The frequency of vibration of string is given by `v = (p)/(2 l) [(F)/(m)]^(1//2)`. Here `p` is number of segment is the string and `l` is the length. The dimension formula for `m` will be

Options

A. `[M^(0) LT^(-1)]`
B. `[M L^(0) T^(-1)]`
C. `[ML^(-1) T^(0)]`
D. `[M^(0) L^(0) T^(0)]`

Answer

To find the dimension formula for \( m \) in the given equation for the frequency of vibration of a string, we can follow these steps:

### Step 1: Write down the given equation
The frequency of vibration of the string is given by:
\[ v = \frac{p}{2l} \left(\frac{F}{m}\right)^{1/2} \]
where \( p \) is the number of segments in the string, \( l \) is the length of the string, \( F \) is the force, and \( m \) is the mass.

### Step 2: Rearrange the equation to isolate \( m \)
To isolate \( m \), we can square both sides of the equation:
\[ v^2 = \left(\frac{p}{2l}\right) \left(\frac{F}{m}\right) \]
Now, rearranging gives:
\[ \frac{F}{m} = \frac{2lv^2}{p} \]
Multiplying both sides by \( m \) gives:
\[ F = \frac{2lv^2}{p} m \]
Now, we can express \( m \):
\[ m = \frac{pF}{2lv^2} \]

### Step 3: Write down the dimensions of each term
1. **Force \( F \)** has dimensions:
\[ [F] = [M][L][T^{-2}] \]
2. **Length \( l \)** has dimensions:
\[ [l] = [L] \]
3. **Frequency \( v \)** has dimensions:
\[ [v] = [T^{-1}] \]

### Step 4: Substitute the dimensions into the equation for \( m \)
Now substituting the dimensions into the equation for \( m \):
\[ m = \frac{pF}{2lv^2} \]
Since \( p \) is a number and has no dimensions, we can ignore it for dimensional analysis:
\[ [m] = \frac{[F]}{[l][v^2]} \]
Substituting the dimensions we have:
\[ [m] = \frac{[M][L][T^{-2}]}{[L][(T^{-1})^2]} \]

### Step 5: Simplify the dimensions
Now simplifying:
\[ [m] = \frac{[M][L][T^{-2}]}{[L][T^{-2}]} = [M][L^{-1}][T^{0}] \]
Thus, the dimension formula for \( m \) is:
\[ [m] = [M][L^{-1}] \]

### Final Answer
The dimension formula for \( m \) is:
\[ [M][L^{-1}] \]

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