}{\pi^2}\), we can substitute this value into the formula:
\[ T = 2\pi \sqrt{\frac{\frac{g}{\pi^2}}{g \cos(60^\circ)}} \]

5. **Calculating \(\cos(60^\circ)\)**: We know that \(\cos(60^\circ) = \frac{1}{2}\). Substituting this value gives:
\[ T = 2\pi \sqrt{\frac{\frac{g}{\pi^2}}{g \cdot \frac{1}{2}}} \]

6. **Canceling Terms**: The \(g\) in the numerator and denominator cancels out:
\[ T = 2\pi \sqrt{\frac{1}{\frac{1}{2} \pi^2}} = 2\pi \sqrt{\frac{2}{\pi^2}} = 2\pi \cdot \frac{\sqrt{2}}{\pi} \]

7. **Final Simplification**: This simplifies to:
\[ T = 2\sqrt{2} \text{ seconds} \]

### Final Answer:
The period of oscillation of the second pendulum attached to the roof of the car is \(2\sqrt{2}\) seconds.