Introduction:
A simple pendulum consists of a mass (bob) attached to a string or rod of fixed length and is free to swing back and forth. When a pendulum is suspended from the ceiling of a moving car, the motion of the car affects the motion of the pendulum. In this case, we need to find the horizontal acceleration of the car in terms of the angle θ that the pendulum makes with the vertical.

Understanding the Forces:
When the car accelerates horizontally, two forces act on the pendulum:
1. The tension force (T) in the string or rod, which always acts towards the pivot point of the pendulum.
2. The inertial force (F) acting horizontally due to the acceleration of the car.

Resolving Forces:
We can resolve the forces acting on the pendulum into their horizontal and vertical components:
1. The tension force (T) can be resolved into Tsinθ (horizontal component) and Tcosθ (vertical component).
2. The inertial force (F) can be resolved into Fsinθ (horizontal component) and Fcosθ (vertical component).

Equating Forces:
The horizontal component of the tension force (Tsinθ) must be equal to the horizontal component of the inertial force (Fsinθ) for the pendulum to remain in equilibrium. Therefore, we can equate these two forces:

Tsinθ = Fsinθ

Substituting Values:
We know that F = ma, where m is the mass of the pendulum bob and a is the acceleration of the car. Substituting this into the equation above, we get:

Tsinθ = masinθ

Canceling out Tension:
Since the tension force does not affect the horizontal acceleration of the car, we can cancel out the T term from both sides of the equation:

sinθ = a

Conclusion:
The horizontal acceleration of the car (a) is equal to sinθ, where θ is the angle the pendulum makes with the vertical. This means that the horizontal acceleration of the car is directly proportional to the sine of the angle θ. As the angle θ increases, the horizontal acceleration of the car also increases.