A Uniform rope having mass m hags vertically from a rigid support. A transverse wave pulse is produced at the lower end. The speed v of wave pulse varies with height h from the lower end as

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Answer

To solve the problem, we need to determine how the speed \( v \) of a transverse wave pulse in a hanging rope varies with the height \( h \) from the lower end of the rope. Here’s a step-by-step solution:

### Step 1: Understand the relationship between wave speed, tension, and linear mass density
The speed \( v \) of a wave in a rope is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension in the rope and \( \mu \) is the linear mass density of the rope.

### Step 2: Determine the tension in the rope at a height \( h \)
Consider a small segment of the rope of length \( x \) from the bottom. The mass of this segment is:
\[ m_x = \mu x \]
The tension \( T \) at height \( h \) is due to the weight of the rope below that point. Therefore, the tension can be expressed as:
\[ T = m_x g = \mu x g \]
where \( g \) is the acceleration due to gravity.

### Step 3: Substitute the tension into the wave speed formula
Substituting the expression for tension into the wave speed formula, we get:
\[ v = \sqrt{\frac{\mu x g}{\mu}} = \sqrt{x g} \]

### Step 4: Relate height \( h \) to the length \( x \)
Since the rope is hanging vertically, the height \( h \) from the bottom of the rope is related to the length \( x \) of the rope below the point of interest. Thus, we can express \( x \) in terms of \( h \):
\[ x = L - h \]
where \( L \) is the total length of the rope.

### Step 5: Substitute \( x \) into the wave speed equation
Now substituting \( x = L - h \) into the wave speed equation gives:
\[ v = \sqrt{(L - h) g} \]

### Step 6: Analyze the relationship between \( v \) and \( h \)
From the equation \( v = \sqrt{(L - h) g} \), we can see that as \( h \) increases (moving up the rope), \( v \) decreases. The relationship is a square root function, which indicates that the speed \( v \) varies with height \( h \) in a parabolic manner.

### Conclusion
Thus, the speed \( v \) of the wave pulse varies with height \( h \) from the lower end of the rope as:
\[ v \propto \sqrt{(L - h)} \]
This indicates that the speed decreases as we move up the rope.