A heavy rope is suspended from a rigid support A wave pulse is set up at the lower end, then

Options

A. the pulse will travel with uniform speed
B. the pulse will travel with increasing speed
C. the pulse will travel with decreasing speed
D. the pulse cannot travel through the rope

Explanation

To solve the problem of finding the speed of a wave pulse traveling up a heavy rope suspended from a rigid support, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a heavy rope suspended vertically from a rigid support.
- A wave pulse is initiated at the lower end of the rope.

2. **Identify Forces Acting on the Rope**:
- At a distance \( x \) from the bottom of the rope, the tension \( T \) in the rope will be affected by the weight of the rope below that point.
- The weight of the rope segment below point \( x \) is given by the mass of that segment multiplied by the acceleration due to gravity \( g \).

3. **Calculate the Tension**:
- The mass of the segment of the rope below point \( x \) can be expressed in terms of the linear density \( \mu \) (mass per unit length) and the length \( x \):
\[ \text{Mass} = \mu \cdot x \]
- Therefore, the tension \( T \) at distance \( x \) is:
\[ T = \text{Weight} = \mu \cdot x \cdot g \]

4. **Use the Wave Speed Formula**:
- The speed \( v \) of a wave pulse in a rope is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
- Substituting the expression for tension \( T \):
\[ v = \sqrt{\frac{\mu \cdot x \cdot g}{\mu}} \]

5. **Simplify the Expression**:
- The \( \mu \) terms cancel out:
\[ v = \sqrt{x \cdot g} \]

6. **Analyze the Result**:
- The expression \( v = \sqrt{x \cdot g} \) indicates that the speed of the wave pulse depends on the distance \( x \) from the bottom of the rope.
- As \( x \) increases (moving upwards), the speed \( v \) also increases.

7. **Conclusion**:
- Therefore, the wave pulse will travel with increasing speed as it moves upwards in the rope.

### Final Answer:
The pulse will travel with increasing speed as it moves upwards in the rope.