Know that friction (drag) acts on an object moving through a liquid

1.5.1 Effects of Forces – Friction (Drag) in Liquids and Gases

Learning objective

Know that a frictional force called drag acts on any object moving through a fluid (liquid or gas) and be able to describe, both qualitatively and quantitatively, the factors that affect its magnitude.

Solid‑friction box (core requirement)

Solid friction is the force that opposes the relative motion of two solid surfaces in contact. It can cause deformation of the surfaces and the work done against friction is converted into heat.

Key definitions

• Friction: a force that opposes relative motion between two bodies.
• Drag: the form of friction experienced by an object moving through a fluid (liquid or gas).
• Viscosity (η): a measure of a fluid’s internal resistance to flow; higher viscosity → greater resistance.
• Density (ρ): mass per unit volume of the fluid (kg m⁻³).
• Drag coefficient (C_d): a dimensionless number that expresses how the shape and surface of an object influence drag.

Why drag occurs

When an object moves through a fluid its surface pushes on the surrounding fluid molecules. By Newton’s third law the fluid exerts an equal and opposite force on the object. This opposite force is the drag force (F_d) and it always acts opposite to the direction of motion.

General drag equation (quadratic form)

For steady, incompressible flow at moderate to high Reynolds numbers (inertial forces dominate) the drag force is

\[ F_d = \tfrac12 \, C_d \, \rho \, A \, v^{2} \]

where

Symbol	Quantity	Units
Fd	drag force	N
Cd	drag coefficient (shape dependent)	–
ρ	density of the fluid (liquid or gas)	kg m⁻³
A	projected cross‑sectional area perpendicular to the flow	m²
v	speed of the object relative to the fluid	m s⁻¹

Assumptions: steady flow, incompressible fluid, Reynolds number large enough that the quadratic term dominates. The same expression is used for liquids and gases; only ρ and C_d differ.

Low‑speed (viscous) drag – Stokes’ law

When the object is very small and moves slowly (low Reynolds number, laminar flow) the drag varies linearly with speed:

\[ F_d = 6\pi \, \eta \, r \, v \]

• η – dynamic viscosity of the fluid (Pa·s)
• r – radius of a spherical particle (m)
• v – speed relative to the fluid (m s⁻¹)

Useful for droplets in oil, pollen in air, or microscopic beads in water.

Factors that influence drag

Factor	Effect on drag
Speed (v)	High‑Re regime: Fd ∝ v² (doubling speed → four‑fold drag). Low‑Re regime (Stokes): Fd ∝ v.
Cross‑sectional area (A)	Drag is directly proportional to the area presented to the flow.
Fluid density (ρ)	Denser fluids (water, air at sea level) produce more drag than less dense fluids (oil, thin air at altitude) for the same A and v.
Shape (drag coefficient Cd)	Streamlined shapes: Cd≈0.04–0.1; blunt shapes: Cd≈0.8–1.2.
Viscosity (η)	Higher viscosity increases drag, especially in the low‑Re (Stokes) regime.
Surface roughness / texture	Rough surfaces can increase turbulence, raising the effective Cd.

Quantitative example – drag in a liquid

Find the drag on a smooth sphere (diameter 0.10 m, C_d≈0.47) moving at 2 m s⁻¹ through water (ρ = 1000 kg m⁻³).

• Radius = 0.05 m → projected area \(A = \pi r^{2}=7.85\times10^{-3}\;{\rm m^{2}}\).
• Insert into the quadratic equation \[ F_d = \tfrac12 (0.47)(1000)(7.85\times10^{-3})(2)^{2} \approx 7.3\;{\rm N} \]

Quantitative example – drag in a gas

A sky‑diver (area ≈ 0.7 m², C_d≈1.0) falls through air at sea level (ρ ≈ 1.2 kg m⁻³). At a speed of 55 m s⁻¹ (≈ 200 km h⁻¹):

\[ F_d = \tfrac12 (1.0)(1.2)(0.7)(55)^{2} \approx 1.1\times10^{3}\;{\rm N} \]

This large drag balances the diver’s weight, giving the terminal velocity.

Link to terminal velocity

When an object falls through a fluid, drag increases with speed until it equals the weight (mg). Setting \(F_d = mg\) and solving for v gives the terminal velocity:

\[ v_t = \sqrt{\frac{2mg}{C_d \rho A}} \]

For very small particles the Stokes form is used instead of the quadratic form.

Everyday examples

1. A swimmer feels increasing resistance as they accelerate – the drag force they must overcome.
2. Raindrops reach a constant falling speed because air drag balances gravity.
3. Boat hulls are shaped to give a low C_d, reducing the power needed to maintain speed.
4. A sky‑diver in a spread‑eagle position experiences much more drag than when diving head‑down.
5. Oil‑filled dampers in car suspensions rely on viscous (Stokes) drag to absorb shocks.

Common misconceptions

• “Drag only occurs in air.” – Drag exists in any fluid, including water, oil, honey, and even thin gases.
• “Drag always increases linearly with speed.” – At high speeds the relationship is quadratic; at very low speeds it is linear (Stokes’ law).
• “All objects of the same size experience the same drag.” – Shape, surface texture and orientation change the drag coefficient.
• “Viscosity matters only for thick liquids.” – Even gases have viscosity; it becomes important for very small or very slow objects.

Practical investigation – Influence of shape and fluid

1. Prepare three objects of equal mass and volume but different shapes (sphere, flat plate, streamlined body).
2. Attach each to a spring balance. Using a pulley, pull the object through a water tank at a constant speed (measure speed with a stopwatch over a known distance). Record the spring‑balance reading – this is the drag force in a liquid.
3. Repeat the same experiment in a transparent acrylic tube filled with air (use a low‑friction guide to keep the object moving straight). Record the drag force in a gas.
4. Compare the forces. Discuss how the differences arise from changes in fluid density, viscosity, and drag coefficient.

Safety reminders

• Keep all electrical equipment away from water; ensure hands are dry before handling the pulley system.
• Wear safety goggles to protect eyes from splashing water or debris.
• Secure the water tank and acrylic tube to prevent tipping.
• Do not lift heavy objects while the system is moving – use a stand or clamp.

Summary

• Drag is a type of friction that opposes motion through any fluid (liquid or gas).
• Its magnitude is given by the quadratic equation \(F_d = \tfrac12 C_d \rho A v^{2}\) for most everyday situations; for very small or slow objects Stokes’ law \(F_d = 6\pi\eta r v\) applies.
• Key influencing factors: speed, projected area, fluid density, shape (drag coefficient), viscosity, and surface texture.
• Understanding drag explains terminal velocity and guides the design of swimmers, boats, aircraft, cars, and sports equipment.