Question

a jet of water of cross sectional area A and velocity v impinges normally on a stationary flat plate. the mass per unitvolume of water is p. by dimensional analysis determine an expression for the force F exerted by the jet against the plate.

Answer 1

We are given,

Area of cross-section = A,

Velocity of jetstream = v,

Mass per unit volume,i.e., density = ρ,

We know,

[F] = [MLT-2]

also,

[​ρ] = [ML3]

[A] = [L2]​

[v] = [LT-1]

Supposing the force depends on A,v and ​​ρ,

F​∝Aa.vb.​ρc

=> [F] = [A]a.[v]b.[​ρ]c

= [L2]a.[LT-1]b.[ML-3]c

so, [MLT-2] = [Mc.L2a + b - 3c​.T-b]

by comparing,

c = 1

2a + b - 3c = 1

-b = -2

=> a = 1; b = 2; c = 1

so,

F​∝Aa.vb.​ρc

F​∝Av2ρ

=> ​F = kAv2ρ

note: k is a proportionality constant... it is a rule that, after using dimensional analysis for evaluation, we have to keep the proportionality symbol,i.e., ​∝, or put = and put k... so, u do not do this x(​F = Av2ρ)x as this might not be always correct.