Question

The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string and its mass per unit length m. Guess the expression for its frequency from dimensional analysis.

Concept of Physics - 1 , HC VERMA , Chapter "Introduction to Physics".

Answer 1

Hello Dear.

According to the Question,

Frequency(f) ∝ L × F × m.

∴ Relation will be,

Frequency(f) = Y (L)ᵃ × (F)ᵇ × (m)ˣ
where, a, b, and x are the Powers of the terms.
Y is the constant.

For L.H.S.

Now,
Frequency = 1/Time Period.
 [f] = 1/T.
 [f] = T⁻¹.
[f] = M⁰ L⁰ T⁻¹.

For R.H.S.

Dimension of length L = L.
Dimension of Mass per unit length m  = M L⁻¹

For the Dimension of Force,
Force = mass × acceleration.
= mass × Velocity/time.
= M L T⁻².

∴ Dimensions of R.H.S. = Y × (L)ᵃ × (M L T⁻²)ᵇ × (ML⁻¹)ˣ
  
Now, 
Putting the Dimensions in the Formula,

M⁰ L⁰ T⁻¹ = Y × (L)ᵃ × (M L T⁻²)ᵇ × (ML⁻¹)ˣ
 M⁰ L⁰ T⁻¹ = Y × Lᵃ.Lᵇ.L⁻ˣ  ×  Mᵇ.Mˣ × T⁻²ᵇ
⇒ M⁰ L⁰ T⁻¹ = Y . Lᵃ⁺ᵇ⁻ˣ . Mᵇ⁺ˣ . T⁻²ᵇ

Equating the Dimensions of both the sides,
We get,
a + b - x  = 0, b + x = 0 &  -2b = -1

b = 1/2.

∴ x = -1/2.

∴ a + b - x = 0.
⇒ a = x - b.
⇒ a = -1/2 - 1/2.
⇒ a = -1.

∴ Frequency(f) = Y (L)ᵃ × (F)ᵇ × (m)ˣ
⇒ Frequency(f) = Y L⁻¹ (F)¹⁾² × (m)⁻¹⁾²
⇒ Frequency(f) = Y L⁻¹ $\sqrt{ \frac{F}{M} }$
⇒ Frequency(f) = Y/L $\sqrt{ \frac{F}{M} }$ 



Hence, the Formula of the Frequency will be $\frac{Y}{L} \sqrt{ \frac{F}{M} }$



Hope it helps.