Question

A metal sphere of radius 'R' and density 'ρ₁' is dropped in a liquid of density 'σ' moves with terminal velocity 'v'. Another metal sphere of same radius and density 'ρ₂' is dropped in the same liquid, its terminal velocity will be ______.

विकल्प

• v [(ρ₁ - σ)/(ρ₂ - σ)]

• v [(ρ₂ + σ)/(ρ₁ + σ)]

• v [(ρ₂ - σ)/(ρ₁ + σ)]

• v [(ρ₁ + σ)/(ρ₂ + σ)]

Answer 1

A metal sphere of radius 'R' and density 'ρ₁' is dropped in a liquid of density 'σ' moves with terminal velocity 'v'. Another metal sphere of same radius and density 'ρ₂' is dropped in the same liquid, its terminal velocity will be v [(ρ₂ - σ)/(ρ₁ + σ)].

Explanation:

The terminal velocity of the sphere

v = `2/9 "R"^2 ((rho_1 - sigma)"g")/eta`   ...(i)

Here, R = radius of sphere, ρ₁ = density of sphere and σ = density of liquid.

Now, if density of metal sphere is changed to ρ₂, then terminal velocity

`"v"_2 = 2/9 ("R"^2 (rho_2 - sigma)"g")/eta`   ...(ii)

Divide Eqs. (i) and (ii), we get

`"v"_2/"v" = (rho_2 - sigma)/(rho_1 - sigma) "or"  "v"_2 = ("v"(rho_2 - sigma))/((rho_1 - sigma))`

Hence, the terminal velocity is `("v"(rho_2 - sigma))/((rho_1 - sigma))`