A solid sphere of density d₁ is released from rest in a long column of a viscous liquid of density d₂ ( d₁ > d₂ ). The sphere eventually attains a constant terminal velocity v₀ . What is the acceleration of the sphere at the instant its downward velocity is v₀ 4 ?

Step-by-step solution

Let the volume of the sphere be V . The forces acting on the sphere are its weight W = V d₁ g downwards, buoyant force F_B = V d₂ g upwards, and viscous drag force F_v = kv upwards. The net downward force is F_ net = W - F_B - F_v ma = V d₁ g - V d₂ g - kv a = g (1 - d₂ d₁ ) - k V d₁ v When the sphere attains terminal velocity v₀ , its acceleration is zero ( a = 0 ). 0 = g (1 - d₂ d₁ ) - k V d₁ v₀ k V d₁ = g v₀ (1 - d₂ d₁ ) We need to find the acceleration a' when the velocity is v = v₀ 4 . Substituting the value o