Forces in Translational Dynamics: How Weight, Friction, and Springs Shape Motion

Gravitational force

The attractive interaction between any two objects that have mass; near Earth it pulls objects toward Earth’s center.

Weight

The gravitational force Earth exerts on an object; measured in newtons (N).

Mass

A measure of how much matter an object has and its resistance to changes in motion (inertia); measured in kilograms (kg).

Near-Earth gravitational force equation

The approximation for weight near Earth’s surface: F_g = mg.

Gravitational field strength (g)

The magnitude of Earth’s gravitational field near the surface, about 9.8 m/s^2.

Newton’s Law of Universal Gravitation

The general gravitational force relationship: F = G(m1 m2)/r^2, where r is the center-to-center distance.

Universal gravitational constant (G)

The constant in Newton’s universal gravitation law that sets the strength of gravitational interaction.

Weight direction

Weight (F_g) always points vertically downward (toward Earth’s center), even on an incline.

Gravity component parallel to an incline

The component of weight along the ramp: F_{g,‖} = mg sinθ, directed down the ramp.

Gravity component perpendicular to an incline

The component of weight into the surface: F_{g,⊥} = mg cosθ, directed into the ramp.

Normal force (F_N)

A contact force exerted by a surface on an object, perpendicular to the surface.

Equilibrium (translational)

A condition where the net force is zero, so acceleration is zero (∑F = 0).

Net force

The vector sum of all forces acting on an object; determines acceleration via Newton’s Second Law.

Free-body diagram (FBD)

A diagram showing all external forces acting on an object, used to apply Newton’s Laws.

Friction

A contact force parallel to surfaces in contact that opposes relative motion or the tendency for relative motion.

Static friction (f_s)

Friction between surfaces that are not sliding; its magnitude adjusts as needed up to a maximum.

Maximum static friction (f_{s,max})

The largest possible static friction magnitude before slipping begins: f{s,max} = μs F_N.

Kinetic friction (f_k)

Friction between sliding surfaces; modeled with constant magnitude fk = μk F_N (for dry friction problems).

Coefficient of static friction (μ_s)

A unitless constant for a pair of surfaces that sets the maximum static friction (typically larger than μ_k).

Coefficient of kinetic friction (μ_k)

A unitless constant for a pair of surfaces that sets kinetic friction during sliding.

Friction direction rule (relative motion method)

Determine how surfaces would (or do) move relative to each other at the contact; friction points opposite that tendency (static) or opposite actual sliding (kinetic).

Threshold of slipping

The instant an object is just about to move, when required static friction equals f_{s,max}.

Spring force

A contact restoring force exerted by a stretched or compressed spring that acts to return the spring to its natural length.

Hooke’s Law

For an ideal spring, spring force magnitude is proportional to displacement from natural length: Fs = kx (often written in 1D as F{s,x} = −kx).

Spring constant (k)

A measure of spring stiffness in N/m; larger k means a stiffer spring that stretches/compresses less for the same force.