Question

solve this problem in terms of dimensions

Answer 1

We have,

F = 6π η rv

We know,

F ∞ ηᵃ

F ∞ rᵇ

F ∞ vˣ

So, F = k ηᵃrᵇvˣ, where k = dimensionless constant

Now, expressing dimentionally, we get

[F] = [k][ηᵃ][rᵇ][vˣ]

Or [MLT⁻²] = [ML⁻¹T⁻¹]ᵃ [L]ᵇ [LT⁻¹]ˣ

Or MLT⁻² = Mᵃ L⁻ᵃ⁺ᵇ⁺ˣ T⁻ᵃ⁻ˣ

Now,

M = Mᵃ or L = L⁻ᵃ⁺ᵇ⁺ˣ or T⁻² = T⁻ᵃ⁻ˣ

Or a = 1 or 1 = -a+b+x or 1 = -a-x ⇒ -2 = -1 -x ⇒ x = 1

⇒ 1 = -1 +b+1
⇒ b = 1

Now,

F = kη¹r¹v¹

Or F = 6π ηrv