A truck is staitonary and has a bob suspended by a lilght string in a frame attached to the truck The truck , suddenly moves to the right with an acceleration of a. The pendulum will tilt

Options

A. to the left and angle of inclination of the penudlum with the verticle is `sin^(-1)((g)/(a))`
B. to the left and angle of inclination of the pendulm with the verticle is `tan^(-1)((alpha)/(g))`
C. to the left and angle of inclination of the pendulum with the veritcal is `sin^(-1)((a)/(g))`
D. to the left and angle of inclination of the pendulum with the veritcal is `tan^(-1)9(g)/(a))`

Answer

To solve the problem of the pendulum tilting when the truck accelerates, we will analyze the forces acting on the bob of the pendulum and apply Newton's laws of motion. Here’s a step-by-step solution:

### Step 1: Understand the Setup
The truck is stationary initially, and a bob is suspended from a light string. When the truck accelerates to the right with acceleration \( a \), the bob will experience a force due to inertia that will cause it to tilt.

**Hint:** Visualize the scenario and identify the forces acting on the bob.

### Step 2: Identify Forces Acting on the Bob
1. **Weight of the bob (mg)**: Acts downward.
2. **Tension in the string (T)**: Acts along the string towards the point of suspension.
3. **Inertial force**: When the truck accelerates, the bob appears to experience a pseudo force to the left (equal to \( ma \)) due to its inertia.

**Hint:** Draw a free body diagram showing these forces.

### Step 3: Set Up the Equations of Motion
We can resolve the forces acting on the bob into two components:
- In the horizontal direction (x-axis): The component of tension \( T \sin \theta \) must balance the inertial force \( ma \).
- In the vertical direction (y-axis): The component of tension \( T \cos \theta \) must balance the weight of the bob \( mg \).

**Hint:** Write down the equations based on the components of forces.

### Step 4: Write the Equations
1. For the horizontal direction:
\[ T \sin \theta = ma \quad \text{(1)} \]
2. For the vertical direction:
\[ T \cos \theta = mg \quad \text{(2)} \]

**Hint:** Remember that \( \theta \) is the angle the pendulum makes with the vertical.

### Step 5: Divide the Equations
To eliminate \( T \), divide equation (1) by equation (2):
\[ \frac{T \sin \theta}{T \cos \theta} = \frac{ma}{mg} \]
This simplifies to:
\[ \tan \theta = \frac{a}{g} \]

**Hint:** This step helps to relate the angle \( \theta \) to the acceleration \( a \) and gravitational acceleration \( g \).

### Step 6: Solve for \( \theta \)
Now, we can find the angle \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{a}{g}\right) \]

**Hint:** This angle indicates how much the pendulum tilts from the vertical.

### Step 7: Determine the Direction of Tilt
Since the truck is accelerating to the right, the bob will tilt to the left from the vertical position.

**Hint:** Visualize the direction of tilt based on the forces acting on the bob.

### Final Answer
The pendulum will tilt to the left with an angle given by:
\[ \theta = \tan^{-1}\left(\frac{a}{g}\right) \]