A string of length L and mass M hangs freely from a fixed point. Then the velocity of transverse waves along the string at a distance x from the free end is

Options

A. `sqrt(gL)`
B. `sqrt(gx)`
C. gL
D. gx

Answer

To find the velocity of transverse waves along a string of length \( L \) and mass \( M \) that hangs freely from a fixed point, we can follow these steps:

### Step 1: Understand the setup
We have a string of length \( L \) and mass \( M \) hanging vertically. The tension in the string varies with the distance from the free end to the fixed point.

### Step 2: Determine the mass per unit length
The mass per unit length \( \mu \) of the string can be calculated as:
\[ \mu = \frac{M}{L} \]

### Step 3: Analyze the tension at a distance \( x \) from the free end
At a distance \( x \) from the free end, the length of the string hanging below this point is \( L - x \). The mass of this segment of the string is:
\[ m' = \mu (L - x) = \frac{M}{L} (L - x) \]

### Step 4: Calculate the weight of the hanging segment
The weight of the hanging segment, which provides the tension \( T \) in the string at distance \( x \), is given by:
\[ T = m' g = \frac{M}{L} (L - x) g \]

### Step 5: Write the expression for the velocity of the wave
The velocity \( v \) of transverse waves on a string is given by:
\[ v = \sqrt{\frac{T}{\mu}} \]
Substituting the expression for \( T \):
\[ v = \sqrt{\frac{\frac{M}{L} (L - x) g}{\frac{M}{L}}} \]

### Step 6: Simplify the expression
The mass per unit length \( \mu \) cancels out:
\[ v = \sqrt{(L - x) g} \]

### Final Result
Thus, the velocity of transverse waves along the string at a distance \( x \) from the free end is:
\[ v = \sqrt{(L - x) g} \]

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