Question

If the period of vibration of a tuning fork depends upon the density '$d$' Young's modulus of the material '$y$' and the length of the spring '$L$' then time period $T$ is proportional to :

• A. $L{d}^2{y}^{\frac{1}{2}}$
• B. $L{d}^{\frac{1}{2}}.y^{-\frac{1}{2}}$
• C. $L{d}^{\frac{3}{2}}.y^{\frac{3}{2}}$
• D. $L{d}^{\frac{-3}{2}}.y^{\frac{-3}{2}}$

Answer 1

Answer: (B)

$L{d}^{\frac{1}{2}}.y^{-\frac{1}{2}}$

$T\propto d^{\alpha }{y}^{\beta }{L}^{\gamma }$

$M^0{L}^0{T}^1=\left(M{L}^{-3}{)}^{\alpha }\right(M^1{L}^{-1}{T}^{-2}{)}^{\beta }(L^1{)}^{\gamma }$
$0=\alpha +\beta$
$0=-3\alpha -\beta +\gamma$
$1=-2\beta$

$\beta =-\frac{1}{2},\alpha =\frac{1}{2},\gamma =1$

$\Rightarrow T\propto L^1{y}^{-\frac{1}{2}}{d}^{\frac{1}{2}}$