1. Acceleration of the vehicle down the incline

Since the vehicle moves without friction down an inclined plane of inclination $\alpha$, its acceleration along the plane is

$a = g \sin \alpha.$

2. Analyze in the non-inertial frame of the vehicle

The pendulum is suspended from the roof of the vehicle, so it is convenient to study the motion in the frame of the vehicle.

In this accelerating frame, a pseudo force acts on the bob opposite to the acceleration of the vehicle. The pseudo-acceleration has magnitude

$a_p = g \sin \alpha$

and is directed up the incline.

3. Effective gravity inside the vehicle

The actual gravitational acceleration $\vec g$ acts vertically downward. The pseudo-acceleration $\vec a_p$ acts up the incline.

Resolve $\vec g$ into components relative to the incline:

• Along the incline: $g \sin \alpha$ downward along the incline
• Perpendicular to incline: $g \cos \alpha$ into the plane

The pseudo-acceleration exactly cancels the component of gravity along the incline:

$g \sin \alpha - g \sin \alpha = 0.$

So only the perpendicular component remains. Hence the effective gravitational acceleration is

$g_{\text{eff}} = g \cos \alpha.$

4. Time period of small oscillations

For a simple pendulum, if the effective gravitational acceleration is $g_{\text{eff}}$, then the time period is

$T = 2\pi \sqrt{\frac{L}{g_{\text{eff}}}}.$

Substituting $g_{\text{eff}} = g \cos \alpha$:

$T = 2\pi \sqrt{\frac{L}{g \cos \alpha}}.$

5. Compare with the options

This matches:

$\boxed{2\pi \sqrt{\frac{L}{g\cos\alpha}}}$

So the correct option is A.

6. Comparison with stored answer

Stored correct answer: A

Our derived answer: A

They agree.