A weight w is suspended from the midpoint of a rope, whose ends are at the same level. In order to make the rope perfectly horizontal, the force applied to each of its ends must be

Options

A. less than w
B. equal to w
C. equal to 2w
D. infinitely large

Explanation

To solve the problem of determining the force applied to each end of a rope in order to make it perfectly horizontal when a weight \( w \) is suspended from its midpoint, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a rope with a weight \( w \) hanging from its midpoint. The ends of the rope are at the same level.
- When the weight is suspended, the rope forms an angle \( \theta \) with the horizontal at both ends.

2. **Identifying Forces**:
- The weight \( w \) acts downward at the midpoint of the rope.
- The tension \( T \) in the rope acts along the rope, directed away from the weight at both ends.

3. **Components of Tension**:
- The tension \( T \) can be resolved into two components:
- A vertical component: \( T \sin(\theta) \) (acting upward)
- A horizontal component: \( T \cos(\theta) \) (acting horizontally)

4. **Equilibrium Condition**:
- For the weight to be in equilibrium, the sum of the vertical components of the tension must equal the weight:
\[ 2T \sin(\theta) = w \]
- Here, \( 2T \sin(\theta) \) accounts for the upward force from both sides of the rope.

5. **Solving for Tension**:
- Rearranging the equation gives:
\[ T = \frac{w}{2 \sin(\theta)} \]

6. **Condition for Horizontal Rope**:
- To make the rope perfectly horizontal, the angle \( \theta \) must be \( 90^\circ \).
- At \( \theta = 90^\circ \), \( \sin(90^\circ) = 1 \).

7. **Substituting the Angle**:
- Substituting \( \theta = 90^\circ \) into the tension equation:
\[ T = \frac{w}{2 \cdot 1} = \frac{w}{2} \]

8. **Conclusion**:
- The force applied to each end of the rope must be \( \frac{w}{2} \) to make the rope perfectly horizontal.

### Final Answer:
The force applied to each end of the rope must be \( \frac{w}{2} \).

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