A simple pendulum is oscillating in a trolley moving on a horizontal straight road with constant acceleration a. If direction of motion of trolley is taken as positive x direction and vertical upward direction as positive y direction then the mean position of pendulum makes an angle

Options

A. `tan^(-1) ((g)/(a))` with y axis in +x direction
B. `tan^(-1) ((a)/(g))` with y axis in -x direction
C. `tan^(-1) ((a)/(g))` with y axis in +x direction
D. `tan^(-1) ((g)/(a))` with y axis in -x direction

Answer

To solve the problem of a simple pendulum oscillating in a trolley that is accelerating horizontally, we can follow these steps:

### Step 1: Understanding the Forces Acting on the Pendulum
When the trolley accelerates with a constant acceleration \( a \), the pendulum experiences two forces:
1. The gravitational force acting downward, \( mg \).
2. The tension \( T \) in the string, which acts along the string.

In the non-inertial frame of the accelerating trolley, a pseudo force \( F_{\text{pseudo}} = ma \) acts on the pendulum in the opposite direction of the trolley's acceleration (to the left if the trolley accelerates to the right).

### Step 2: Setting Up the Equilibrium Conditions
At the mean position of the pendulum, the net force in both the x (horizontal) and y (vertical) directions must be zero.

#### In the vertical direction (y-axis):
\[ T \cos \theta - mg = 0 \]
This implies:
\[ T \cos \theta = mg \quad \text{(1)} \]

#### In the horizontal direction (x-axis):
\[ T \sin \theta - ma = 0 \]
This implies:
\[ T \sin \theta = ma \quad \text{(2)} \]

### Step 3: Dividing the Two Equations
To eliminate \( T \), we can divide equation (2) by equation (1):
\[ \frac{T \sin \theta}{T \cos \theta} = \frac{ma}{mg} \]
This simplifies to:
\[ \tan \theta = \frac{a}{g} \]

### Step 4: Finding the Angle \( \theta \)
To find the angle \( \theta \) that the pendulum makes with the vertical, we take the inverse tangent:
\[ \theta = \tan^{-1} \left(\frac{a}{g}\right) \]

### Conclusion
Thus, the mean position of the pendulum makes an angle \( \theta \) with the vertical given by:
\[ \theta = \tan^{-1} \left(\frac{a}{g}\right) \]