Newton's Laws of Motion

Before Newton, the prevailing view held that moving objects naturally slow down — that rest is the natural state and a sustained force is required to keep anything in motion. Galileo overturned this: a ball rolling on a perfectly frictionless surface would roll forever. Newton formalized this insight as his First Law, also called the Law of Inertia.

The resistance of a body to any change in its velocity is called inertia. Mass (SI unit: kg) is the quantitative measure of inertia. A 10 kg block requires twice the net force of a 5 kg block to produce the same acceleration — it has twice the inertia. Mass is an intrinsic property of matter, independent of gravity or location.

Inertial Reference Frames

Newton's laws hold only in inertial reference frames — frames that are not themselves accelerating. Any frame moving at constant velocity relative to an inertial frame is also inertial. An accelerating car, a spinning carousel, or a rocket under thrust are non-inertial frames where fictitious forces appear (the "centrifugal force," the Coriolis force). Near Earth's surface, the ground is an excellent approximation to an inertial frame for almost all introductory problems.

A force is a push or pull exerted by one object on another. Forces are vectors — they have both magnitude and direction. The SI unit of force is the newton (N), defined as the force that gives a 1 kg mass an acceleration of 1 m/s²: $1\,\text{N} = 1\,\text{kg}\cdot\text{m/s}^2$. The weight of a medium apple is about 1 N.

When multiple forces act on one body, the effect is identical to that of a single net force equal to their vector sum. This is the superposition principle:

Superposition holds for any number of forces of any type acting simultaneously. You add them as vectors, component by component.

Free-Body Diagrams

A free-body diagram (FBD) isolates one object and shows every external force acting on it as a labeled arrow from a common point. It is the single most powerful tool for applying Newton's Laws. Rules: (1) draw only one object; (2) include every external force — gravity, normal, friction, tension, applied forces; (3) label each force with its symbol and show its direction; (4) draw and label your coordinate axes.

This is a vector equation — it holds simultaneously in every direction. In component form, the equation separates into independent scalar equations along each axis:

Two critical points: (1) The law involves the net force — the vector sum of all forces on the body, not any individual force. (2) The acceleration $\vec{a}$ is the acceleration of the entire body, treated as a particle, at the instant the forces act.

Units: From $F = ma$, force has units of $\text{kg}\cdot\text{m/s}^2 = \text{N}$. Mass is a scalar measured in kg. It is an intrinsic property of the object, independent of gravity — the same block has mass 5 kg on Earth, the Moon, and in deep space, though its weight changes in each place.

Gravitational Force and Weight

Every object near Earth's surface experiences a downward gravitational force — its weight. Near Earth's surface:

Weight is a force (measured in N), fundamentally different from mass (measured in kg). On the Moon, $g \approx 1.6\,\text{m/s}^2$, so a 70 kg astronaut weighs only 112 N there — but their mass remains 70 kg. Weight depends on location; mass does not.

Normal Force

When a body presses against a surface, the surface exerts a normal force $\vec{N}$ perpendicular to the contact surface, preventing penetration. The normal force adjusts to whatever value is needed. On a horizontal surface with only gravity and the normal force present, $N = mg$. But on an incline, with a vertical applied force, or inside an accelerating elevator, $N \neq mg$. Never assume $N = mg$ without analyzing the forces.

Friction

Kinetic friction $f_k$ acts on a sliding object, opposing the direction of sliding. Static friction $f_s$ acts on a stationary object, opposing the tendency to slide. The coefficients $\mu_k$ (kinetic) and $\mu_s$ (static) depend on the two surfaces in contact, with $\mu_s > \mu_k$ — meaning it takes more force to start an object moving than to keep it moving.

Tension

A tension $T$ is the pulling force exerted by a string, rope, or cable along its length. For a massless, inextensible rope, the tension is the same throughout. Ropes can only pull — they cannot push. If a calculation yields $T < 0$, the rope has gone slack and $T = 0$.

These are called a Newton's Third Law pair (or action–reaction pair). Three key facts about every such pair: (1) The two forces are always exactly equal in magnitude. (2) They always act on different bodies — never on the same object. (3) They are always the same type of force (both gravitational, both contact, etc.). Because they act on different bodies, they can never cancel each other.

A Classic Mistake: Weight and Normal Force

Students often claim that weight and normal force form a Newton's Third Law pair. They do not. Consider a book at rest on a table: (1) Earth's gravity pulls the book down (weight $W = mg$, a gravitational force on the book). (2) The table pushes the book up (normal force $N$, a contact force on the book). Both forces act on the same object (the book), so they cannot be a Third Law pair. The real pairs are: Earth pulls book down ↔ book pulls Earth up (gravitational pair); and table pushes book up ↔ book pushes table down (contact pair).

Choosing coordinates wisely can eliminate algebra. For an inclined plane, tilting the axes so that $x$ runs along the slope and $y$ is perpendicular to it means the normal force equation becomes $N - mg\cos\theta = 0$, giving $N = mg\cos\theta$ with no trigonometry beyond the initial decomposition. The along-slope equation then yields $a = g\sin\theta$ directly.

Remarkably, $a = g\sin\theta$ is completely independent of mass — every object slides down a frictionless incline at the same rate regardless of how heavy it is (a result Galileo famously demonstrated). At $\theta = 0°$, $a = 0$ (flat surface, no acceleration). At $\theta = 90°$, $a = g$ — pure free fall.