A small ball of mass m and density ρ is dropped in a viscous liquid of density ρ₀. After sometime, the ball falls with constant velocity. The viscous force on the ball is :

Solution

Physics sir · 4 steps

Constant velocity means zero net force, so the viscous drag balances weight minus buoyancy.

1. Forces on the ball. Weight mg acts down. Buoyancy F_B and the viscous force Fᵥ both act up, because the ball moves down and drag opposes motion.
2. Terminal velocity. At constant velocity the acceleration is zero, so the forces balance: Fᵥ + F_B = mg.
3. Write the buoyancy. The ball's volume is V = m/ρ. The displaced liquid has mass Vρ₀, so F_B = m/ρρ₀ g.

   Fᵥ = mg - mρ₀/ρg = mg(1 - ρ₀/ρ)

4. So. The viscous force is mg(1-ρ₀/ρ), which is option (a).

The idea to keep. Terminal velocity always means net force zero. Write buoyancy as Vρ₀ g with V = m/ρ, and the drag is whatever is left of the weight.

Answer from the NTA final answer key. Solution by PrepWiser.