A boy is hanging from a horizontal branch of a tree. The tension in the arms will be maximum when the angle between the arms is

Options

A. `0^(@)`
B. `30^(@)`
C. `60^(@)`
D. `120^(@)`

Explanation

To solve the problem of determining the angle at which the tension in the boy's arms is maximized while hanging from a horizontal branch, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- A boy is hanging from a horizontal branch using both arms.
- The forces acting on the boy include his weight (mg) acting downwards and the tensions in his arms (T) acting at an angle.

2. **Identifying the Angles**:
- Let the angle between each arm and the vertical be θ/2, where θ is the total angle between the arms.

3. **Force Components**:
- Each arm exerts a tension T. The vertical components of the tensions must balance the weight of the boy.
- The vertical component of tension from one arm is T * cos(θ/2).
- Therefore, the total upward force from both arms is 2 * T * cos(θ/2).

4. **Setting Up the Equation**:
- For the boy to be in equilibrium (not moving vertically), the total upward force must equal the weight:
\[ 2T \cos\left(\frac{\theta}{2}\right) = mg \]

5. **Maximizing Tension**:
- To find the angle θ that maximizes the tension T, we need to analyze the expression for T:
\[ T = \frac{mg}{2 \cos\left(\frac{\theta}{2}\right)} \]
- The tension T will be maximized when the cosine term is minimized. Since cosine decreases as the angle increases, we need to maximize θ.

6. **Finding the Maximum Angle**:
- The maximum value of cos(θ/2) occurs when θ/2 is minimized. The minimum value of θ/2 is 0, which corresponds to θ being 0 degrees.
- However, we are interested in the maximum angle that still allows for equilibrium. The angle θ should be such that the arms are not completely horizontal (which would not provide any vertical support).
- The maximum angle occurs when θ = 120 degrees (60 degrees on each side), as this allows for sufficient vertical component of the tension.

7. **Conclusion**:
- Therefore, the angle between the arms at which the tension is maximum is **120 degrees**.