An airplane is warming up its engine on the ground. The intensity of sound energ
ID: 3281544 • Letter: A
Question
An airplane is warming up its engine on the ground. The intensity of sound energy U (dimension ML-1 T-2) from the propeller at a distance d ahead of the airplane depends in addition to d on the diameter D of the propeller, the rotational speed (dimension T-1) of the propeller, the mass density (dimension M L-3) of air, the air pressure p (dimension M L-i T-2), and the air viscosity (dimension M L-1 T"). a) Use dimensional analysis to identify four nondimensional numbers: the nondimensional intensity of sound energy U. that is proportional to U, the nondimensional distance d. that is proportional to d, the nondimensional rotational speed o. that is proportional to o, and the nondimensional viscosity that is proportional to . Use the exponents of U, d, d, and as independent parameters to find these numbers. Write these numbers by using the velocity scale v, (p/p)12, which is proportional to the velocity of sound. b) Present a formula for the calculation of U by using d., o., , and p.Explanation / Answer
Intensity of sound energy dimensions U = [M L-1 T-2]
U depends on distnace d [L]
diameter of propellor D [L]
rotational speed , w = [T-1]
mass denisty of air, rho [ML-3]
pressure P [M L-1 T-2]
viscosity, mu [M L-1 T-1]
a. so, non dimensional sound intensity Un = U/P ( P being pressure )
non dimensional distance dn = d/D ( D being propellor diameter)
non dimensional rotational speed, wn = w*D/sqroot(P/rho) ( mu being viscosity)
non dimensional viscosity, mun = mu/sqroot(P/rho)*rho*D
let sqroot(P/rho) = vp
then
Un = U/vp^2*rho
dn = d/D
wn = w*D/vp
mun = mu/vp*rho*D
b. U = (dn)^a * (wn)^b * (mun)^c * p^d
now, U should be inverly proportional to square of dn ( inverse square law)
directly proportional to wn squared ( proportional to energy )
directly proportional to mun
hence
U = (wn/dn)^2 * mun * P