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A solid sphere of radius r and density rhoₛ is released from rest in a liquid of density rhoₗ and coefficient of viscosity eta, where rhoₛ > rhoₗ and the liquid is deep enough. Which of the following statements are correct? (A) The buoyant force on the ball is (4/3)*pi*r³*rhoₗ*g. (B) The viscous drag on the sphere when moving at speed v is 6*pi*r*eta*v. (C) The speed of the sphere first increases and then becomes constant at terminal velocity. (D) The terminal velocity is 2*r²*(rhoₛ - rhoₗ)*g / (9*eta).
- (A) Buoyant force = (4/3)*pi*r³*rhoₗ*g
- (B) Viscous drag = 6*pi*r*eta*v
- (C) Speed increases then becomes constant
- (D) Terminal velocity = 2*r²*(rhoₛ - rhoₗ)*g / (9*eta)
Correct answer: (A) Buoyant force = (4/3)*pi*r³*rhoₗ*g
Solution
All four options (A), (B), (C), and (D) are correct. (A) follows from Archimedes' principle. (B) is Stokes' law. (C) describes the standard motion where net downward force decreases as speed increases until terminal velocity is reached. (D) is derived by setting gravity minus buoyancy equal to Stokes drag at terminal velocity.
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