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. **Understand the Setup**:
- We have a rope with a weight \( W \) suspended at its midpoint. The ends of the rope are at the same level.
- The goal is to make the rope perfectly horizontal.

2. **Identify the Forces**:
- When the rope is not horizontal, the tension \( T \) in the rope creates two components: a vertical component \( T \cos \theta \) and a horizontal component \( T \sin \theta \).
- The vertical components of the tension from both sides must balance the weight \( W \).

3. **Set Up the Equilibrium Condition**:
- Since the system is in equilibrium, the sum of the vertical forces must equal zero. Therefore, we can write:
\[ 2T \cos \theta = W \]
- This equation states that the total upward force (from both sides of the rope) equals the downward force (the weight).

4. **Solve for Tension \( T \)**:
- Rearranging the equation gives:
\[ T = \frac{W}{2 \cos \theta} \]

5. **Consider the Case of a Perfectly Horizontal Rope**:
- For the rope to be perfectly horizontal, the angle \( \theta \) must approach \( 90^\circ \).
- The cosine of \( 90^\circ \) is \( 0 \), which leads to:
\[ T = \frac{W}{2 \cdot 0} \]
- This implies that \( T \) approaches infinity.

6. **Conclusion**:
- Therefore, to make the rope perfectly horizontal, the force applied to each end must be infinitely large.

### Final Answer:
The force applied to each of its ends must be infinitely large.