Using dimensions method show that the speed of sound in a material medium of modulus of elasticity E and density rho is v prop sqrt(E/(rho))

Using dimensions method show that the speed of sound in a material med

1.	Using dimensions method show that the speed of sound in a material medium of modulus of elasticity E and density rho is v prop sqrt(E/(rho))
Answer» <P> Solution :`vprop E^(x)rho^(y)` `v=kE^(x)rho^(y)` ……1 k is a DIMENSIONLESS constant Dimensions of `E=("Stress")/("Strain")=ML^(-1)T^(-2)`. Strain has no dimensions Taking the dimensions of equation (1) on both SIDES `LT^(-1)=[ML^(-1)T^(-2)]^(x)[ML^(-3)]^(y)=M^(x)L^(-x)T^(-2x)M^(y)L^(-3y)` `M^(0)LT^(-1)=M^(x+y)L^(-x-3y)T^(-2x)` Equating the dimensios on both sides of T `-1=-2x,x=1//2` of M `x+y=0, y=-x-1//2` Substituting the values of x and y in equation (1) we get `v=k E^(1//2)p^(-1//2)` or `v prop sqrt(E/(rho))`