A metal ball of weight W falls through a column of glycerine of viscosity V. If the ball experiences an upthrust U and terminal velocity is attained, then

Analyze the question A metal ball falls through glycerine. Three forces act on it:

• Weight W, acting downward
• Upthrust U, acting upward
• Viscous drag, acting upward (it opposes the motion)

The question calls the viscous force "V" (it says "viscosity V", but in this context V stands for the viscous drag force).

Concept At terminal velocity, acceleration is zero. By Newton's first law, the net force is zero, so the total upward force equals the total downward force.

Solution

• Downward force: W
• Upward forces: U + V (upthrust plus viscous drag)

At terminal velocity:

W = U + V

Checking the other options

• A (W > U + V): the net force would be downward, so the ball would still be accelerating. This is not terminal velocity.
• C (W = U - V): this would treat drag as acting downward, which is wrong because drag opposes the downward motion.
• D (W < U + V): the net force would be upward, so the ball would be slowing down.

Exam tip For terminal velocity problems, remember "forces balance, acceleration is zero." Write the downward forces on one side and all the upward forces (upthrust and drag) on the other. Drag and upthrust always oppose the weight of a falling object.