Force

A force is an interaction that changes, or tends to change, the motion of a body. In Newtonian mechanics the net force is related to acceleration by Newton’s Laws of Motion:

$$\mathbf{F}=m\mathbf{a}=m\frac{d^2\mathbf{x}}{d{t}^2}.$$

So force is the bridge from kinematics to dynamics: Velocity and acceleration describe motion, while force explains why that motion changes.

Units

The SI unit is the Newton,

$$1\mathrm{N}=1\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-2}.$$

This also links to Kilogram, Metre, and Second.

Work and energy

A force applied through a displacement does Work:

$$W=\int \mathbf{F}\cdot d\mathbf{x}.$$

The dot product means only the component of force along the displacement transfers Energy. If the force is perpendicular to the motion, it changes direction but does no work at that instant.

Common examples

• Gravity near Earth’s surface: $F=mg$.
• Spring force: $F=-kx$, which points back toward equilibrium.
• Friction: often modelled as opposing relative motion.

Common pitfalls

• Confusing force with energy; force has units of newtons, energy has units of joules.
• Forgetting force is a vector.
• Using $F=ma$ with a non-inertial frame without adding the appropriate pseudo-forces.