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 finding the angle that the mean position of a simple pendulum makes when the pendulum is oscillating in a trolley that is accelerating, we can follow these steps:

### Step 1: Understand the Forces Acting on the Pendulum
In the accelerating trolley, the pendulum experiences two forces:
- The gravitational force acting downward (mg).
- The pseudo force acting horizontally in the opposite direction of the trolley's acceleration (ma), where 'm' is the mass of the pendulum bob and 'a' is the acceleration of the trolley.

### Step 2: Draw a Free Body Diagram
Draw a diagram showing the pendulum bob. Label the forces:
- The gravitational force (mg) acting downward.
- The pseudo force (ma) acting horizontally towards the left (if the trolley accelerates to the right).

### Step 3: Set Up the Right Triangle
From the free body diagram, we can see that the forces form a right triangle:
- The vertical side represents the gravitational force (mg).
- The horizontal side represents the pseudo force (ma).

### Step 4: Determine the Angle θ
The angle θ that the pendulum makes with the vertical can be determined using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{ma}{mg} \]
This simplifies to:
\[ \tan(\theta) = \frac{a}{g} \]

### Step 5: Solve for θ
To find the angle θ, take the arctangent (inverse tangent) of both sides:
\[ \theta = \tan^{-1}\left(\frac{a}{g}\right) \]

### Step 6: Consider the Direction of the Angle
Since the trolley is accelerating in the positive x direction, the angle θ will be measured from the vertical down towards the direction of the acceleration (which is to the right). Thus, we can express the angle as:
\[ \theta = -\tan^{-1}\left(\frac{a}{g}\right) \]
This indicates that the angle is measured in the negative x direction.

### Final Answer
The mean position of the pendulum makes an angle:
\[ \theta = -\tan^{-1}\left(\frac{a}{g}\right) \]