Try pushing a heavy box across the floor. At first it won't budge — you push harder and harder, and then suddenly it breaks free and slides. Once it's moving, keeping it going feels easier than starting it did. That stubborn, grippy force between the box and the floor is friction, and the whole story of this page is hidden in that one everyday shove.
Naming what's happening
Two things set how strong friction is: how hard the surfaces are pressed together, and how grippy they are. The "pressed together" part is the normal force $N$ — for a block sitting on a flat floor, that's just its weight, $N = mg$. The "grippiness" is a single number called the coefficient of friction $\mu$ (mu), which you read off a slider in the sim. The simplest rule for a sliding object is beautifully short:
Kinetic friction
$$f_k = \mu_k\, N$$
Put numbers in: a $2\ \text{kg}$ book on a table presses down with $N = mg \approx 2 \times 9.8 = 20\ \text{N}$. If the table's kinetic coefficient is $\mu_k = 0.3$, then friction pulls back with $f_k = 0.3 \times 20 = 6\ \text{N}$. Push with less than that while it's at rest and it stays put; once it slides, only $6\ \text{N}$ of drag opposes you.
Getting precise
Friction actually comes in two flavours. While the object is still, static friction silently grows to match whatever you apply, but only up to a ceiling: $f_s \le \mu_s N$. Cross that ceiling and it breaks loose. Because $\mu_s > \mu_k$, starting is always harder than keeping going. Tilt the surface instead of pushing, and the block slips exactly when the slope reaches the critical angle given by $\tan\theta_c = \mu_s$ — a neat way to measure $\mu_s$ with nothing but a protractor. Friction is also a vector: it always points opposite to the way the contact would slide, which is why the friction under your shoe actually pushes you forward when you walk. In the sim, the Applied Force, Mass, μ_k, μ_s, Gravity, and Incline θ sliders feed straight into these relationships and the force arrows update live.
Try this in the sim above
First, drag μ_k to $0$ and give the block a push — with no kinetic friction it never slows down, just like in deep space. Next, in Static Limit mode raise the applied force slowly and watch the block refuse to move until you exceed $\mu_s N$. Finally, switch to Critical Angle mode and increase $\theta$ until the block lets go — compare that tipping angle to $\arctan(\mu_s)$ and see how closely they agree.
Section 03
Equations & Derivation
Friction Forces
$$f_s \leq \mu_s N \quad\text{(static)},\qquad f_k = \mu_k N \quad\text{(kinetic)}$$
$$\tan\theta_c = \mu_s,\qquad W_{\text{friction}} = -f_k d = -\mu_k N d$$
Symbol Definitions
Symbol
Quantity
SI Unit
$f_s$
Static friction force (prevents sliding)
N
$f_k$
Kinetic friction force (opposes sliding)
N
$\mu_s$
Coefficient of static friction
dimensionless
$\mu_k$
Coefficient of kinetic friction (< μ_s)
dimensionless
$N$
Normal force (perpendicular to surface)
N
$\theta_c$
Critical angle for onset of sliding
°
1
Static friction is self-adjusting. $f_s$ adjusts to match the applied force up to $f_{s,\max}=\mu_s N$. Once this maximum is exceeded, sliding begins and kinetic friction $f_k=\mu_k N
2
Critical angle. On an incline, the object starts sliding when $mg\sin\theta > \mu_s mg\cos\theta$, i.e. $\tan\theta_c=\mu_s$. This is how $\mu_s$ is measured in the lab: slowly increase the angle until sliding begins.
3
Friction as non-conservative force. $W_f=-f_k d<0$: friction removes energy from the mechanical system, converting it to heat. The magnitude of heat generated equals $|W_f|=\mu_k mg\cos\theta\cdot d$.
4
Amontons's Laws (1699). Kinetic friction (1) is proportional to normal force, (2) is independent of contact area, (3) is independent of sliding speed (approximately). These are empirical — explained microscopically by adhesion and deformation of asperities.
At rest, surface asperities (microscopic bumps) interlock and form micro-welds. Once sliding begins, these junctions break repeatedly without fully re-forming, giving lower kinetic friction. This means it takes more force to start sliding than to maintain it.
Key takeaway: $\mu_s > \mu_k$ always: starting is harder than sliding.
Friction opposes relative sliding tendency, not motion itself. Static friction on a walking foot points forward — in the direction of motion. Friction drives a car forward (at the driving wheels). The key is: friction opposes the direction the surface would slide if friction were absent.
Key takeaway: Friction opposes impending or actual relative sliding — not necessarily the direction of motion.
Car braking (kinetic friction between brake pads and rotors), ABS systems (maintaining static friction), walking (static friction at shoe-ground contact prevents slipping), writing (friction between pen and paper), machine bearings (friction must be minimised), rock climbing (grip depends on static friction coefficient).
Key takeaway: Friction enables walking, braking, and gripping — but must be minimised in bearings and engines.
Inclined plane method: place object on surface, slowly increase angle $\theta$ until sliding begins. Then $\tan\theta_c=\mu_s$. For $\mu_k$: push object on flat surface with known force and measure acceleration; $\mu_k=(F_{\text{applied}}-ma)/(mg)$. Or measure deceleration of sliding object: $\mu_k=a/g$.
Key takeaway: $\mu_s=\tan\theta_c$ at the critical angle; $\mu_k$ from deceleration measurement.
Resources: Khan Academy; HyperPhysics (hyperphysics.phy-astr.gsu.edu); MIT OCW 8.01/8.02.
Section 05
Common Misconceptions
❌ Friction always acts opposite to the direction of motion.
✅ Friction opposes relative sliding tendency, not motion. The friction on a car's driving wheels points forward — in the direction of motion — enabling the car to accelerate. The contact point would slide backward without friction, so friction acts forward.
❌ A heavier object experiences more friction and therefore decelerates faster.
✅ Heavier objects have larger friction force ($f_k=\mu_k mg$) but also larger inertia ($F=ma$ → $a=f_k/m=\mu_k g$). The mass cancels: all objects decelerate at the same rate $a=\mu_k g$ regardless of mass (same $\mu_k$, flat surface).
📖 HRW 10th Ed., §6-1.
❌ Friction does not depend on the smoothness of surfaces.
✅ This is only approximately true for rigid surfaces (Amontons's Laws). For very smooth surfaces, molecular adhesion increases friction. For very rough surfaces, normal force concentrations affect it. For soft materials (rubber, skin), real contact area changes with load, violating simple Coulomb friction.
📖 Serway & Jewett 8th Ed., §5.8; Dowson — History of Tribology.
Misconception research: Arons — A Guide to Introductory Physics Teaching; Halloun & Hestenes (1985), AJP 53.