Question

The speed of sound in air depends upon elastic constant E and density d of air using diamensional method find the formula for sound in air.

Answer 1

Given info : The speed of sound in air depends upon elastic constant E and density of air d.

To find : the formula of speed of sound in air using dimensional method is...

solution : elastic constant, E = stress/strain

so dimension of E = dimension of stress/strain.

we know, dimension of stress = dimension of pressure = [ML¯¹T¯²]

strain is dimensionless quantity.

so, dimension of E = [ML¯1T¯²]

now dimension of density = [ML¯³]

dimension of velocity of sound = [LT¯¹]

from dimensional analysis,

v ∝ E^x d^y

⇒[LT¯¹] ∝ [ML¯¹T¯²]^x [ML¯³]^y

⇒[LT¯¹] ∝ [M^(x + y) L^(-x - 3y) T^(-2x)]

on comparing, we get

x + y = 0

-x - 3y = 1

-2x = -1 ⇒x = 1/2

and y = -1/2

so, v ∝ E½ d¯½

⇒v ∝ √(E/d)

Therefore the formula of speed of sound in air is v = k√(E/d) , where k is proportionality constant.