Viscosity, Terminal Velocity & Stoke’s Law Explained (JAMB/WAEC Physics, Friction 2)
This is Part 2 of our Friction series, going deeper into viscosity, terminal velocity, and Stoke’s Law than the introduction covered in Part 1.
These three ideas are closely connected: viscosity is a fluid’s internal resistance to flow, Stoke’s Law quantifies the drag force this resistance creates on a moving object, and terminal velocity is the constant speed an object reaches once that drag force balances the forces pulling it through the fluid.
This topic is a regular feature in JAMB and WAEC Physics, usually tested through terminal velocity calculations. This lesson works through the full derivation with worked examples.
Quick takeaways
- Viscosity measures a fluid’s internal resistance to flow, and it directly determines how much drag force a fluid exerts on an object moving through it.
- Stoke’s Law, F = 6πηrv, gives the viscous drag force on a small sphere moving through a fluid at low speed.
- Terminal velocity is the constant maximum speed an object reaches when falling through a fluid, once the net force acting on it becomes zero.
- At terminal velocity, the object’s weight is exactly balanced by the sum of the upward forces: the viscous drag force and the upthrust (buoyant force).
- Exam questions typically ask you to calculate terminal velocity by setting up and solving the force-balance equation at that point.
Timestamps
- 0:00 Introduction
- 0:30 Definition of Viscosity
- 1:02 Viscosity and Thermal Velocity — Overview
- 1:16 Nature of the Liquid
- 2:20 Molecular Interactions
- 3:09 Effect of Temperature on Viscosity
- 3:56 Applications of Viscosity
- 5:15 Definition of Thermal Velocity
- 5:48 Factors Affecting Thermal Velocity
- 9:15 Applications of Thermal Velocity
- 11:15 Viscosity vs Thermal Velocity — Comparison
- 12:23 Stoke’s Law
- 13:50 Mathematical Formulation of Stoke’s Law
What forces act on an object falling through a fluid?
When an object falls through a fluid (such as a small ball dropped into oil), three forces act on it:
- Weight (W) — the downward gravitational force, W = mg
- Upthrust (U) — the upward buoyant force exerted by the displaced fluid, described by Archimedes’ principle
- Viscous drag force (F) — the upward resistive force due to the fluid’s viscosity, described by Stoke’s Law: F = 6πηrv
As the object accelerates downward, the viscous drag force increases with velocity, until eventually the upward forces balance the downward weight.
What is terminal velocity, and how is it reached?
Terminal velocity is the constant, maximum velocity an object reaches when falling through a fluid, occurring once the net force acting on it becomes zero — at this point, the object stops accelerating and continues falling at a steady speed. This happens when:
Weight = Upthrust + Viscous drag force mg = U + 6πηrv
Before reaching terminal velocity, the object accelerates because its weight exceeds the combined upward forces; once these forces balance exactly, acceleration becomes zero and the velocity remains constant from that point onward.
How do you calculate terminal velocity using Stoke’s Law?
To find the terminal velocity of a small sphere falling through a fluid:
- Write the force-balance equation at terminal velocity: mg = U + 6πηrv
- Substitute known expressions, such as weight (mg) and upthrust (using the volume of the sphere and fluid density, from Archimedes’ principle)
- Rearrange the equation to solve for v (terminal velocity), isolating it algebraically
- Substitute the given numerical values for viscosity (η), radius (r), and the other known quantities to calculate the final terminal velocity
Why does viscosity affect how quickly an object reaches terminal velocity?
A more viscous fluid produces a larger drag force for a given velocity, since viscosity (η) appears directly in Stoke’s Law. This means an object falling through a highly viscous fluid (like honey) reaches terminal velocity more quickly and at a lower speed, compared to the same object falling through a less viscous fluid (like water), where it must reach a higher velocity before the drag force becomes large enough to balance its weight.
Common mistakes students make with this topic
- Forgetting to include upthrust in the force-balance equation, and only balancing weight against viscous drag
- Rearranging the terminal velocity equation incorrectly when solving for v
- Confusing the role of viscosity — higher viscosity increases drag force at a given speed, it doesn’t directly change the object’s weight or upthrust
- Assuming an object reaches terminal velocity immediately, rather than understanding it accelerates until the forces balance
Frequently asked questions
Q: What is terminal velocity? Terminal velocity is the constant, maximum speed an object reaches when falling through a fluid, occurring once the net force acting on it becomes zero because the upward forces (upthrust and viscous drag) exactly balance its weight.
Q: What is the equation used to find terminal velocity? The force-balance equation at terminal velocity is mg = U + 6πηrv, where mg is the object’s weight, U is the upthrust, and 6πηrv is the viscous drag force from Stoke’s Law.
Q: Why does a more viscous fluid affect terminal velocity? A more viscous fluid produces a greater drag force at a given speed, meaning an object reaches terminal velocity more quickly and at a lower overall speed compared to a less viscous fluid.
Q: What three forces act on an object falling through a fluid? The three forces are weight (acting downward), upthrust (acting upward, from the displaced fluid), and viscous drag force (acting upward, opposing motion through the fluid).
New to this series? Start with Friction 1 (static, dynamic friction and coefficient of friction) →
This is one topic from our full JAMB & WAEC Physics course — see everything covered →
