Question

Give that the time period T of ossilation of a gas bubble from an explosion under water depends uopon P, D and E ehere P is the static pressure, D is the density of water and E is the total energy of explosion, find dimensionally the relation of T.

Answer 1

period T → sec → T
pressure P → N/m² → ML⁻¹T⁻²
density d → kg/m³ → ML⁻³
energy E → Joule → ML²T⁻²
constanta k → has no dimension

assume : T = k(P^α)(d^β)(E^γ) , so as dimensionally we find
(M^0)(L^0)(T^1) = (ML⁻¹T⁻²)^α • (ML⁻³)^β • (ML²T⁻²)^γ
(M^0)(L^0)(T^1) = M^(α + β + γ) • L^(-α - 3β + 2γ) • T^(-2α - 2γ)

then
0 = α + β + γ
0 = -α - 3β + 2γ
1 = -2α - 2γ

solve for α,β and γ until we get
α = -5/6
β = 1/2
γ = 1/3

conclusion
T = k(P^-5/6)(d^1/2)(E^1/3)
T = k(d^3 E^2 / P^5)^(1/6)