To determine the dimensional formula for [tex]\( m \)[/tex] in the given equation for the frequency of vibration of a string, we will analyze the dimensions of each term in the equation:
[tex]\[
v = \frac{P}{2 l}\left[\frac{F}{m}\right]^{1 / 2}
\][/tex]
1. Identify the dimensions of each variable:
- [tex]\( v \)[/tex]: frequency, which has the dimensions [tex]\([T^{-1}]\)[/tex].
- [tex]\( P \)[/tex]: number of segments, which is dimensionless.
- [tex]\( l \)[/tex]: length, which has the dimensions [tex]\([L]\)[/tex].
- [tex]\( F \)[/tex]: force, which has the dimensions [tex]\([M L T^{-2}]\)[/tex].
- [tex]\( m \)[/tex]: mass per unit length, which we need to find the dimensions of.
2. Rewrite the equation with the dimension notation:
[tex]\[
\left[T^{-1}\right] = \frac{\left[\text{dimensionless}\right]}{\left[L\right]} \left[\frac{\left[M L T^{-2}\right]}{m}\right]^{1 / 2}
\][/tex]
Simplifying, the equation becomes:
[tex]\[
\left[T^{-1}\right] = \frac{1}{\left[L\right]} \left[\frac{\left[M L T^{-2}\right]}{m}\right]^{1 / 2}
\][/tex]
3. Simplify the dimensions inside the square root:
[tex]\[
\left[T^{-1}\right] = \frac{1}{\left[L\right]} \left[\left[\frac{M L T^{-2}}{m}\right]^{1 / 2}\right]
\][/tex]
4. Isolate the dimension of [tex]\( m \)[/tex]:
To simplify further, let's isolate the square root term:
[tex]\[
\left[\frac{M L T^{-2}}{m}\right]^{1 / 2} = [L T^{-1}]
\][/tex]
Squaring both sides to remove the square root:
[tex]\[
\frac{M L T^{-2}}{m} = L^2 T^{-2}
\][/tex]
5. Solve for [tex]\( m \)[/tex]:
Multiply both sides by [tex]\( m \)[/tex] and divide by [tex]\( L^2 T^{-2} \)[/tex]:
[tex]\[
m = \frac{M L T^{-2}}{L^2 T^{-2}}
\][/tex]
Simplify the expression:
[tex]\[
m = M L^{-1}
\][/tex]
6. Write the dimensional formula for [tex]\( m \)[/tex]:
[tex]\[
m = [M L^{-1} T^0]
\][/tex]
Therefore, the dimensional formula for [tex]\( m \)[/tex] is [tex]\([M L^{-1} T^0]\)[/tex].
The correct choice is [tex]\( \boxed{3} \)[/tex].
