A ball bearing falls through a viscous liquid (a) Using a labelled diagram of a tall vessel, show all the forces acting on it (b) When will it attain termin...
Question 1 Report
A ball bearing falls through a viscous liquid
(a) Using a labelled diagram of a tall vessel, show all the forces acting on it
(b) When will it attain terminal velocity?
Answer Details
Ball bearing falling through a viscous liquid
(a) Labelled diagram showing the forces acting on the ball
Inside a tall vessel of viscous liquid, a spherical ball bearing falling vertically downwards experiences three forces:
- Weight, \(W = mg\), acting vertically downwards through the centre of the ball.
- Upthrust (buoyancy), \(U\), acting vertically upwards, equal to the weight of liquid displaced.
- Viscous drag, \(F\), a fluid-friction force given by Stokes' law \(F = 6\pi\eta r v\), acting vertically upwards, opposing the motion.
(b) When terminal velocity is attained
As the ball accelerates downwards its speed \(v\) increases, and because the viscous drag \(F = 6\pi\eta r v\) grows with speed, the total upward force \((U + F)\) increases too. The ball attains its terminal velocity at the instant the upward forces exactly balance the downward weight, so the resultant force and hence the acceleration become zero:
\[ W = U + F \quad\Rightarrow\quad mg = U + 6\pi\eta r v \]From that moment onwards there is no further acceleration, and the ball continues to fall through the liquid with a constant (maximum, uniform) velocity, which is the terminal velocity.
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