Question

A spherical solid ball of volume V is made up of a material of density ρ₁. It is falling through the liquid of density ρ₂(ρ₂ < ρ₁). Assume that, the liquid applies a viscous force on the ball that is proportional to the square of the speed v_t, i.e., `["F"_"viscous" = "KV"_"t"^2]`, then the terminal speed of the ball is ______.

(g = acceleration due to gravity)

पर्याय

• `sqrt((K(rho_1 - rho_2))/(Vg))`

• `sqrt((V(rho_1 -rho_2)g)/K)`

• `sqrt((Vg(rho_1 - rho_2))/K)`

• `sqrt((Vg(rho_1 - rho_2))/(2K))`

Answer 1

A spherical solid ball of volume V is made up of a material of density ρ₁. It is falling through the liquid of density ρ₂(ρ₂ < ρ₁). Assume that, the liquid applies a viscous force on the ball that is proportional to the square of the speed v_t, i.e., `["F"_"viscous" = "KV"_"t"^2]`, then the terminal speed of the ball is `underlinebb(sqrt((V(rho_1 -rho_2)g)/K))`.

Explanation:

The given situation is shown below.

When the ball moves with terminal velocity (v_t), then

`"F"_"viscous" + "F"_"upthrust" = "w" ⇒ "Kv"_"t"^2 + "V"rho_2"g" = "mg"`

⇒ `"KV"_"t"^2 + "V"_{rho_2"g"} = "V"_{rho_1"g"} ⇒ "Kv"_"t"^2 = "V"(rho_1 - rho_2)"g"`

v_t = `sqrt(("V"(rho_1 - rho_2)"g")/"K")`