Question

The velocity of sound V in a metal is dependent in the young modulus E and the density e. Use the method of dimension to derive the relationship between the parameters

Answer 1

Dimensions of young modulus:

$$E=[M]^1[L]^{-1}[T]^{-2}$$

Dimensions of density:

$$\rho=[M]^1[L]^{-3}$$

Speed of sound:

$$v=[L][T]^{-1}$$

Obtain the relationship:

[L]^1[T]^{-1}=([M]^1[L]^{-1}[T]^{-2})^a·([M]^1[L]^{-3})^b,\\ [L]^1[T]^{-1}=[M]^{a+b}[L]^{-a-3b}[T]^{-2a}.

Equate the power of each dimension (powers on the left with powers on the right):

[T]^{-1}=[T]^{-2a},\\ -1=-2a→a=\frac12.\\\space\\ [M]^0=[M]^{a+b},\\ 0=a+b→b=-\frac12.

Therefore, we have

$$\Big([L]^1[T]^{-1}\Big)=\\=\Big([M]^1[L]^{-1}[T]^{-2}\Big)^{\frac12}·\Big([M]^1[L]^{-3}\Big)^{-\frac12},$$

which corresponds to

$$v=\sqrt\frac E\rho.$$