A particle of mass m is projected with speed u at an angle `theta` with the horizontal. Find the torque of the weight of the particle about the point of projection when the particle is at the highest point.

Explanation

To find the torque of the weight of a particle about the point of projection when it is at the highest point, we can follow these steps:

### Step 1: Understand the Situation
A particle of mass \( m \) is projected with an initial speed \( u \) at an angle \( \theta \) with the horizontal. At the highest point of its trajectory, we need to analyze the forces acting on the particle and their effect on torque about the point of projection.

**Hint:** Visualize the projectile motion and identify the forces acting on the particle at the highest point.

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

**Hint:** Remember that at the highest point, the vertical component of the velocity is zero, but the weight still acts downward.

### Step 3: Determine the Position of the Particle
At the highest point of the projectile's motion, the horizontal distance traveled can be determined using the range formula. The range \( R \) of the projectile is given by:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
Since the highest point is at half the range, the horizontal distance \( x \) from the point of projection \( O \) to the highest point is:
\[ x = \frac{R}{2} = \frac{u^2 \sin(2\theta)}{2g} \]

**Hint:** Use the range formula to find the horizontal position of the particle when it is at the highest point.

### Step 4: Calculate the Perpendicular Distance
The torque \( \tau \) about point \( O \) due to the weight of the particle is given by the formula:
\[ \tau = F \cdot d \]
where \( F \) is the force (weight \( mg \)) and \( d \) is the perpendicular distance from the line of action of the force to the point of rotation. At the highest point, this perpendicular distance is equal to the vertical height \( h \) of the projectile, which can be calculated using the formula:
\[ h = \frac{u^2 \sin^2(\theta)}{2g} \]

**Hint:** Remember that the perpendicular distance is the vertical height of the projectile at the highest point.

### Step 5: Calculate the Torque
Now substituting the values into the torque equation:
\[ \tau = mg \cdot h = mg \cdot \frac{u^2 \sin^2(\theta)}{2g} \]
Simplifying this gives:
\[ \tau = \frac{m u^2 \sin^2(\theta)}{2} \]

**Hint:** Make sure to simplify the expression correctly and check the units for consistency.

### Final Answer
The torque of the weight of the particle about the point of projection when the particle is at the highest point is:
\[ \tau = \frac{m u^2 \sin^2(\theta)}{2} \]