First slide

Dimensions of physical quantities

Question

The velocity of sound in air $(v)$ pressure $(p)$ and density of air $(d)$ are related as $v α p^{x} d^{y}$ the value of $' x'$ and $' y'$ respectively are

Moderate

A

$1, \frac{1}{2}$
B

$- \frac{1}{2}, - \frac{1}{2}$
C

$\frac{1}{2}, \frac{1}{2}$
D

$\frac{1}{2}, - \frac{1}{2}$

Solution

$v α p^{x} d^{y} - - - (1) p r e s s u r e = \frac{F}{A} = M L T^{- 2} / L^{2} = M L^{- 1} T^{- 2} d e n s i t y = m a s s / v o l u m e = M L^{- 3} v e l o c i t y = L T^{- 1} L T^{- 1} = {(M L^{- 1} T^{- 2})}^{x} . {(M L^{- 3})}^{y} M^{0} L T^{- 1} = M^{x + y} . L^{- x - 3 y} . T^{- 2 x} c o m p a f r i n g t h e p o w e r s o f M, L, T M \to x + y = 0 \Rightarrow x = - y T \to - 2 x = - 1 \Rightarrow x = \frac{1}{2} \Rightarrow y = - \frac{1}{2} S u b s t i t u t e o b t a i n e d v a l u e s i n e q n (1) v = p^{\frac{1}{2}} d^{- \frac{1}{2}}$

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