Question

The displacement of air particle in a flute is given by x= a sin(wt) cos(kx) where w is a quantity which depends on the length of the flute(L), gas constant(R), temperature of air(T) and mass per unit mole of the air(u). Find an expression for w in terms of all these quantities. [Use PV= nRT]

Answer 1

length of flute = L => dimension of L = [L]

gas constant = R => dimension of R = [ML²T^-2K^-1]

temperature of air = T => dimension of T = [K]

mass per unit mole of air = u => dimension of u = [M]

angular frequency = w => dimension of w = [T^-1]

from dimensional analysis,

w=kL^xR^yT^zu^s

or, [T^-1] = k[L]^x [ML²T^-2K^-1]^y [K]^z [M]^s

= k[M^(y+s)] [L^(x+2y)] [T^-2y] [K^(-y+z)]

comparing both sides,

y + s = 0.....(1)

x + 2y = 0 .....(2)

-2y = -1 => y = 1/2

-y + z = 0 => z = y = 1/2

putting y in equation (1), s = -1/2

from equation (2), x = -2y = -1

so, $w=kL^{-1}R^{1/2}T^{1/2}u^{-1/2}$

or, $w=k\sqrt{\frac{RT}{uL^2}}$This is required expression

Answer 2

Explanation:

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