Friction and Drag: Static & Kinetic Friction, Drag Forces, and Terminal Speed

Friction and Drag

Introduction

Friction and drag are resistive forces that oppose the motion of objects. Friction acts between solid surfaces, while drag acts on objects moving through fluids (liquids or gases). Understanding these forces is essential for analyzing motion in real-world systems, from sliding blocks to falling objects and biological locomotion.

Friction

Static Friction

Static friction is the force that prevents an object from moving when a force is applied. It acts in the direction opposite to the impending motion and adjusts up to a maximum value to prevent slipping.

• Maximum static friction: $f_{s,\text{max}} = \mu_s n$

• $\mu_s$ is the coefficient of static friction, dependent on the materials in contact.

• $n$ is the normal force (perpendicular contact force between surfaces).

• Static friction force points opposite to the direction the object would move if friction were absent.

Example: Penguins sliding on ice experience very low static friction, allowing them to travel long distances efficiently.

Comparing Static Friction on Different Surfaces

When identical blocks rest on ramps with different surface materials but the same angle, the magnitude of static friction is the same for each block, as it balances the component of gravity parallel to the ramp. The coefficient of static friction only determines the maximum possible value, not the actual value unless the block is on the verge of sliding.

Critical Angle and Coefficient of Static Friction

The critical angle is the angle at which a block just begins to slide down an incline. At this point, the maximum static friction is reached:

• Decompose gravity: $w_x = -mg\sin\theta$, $w_y = -mg\cos\theta$

• At the threshold of motion: $f_{s,\text{max}} = mg\sin\theta$ and $n = mg\cos\theta$

• So, $\mu_s = \tan\theta$

Example: If a block starts sliding at $30^\circ$, $\mu_s = \tan 30^\circ = 0.58$.

Kinetic Friction

Kinetic friction acts when two surfaces are sliding past each other. It is generally less than the maximum static friction for the same materials.

• Kinetic friction force: $f_k = \mu_k n$

• $\mu_k$ is the coefficient of kinetic friction.

• Once motion starts, friction drops from its maximum static value to the kinetic value.

Key Properties

• Kinetic friction does not depend on the speed of sliding or the contact area (for most practical cases).

• For identical blocks moving at different speeds, the kinetic friction force is the same if the normal force is unchanged.

Coefficients of Friction for Common Materials

Materials	Static ($\mu_s$)	Kinetic ($\mu_k$)
Rubber on concrete	1.00	0.80
Steel on steel (dry)	0.80	0.60
Steel on steel (lubricated)	0.10	0.05
Wood on wood	0.50	0.20
Wood on snow	0.12	0.06
Ice on ice	0.10	0.03

Friction in Human Joints

Human joints are covered by cartilage and lubricated by synovial fluid, resulting in extremely low coefficients of friction ($\mu_s \approx 0.016$, $\mu_k \approx 0.015$). Artificial joints have higher friction, but still much lower than most engineering materials.

Drag

Introduction to Drag

Drag is a resistive force experienced by objects moving through fluids. It acts opposite to the direction of motion and increases with speed. The nature of drag depends on the flow regime, characterized by the Reynolds number.

Reynolds Number

The Reynolds number ($Re$) is a dimensionless quantity that predicts flow patterns in different fluid flow situations:

• $Re = \frac{\rho v L}{\eta}$

• $\rho$ = fluid density, $v$ = object speed, $L$ = characteristic length, $\eta$ = fluid viscosity

• High $Re$ ($>1000$): Inertial forces dominate (turbulent flow)

• Low $Re$ ($<1$): Viscous forces dominate (laminar flow)

Drag at High Reynolds Number (Quadratic Drag)

For high Reynolds numbers, drag force is proportional to the square of the speed:

• $D = \frac{1}{2} C_D \rho A v^2$

• $C_D$ = drag coefficient (depends on shape), $A$ = cross-sectional area

• Direction is opposite to motion

Object	$C_D$
Commercial airliner	0.024
Toyota Prius	0.24
Pitched baseball	0.35
Racing cyclist	0.88
Running person	1.2

Drag at Low Reynolds Number (Viscous Drag)

For low Reynolds numbers, drag force is proportional to speed (Stokes' Law):

• $D = 6 \pi \eta r v$

• $\eta$ = viscosity, $r$ = radius of sphere, $v$ = speed

• Direction is opposite to motion

Terminal Speed

Terminal speed is reached when the drag force equals the gravitational force, resulting in zero net acceleration and constant velocity.

• At terminal speed: $D = mg$

• For high $Re$: $v_t = \sqrt{\frac{2mg}{C_D \rho A}}$

Example: Parachutists and gliding animals reach terminal speed when the upward drag balances their weight.

Comparing Terminal Speeds

For spheres of different sizes and masses, terminal speed depends on both mass and cross-sectional area. For example, if Sphere 1 has twice the radius and eight times the mass of Sphere 2, its terminal speed is twice as large ($v_1 = 2v_2$).

Summary Table: Key Equations

Force	Equation	Variables
Static friction (max)	$f_{s,\text{max}} = \mu_s n$	$\mu_s$ = static friction coefficient, $n$ = normal force
Kinetic friction	$f_k = \mu_k n$	$\mu_k$ = kinetic friction coefficient, $n$ = normal force
Quadratic drag (high $Re$)	$D = \frac{1}{2} C_D \rho A v^2$	$C_D$ = drag coefficient, $\rho$ = fluid density, $A$ = area, $v$ = speed
Viscous drag (low $Re$)	$D = 6 \pi \eta r v$	$\eta$ = viscosity, $r$ = radius, $v$ = speed
Terminal speed (high $Re$)	$v_t = \sqrt{\frac{2mg}{C_D \rho A}}$	$m$ = mass, $g$ = gravity, $C_D$ = drag coefficient, $\rho$ = fluid density, $A$ = area

Additional info: The notes above expand on the original lecture content by providing definitions, formulas, and examples for each concept, as well as relevant tables and diagrams to reinforce understanding.