Question

: The period of vibration of a tuning fork depends on the length l of its prong, density ρ and young modulus Y of its material. Then, the equation for the period of vibration will be.
A $k\rho \sqrt{{\displaystyle \frac{l}{Y}}}$
B $kl\sqrt{{\displaystyle \frac{\rho }{Y}}}$
C ${\displaystyle \frac{k}{l}}\sqrt{{\displaystyle \frac{Y}{\rho }}}$
D ${\displaystyle \frac{l}{k}}\sqrt{{\displaystyle \frac{\rho }{Y}}}$

Answer 1

Hint: We can use dimension formula to derive relations between physical quantities based on their interdependence. The period of vibration of the tuning fork depends on the length, density and young modulus of its material.
[period]= [length]${}^a$[density]${}^b$[young modulus]${}^c$.

Complete step by step answer:
 Given that the time period T of the tuning fork depends on the length l, density ρ and young modulus Y
So, T= kl${}^a$ρ${}^b$Y${}^c$

Where, k=dimensionless constant.
Substituting the dimensions of each quantity in the equation we get,
[M${}^0$L${}^0$T${}^1$] =k[M${}^0$L${}^1$T${}^0$] ${}^a$[M${}^1$L${}^{-3}$T${}^{{}^0}$]${}^b$ [M${}^1$L${}^{-1}$T${}^{-2}$]${}^c$
[M${}^0$L${}^0$T${}^1$] =k[M${}^{b+c}$L${}^{a-3b-c}$T${}^{-2c}$]
Now we equate the powers of M, L and T on both sides, we get
-2c=1
$\Rightarrow$c=$-{\displaystyle \frac{1}{2}}$
b+c=0
$\Rightarrow$b=-c =${\displaystyle \frac{1}{2}}$
a- 2b- c =0
a= 2b+c
  =2×${\displaystyle \frac{1}{2}}$+2×$\left(-{\displaystyle \frac{1}{2}}\right)$=1-1=0
Therefore we get the value of a=0, b=${\displaystyle \frac{1}{2}}$and c=$-{\displaystyle \frac{1}{2}}$.
Substituting these values in the equation,
T=kl${}^1$ρ${}^{{\displaystyle \frac{1}{2}}}$Y${}^{-{\displaystyle \frac{1}{2}}}$
Therefore time period T=$kl\sqrt{{\displaystyle \frac{\rho }{Y}}}$

So, the correct answer is “Option B”.

Additional Information:
 The equations obtained when we equal a physical quantity with its dimensional formulae are called Dimensional Equations.
The expressions or formulae which tell us how and which of the fundamental quantities are present in a physical quantity are known as the Dimensional Formula of the physical quality.

Note:
Students should know the dimensional formula of common physical terms used in physics. Formulae should be memorised to obtain the dimensional formula of derived quantity using fundamental quantity.