A simple pendulum of length l ils pulled aside to make an angle `theta` with the vertical. Find the magnitude of the torque of the weitht w of the bob about the point of suspension. When is the torque zero?

Explanation

To find the magnitude of the torque of the weight \( W \) of the bob about the point of suspension when the pendulum is pulled aside to make an angle \( \theta \) with the vertical, we can follow these steps:

### Step 1: Understand the Setup
We have a simple pendulum consisting of a bob of mass \( m \) attached to a string of length \( L \). The bob is displaced to an angle \( \theta \) from the vertical.

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

### Step 2: Identify the Forces
The only force acting on the bob is its weight \( W \), which acts vertically downward. The weight can be expressed as:
\[ W = mg \]
where \( g \) is the acceleration due to gravity.

**Hint:** Remember that the weight acts downward regardless of the pendulum's position.

### Step 3: Determine the Torque
Torque (\( \tau \)) about the point of suspension is given by the formula:
\[ \tau = r \times F \]
where \( r \) is the distance from the pivot point to the point of application of the force, and \( F \) is the force.

In this case, the torque due to the weight of the bob is:
\[ \tau = W \cdot D \]
where \( D \) is the perpendicular distance from the line of action of the weight to the pivot point.

**Hint:** The torque is maximized when the force is applied at the maximum distance from the pivot.

### Step 4: Calculate the Perpendicular Distance \( D \)
The perpendicular distance \( D \) can be found using trigonometry. When the bob is at an angle \( \theta \), the perpendicular distance from the weight to the pivot is:
\[ D = L \sin \theta \]

**Hint:** Use the sine function to relate the angle to the opposite side of the triangle formed by the pendulum.

### Step 5: Substitute \( D \) into the Torque Formula
Now substituting \( D \) into the torque equation, we get:
\[ \tau = W \cdot (L \sin \theta) \]
Thus, the magnitude of the torque is:
\[ \tau = W L \sin \theta \]

**Hint:** This equation shows how torque depends on both the weight of the bob and the angle \( \theta \).

### Step 6: Determine When the Torque is Zero
The torque will be zero when:
\[ \sin \theta = 0 \]
This occurs when \( \theta = 0 \) (i.e., when the pendulum is in the vertical position).

**Hint:** Think about the physical meaning: when the pendulum is vertical, the weight acts directly downward through the pivot, resulting in no torque.

### Final Answer
The magnitude of the torque of the weight \( W \) of the bob about the point of suspension is:
\[ \tau = W L \sin \theta \]
The torque is zero when the pendulum is in the vertical position, i.e., \( \theta = 0 \).