Question

The velocity of sound `(upsilon)` in a gas depends upon coefficint of volume elesticity E of the gas and density d of the gas. Use method of dimensions to derive the formula for `upsilon.`

Answer 1

Let `upsilon = KE^a d^b …(i)`
where K is dimensionless constant of
proportionality.
`:. [ M^0 L^1 T^(-1)] = (ML^(-1) T^(-2))^a (ML^(-3))^b`
` = M^(a +b) L^(-a -3b) T^(-2a)`
Applying the principle of homogeneity of
dimensions, we get
`a +b = 0 , - a-3b b =1 , -2a = -1`
`:. a = (1)/(2), b = -a =-(1)/(2)`
From (i), `upsilon = KE^(1//2) d^(-1//2) =Ksqrt(E//d)`