When a body is projected at an angle with the horizontal in the uniform gravitational field of the earth, the angular momentum of the body about the point of projection, as it proceeds along its path

Options

A. remains constant
B. increases
C. decreases
D. initially decreases and after its highest point increases.

Explanation

To solve the problem of angular momentum of a body projected at an angle with the horizontal in a uniform gravitational field, we can follow these steps:

### Step 1: Understand the Setup
When a body is projected at an angle θ with the horizontal, it follows a parabolic trajectory under the influence of gravity. The point of projection is considered as the origin (0,0) in our coordinate system.

**Hint:** Visualize the trajectory of the projectile and the forces acting on it.

### Step 2: Identify Forces Acting on the Body
The only force acting on the body after it is projected is the gravitational force (weight), which acts downward with a magnitude of mg, where m is the mass of the body and g is the acceleration due to gravity.

**Hint:** Remember that the gravitational force acts vertically downwards.

### Step 3: Define Angular Momentum
The angular momentum (L) of a body about a point is given by the formula:
\[ L = r \times p \]
where \( r \) is the position vector from the point of rotation to the body, and \( p \) is the linear momentum of the body. The linear momentum \( p \) is given by \( p = mv \), where \( v \) is the velocity of the body.

**Hint:** Angular momentum depends on both the position vector and the linear momentum.

### Step 4: Calculate Angular Momentum
As the body moves along its path, the position vector \( r \) changes. The angular momentum about the point of projection can be expressed as:
\[ L = r \cdot mv \]
Since the body is moving in a parabolic path, the distance \( r \) from the point of projection increases as the body moves upward and then downward.

**Hint:** Consider how the position vector changes with time as the projectile moves.

### Step 5: Analyze the Torque
The torque (\( \tau \)) about the point of projection due to the gravitational force is given by:
\[ \tau = r \times F \]
where \( F = mg \). The torque will change as the distance \( r \) increases.

**Hint:** Torque depends on the perpendicular distance from the line of action of the force to the point of rotation.

### Step 6: Relate Torque to Angular Momentum
From the relationship between torque and angular momentum:
\[ \tau = \frac{dL}{dt} \]
If the torque is increasing (as \( r \) increases), it implies that the angular momentum \( L \) is also increasing with time.

**Hint:** Remember that if torque is non-zero, angular momentum will change.

### Step 7: Conclusion
As the body proceeds along its path, the angular momentum about the point of projection increases due to the increasing distance \( r \) and the constant gravitational force acting on the body.

**Final Answer:** The angular momentum of the body about the point of projection increases as it proceeds along its path.

---