Newtonian mechanics

Newton's second law

Net force equals mass times acceleration: ΣF = ma, the quantitative rule behind every dynamics problem.

Newton's second law answers the question: when a force acts on an object, how quickly does its motion change? The answer fits in one short equation, yet it underlies the whole of classical mechanics.

\vec{F} = m\vec{a}

Here $\vec{F}$ is the net force (the vector sum of every force acting on the object), $m$ is mass — a measure of how much the object resists a change in velocity — and $\vec{a}$ is acceleration, how quickly velocity changes over time.

Definition: Inertial mass

Mass in the second law isn't a vague 'amount of stuff' — it is defined through this very equation: for the same force, a larger $m$ means a smaller $a$ . We measure mass by seeing how much a known force accelerates the object.

SchoolWhy ΣF = ma and not ΣF = mv?

Before Newton, the Aristotelian view held that a force was needed to sustain motion — no force, no motion. But experiments (notably Galileo's with inclined planes) showed that on a frictionless surface, an object keeps moving in a straight line forever with no force at all. So force does not cause velocity — it causes a change in velocity, i.e. acceleration. That is the first law (inertia), and the second law quantifies the relationship.

Section 1 of the lab directly illustrates ΣF = ma on an inclined plane: drag the force F, friction μ, and angle θ sliders and watch the acceleration respond instantly. See work-energy and collisions in sections 2 and 3.

Three cases on an inclined plane

Consider a mass $m$ on an incline of angle $\theta$ , with an additional pulling force $F$ along the incline and a kinetic friction coefficient $\mu$ . Decompose gravity $P = mg$ into two components: $mg\sin\theta$ (along the incline, pulling the object downhill) and $mg\cos\theta$ (perpendicular to the incline, balanced by the normal force $N$ ).

N = mg\cos\theta

\sum F_{\parallel} = F - mg\sin\theta \mp \mu N = ma

The sign of friction depends on the direction of motion: friction always opposes relative motion, so if the object slides down, friction points uphill, and vice versa. When the object is at rest, static friction self-adjusts its magnitude (never exceeding $\mu N$ ) to keep $\Sigma F = 0$ — the limiting case of the first law.

Example: Inclined plane with no pulling force

A 2 kg box sits on a 30° incline with friction coefficient $\mu = 0.2$ , no extra pulling force. Does it slide down, and if so, what is its acceleration?

Solution

Compare $\tan\theta = \tan 30° \approx 0.577$ with $\mu = 0.2$ . Since $\tan\theta > \mu$ , gravity's along-slope component beats maximum static friction, so the box slides down. Acceleration: $a = g(\sin\theta - \mu\cos\theta) = 9.8 \times (0.5 - 0.2 \times 0.866) \approx 3.2\ \text{m/s}^2$ .

Apply the second law axis by axis: identify every force, choose a positive direction, and find the net force along that axis. A 10 kg cart pulled by 35 N against 15 N friction has a 20 N net force and accelerates at 2 m/s². The same 20 N net force on a 5 kg cart gives 4 m/s²; greater mass resists changes in motion more.

Example: Acceleration of a cart

A 4 kg cart has an 18 N force right and 6 N friction left. Find its acceleration.

Solution

ΣF=12 N, so a=ΣF/m=12/4=3 m/s² to the right.

Quick check

An object is at rest on a frictionless horizontal floor. What is the net force on it?

If you double an object's mass while keeping the net force the same, its acceleration will:

References

Isaac Newton (1687). Principia Mathematica
Young, Freedman (2019). University Physics