Applications of Newton’s Laws in AP Physics C Mechanics: Modeling Real Forces

Friction

A contact force along the surface of contact that opposes relative motion (kinetic) or impending relative motion (static) between two surfaces.

Static friction (fs)

Friction acting when surfaces are not sliding; it self-adjusts to whatever value is needed to prevent slipping, up to a maximum.

Maximum static friction (fs,max)

The largest possible static friction magnitude before slipping begins: fs,max = μs N.

Coefficient of static friction (μs)

A dimensionless constant that sets the upper limit of static friction via fs ≤ μs N.

Kinetic friction (fk)

Friction acting when surfaces are sliding; in the standard AP model its magnitude is fixed for a given normal force: fk = μk N.

Coefficient of kinetic friction (μk)

A dimensionless constant that determines kinetic friction magnitude via fk = μk N; typically μk < μs.

Friction direction rule (static)

Static friction points opposite the motion that would occur if friction were absent (opposes the tendency to slip).

Normal force (N)

The contact force exerted perpendicular to a surface; it must be found from Newton’s 2nd law perpendicular to the surface and is not automatically equal to mg.

Free-body diagram (FBD)

A diagram showing only the external forces acting on an object (e.g., weight, normal, friction, tension, applied force) used to apply Newton’s laws.

Incline weight components

For a block on an incline: component parallel to plane is mg sinθ (down the slope) and perpendicular is mg cosθ (into the plane).

Normal force on an incline

If no acceleration perpendicular to the plane, N = mg cosθ for a block resting/moving on an incline.

Static no-slip condition on an incline

A block can remain at rest if mg sinθ ≤ μs mg cosθ, equivalently tanθ ≤ μs.

Sliding acceleration on an incline (with kinetic friction)

If the block slides down, a = g( sinθ − μk cosθ ) along the incline (down-slope taken positive).

Angled pull reduces normal force

For a block pulled by force F at angle φ above horizontal (no vertical acceleration): N = mg − F sinφ, reducing friction.

Drag (air resistance)

A resistive force from a fluid that opposes an object’s motion relative to the fluid; it typically makes acceleration decrease as speed increases.

Linear drag model

A drag model (often low-speed/viscous) where drag magnitude is proportional to speed: FD = b v.

Quadratic drag model

A drag model (often higher-speed in air) where drag magnitude is proportional to speed squared: FD = c v².

Drag direction

Drag always points opposite the velocity vector (opposite the direction of motion relative to the fluid).

Terminal velocity (vt)

The constant speed reached when net force becomes zero, so acceleration becomes zero (forces balance, not vanish).

Terminal speed with linear drag

For a falling object with linear drag: vt = mg/b (from setting mg = b vt).

Terminal speed with quadratic drag

For a falling object with quadratic drag: vt = sqrt(mg/c) (from setting mg = c vt²).

Velocity vs. time with linear drag

For a drop from rest under linear drag: v(t) = (mg/b)(1 − e^{-(b/m)t}), approaching vt asymptotically.

Hooke’s law

The ideal spring-force model in 1D: Fs = −k x, where x is displacement from natural length and the minus sign indicates a restoring force.

Spring constant (k)

A measure of spring stiffness (units N/m) relating displacement to spring force magnitude in Hooke’s law.

Static equilibrium with a vertical spring

For mass m hanging at rest: kx = mg, so the extension is x = mg/k (spring stretches until it balances weight).