A ball of radius r and density ρ dropped through a viscous liquid of density σ and viscosity η attains its terminal velocity at time t, given by t = Aρᵃrᵇηᶜσᵈ, where A is a constant and a, b c and d are integers. The value of (b + c)/(a + d) i s ______.
Physics · Unit dimension and error · Dimension and its application
JEE Mains 2026 23 January Evening Shift

Question

A ball of radius r and density ρ dropped through a viscous liquid of density σ and viscosity η attains its terminal velocity at time t, given by t = Aρᵃrᵇηᶜσᵈ, where A is a constant and a, b c and d are integers. The value of (b + c)/(a + d) i s ______.

Physics · Unit dimension and error · Dimension and its application

JEE Mains 2026 23 January Evening Shift

G1 Physics · Accepted Answer

Given, t = Aρᵃrᵇηᶜσᵈ. Write dimensions: [t] = T, [ρ] = M L⁻³, [r] = L, [η] = M L⁻¹ T⁻¹, [σ] = M L⁻³. Therefore, T = (M L⁻³)ᵃ Lᵇ (M L⁻¹ T⁻¹)ᶜ (M L⁻³)ᵈ. Compare the power of T: -c = 1. Hence, c = -1. Compare the power of M: a+c+d = 0. Therefore, a+d = 1. Compare the power of L: -3a+b-c-3d = 0. Using c = -1 and a+d = 1, we get -3(a+d)+b+1 = 0. Thus, -3+b+1 = 0, so b = 2. Therefore, (b+c)/(a+d) = (2-1)/1 = 1. Hence the answer is 1.