The Timpson Lyceum · University Physics I (2025)

Newton's Laws of Motion – Lecture 4 (University Physics I)

Overview

This lecture moves us out of kinematics — describing motion — and into dynamics, the study of the forces that cause motion. It lays out Newtonian mechanics as an axiomatic system: a handful of undefined primitives (space, time, particle, mass, force) plus Newton's laws as the postulates we validate by observation. From these we get the modern, careful statements of Newton's three laws of motion, the idea of an inertial reference frame, the fundamental sources of force in nature, and the everyday force types you will meet in problems. The math is light here, but conceptually this is one of the most important lectures in the course because every later topic rests on $\sum \vec F = m\vec a$ .

Key Concepts

Primitives (undefined terms). Newtonian mechanics presupposes, without defining, five things: space, time, particle, mass, and force. Space is modeled by ordinary 3-D Euclidean space $\mathbb{R}^3$ (Pythagorean theorem holds, no curvature); time is modeled by the real numbers. A particle is a point-like model of a real object, carrying a position $\vec r$ at a time $t$ ; its trajectory is $\vec r(t)$ . We model real objects (a book, a car) as particles. Because $\mathbb{R}^3$ comes pre-packaged, we inherit vectors, Euclidean geometry, and calculus for free.

Newton's "zeroth law" — forces are vectors. A real force is modeled by a vector. The consequence is superposition: the net (resultant) force on an object is the vector sum of the individual forces, $\vec F_{\text{net}}=\sum_i \vec F_i$ . This is an observed fact about our universe, not a logical necessity.

Newton's first law (law of inertia). There exists a reference frame in which an object with zero net force has constant velocity (zero acceleration): at rest it stays at rest, in motion it stays in motion. The crucial, often-omitted clause is the existence of such a frame.

Inertial reference frame: a frame in which the first law holds. Equivalently, a frame in which an object feeling no force shows no acceleration ("a frame where a juggler can still juggle just fine").
A frame accelerating relative to an inertial one is non-inertial — e.g., taped to the back of an accelerating bus, you'd see a frictionless skater "spontaneously" accelerate toward you with no force acting.
Standing on Earth's surface is a near-perfect inertial frame (tiny corrections from Earth's spin and orbit).
Any frame moving at constant velocity relative to an inertial frame is also inertial. Proof sketch: if $\vec v_{C/A}$ and $\vec v_{B/A}$ are each constant (no net force in inertial frame $A$ ), then $\vec v_{C/B}=\vec v_{C/A}-\vec v_{B/A}$ is a constant minus a constant, hence constant — so $B$ is inertial too.

Newton's second law. In an inertial frame, the sum of the external forces on an object equals its mass times its acceleration: $\sum \vec F_{\text{ext}} = m\vec a.$ Because forces are vectors, this splits into independent component equations: $\sum F_x = ma_x,\quad \sum F_y = ma_y,\quad \sum F_z = ma_z.$ Only forces acting on the object count — never the forces the object exerts on something else, and never "internal" forces (you cannot accelerate yourself by pushing on yourself; running works via the external friction force from the ground).

Newton's third law. In an inertial frame, the force object $A$ exerts on object $B$ is equal and opposite to the force $B$ exerts on $A$ : $\vec F_{A\text{ on }B} = -\,\vec F_{B\text{ on }A}$ . These action–reaction pairs act on different objects, so they never both appear in a single object's second-law sum. (Lean on a wall: you push it, it pushes back equally — no motion.)

Sources of force. All forces ultimately reduce to four fundamental interactions: gravity, the electromagnetic force (responsible for nearly every contact-type force in this course — why your hand doesn't pass through a wall), the weak nuclear force (radioactive decay), and the strong nuclear force (binds the nucleus). In practice we use only gravity and electromagnetism.

Everyday force types.

Contact / applied force — direct pushing or pressing.
Normal force $\vec N$ — surface push, always perpendicular to the surface (not necessarily opposite to weight).
Friction $\vec f$ — resistive force opposing relative sliding; on a held ramp, $\vec N+\vec f$ vector-sum to cancel weight until friction is overcome.
Tension $\vec T$ — pull transmitted along a rope; an ideal rope/spring exerts a force independent of the mass attached.
Weight $\vec w$ — the gravitational pull, magnitude $w=mg$ with $g\approx 9.8\ \text{m/s}^2$ .
Long-range forces — gravity and electromagnetism act without a material medium.

Units. Force is measured in newtons: $1\ \text{N} = 1\ \text{kg}\cdot\text{m/s}^2$ , the force that gives a $1\ \text{kg}$ mass an acceleration of $1\ \text{m/s}^2$ .

Mass vs. weight. Mass measures an object's resistance to acceleration (inertia) and the amount of matter; it is intrinsic. Weight is the gravitational force on it, $w=mg$ , and vanishes far from any mass. Remarkably, inertial mass and gravitational mass are experimentally identical to about 1 part in $10^{12}$ . Mass can be measured without gravity: pull an ideal spring the same distance with the unknown mass and with a $1\ \text{kg}$ standard; equal forces give $m\,a_1 = (1)\,a_2$ , so $m = a_2/a_1$ .

Worked Example

Two wrestlers tug a championship belt. Three forces act on the belt: $F_1 = 250\ \text{N}$ at $53^\circ$ above the $-x$ axis, $F_2 = 50\ \text{N}$ along $+x$ , and $F_3 = 120\ \text{N}$ straight down. Find the net force's components, magnitude, and direction. Take right as $+x$ and up as $+y$ .

Resolve $\vec F_1$ into components and sum each axis:

\begin{aligned} \sum F_x &= F_2 - F_1\cos 53^\circ = 50 - 250\cos 53^\circ \approx -100\ \text{N} \\ \sum F_y &= F_1\sin 53^\circ - F_3 = 250\sin 53^\circ - 120 \approx 80\ \text{N} \end{aligned}

Magnitude and direction:

\begin{aligned} F_{\text{net}} &= \sqrt{(\textstyle\sum F_x)^2 + (\sum F_y)^2} = \sqrt{(-100)^2 + 80^2} \approx 130\ \text{N} \\ \theta &= \tan^{-1}\!\left(\frac{|\sum F_y|}{|\sum F_x|}\right) = \tan^{-1}\!\left(\frac{80}{100}\right) \approx 39^\circ \end{aligned}

Since $\sum F_x<0$ and $\sum F_y>0$ , the resultant lies in the second quadrant: $39^\circ$ above the $-x$ direction, or equivalently $180^\circ-39^\circ = 141^\circ$ measured from $+x$ .

Short example — friction on a sliding bottle. A $0.45\ \text{kg}$ ketchup bottle leaves a hand at $v_0 = 2.8\ \text{m/s}$ and slides $1.0\ \text{m}$ to rest. Find the friction force. First get the acceleration from kinematics, then apply the second law:

\begin{aligned} v^2 &= v_0^2 + 2a\,\Delta x \;\Rightarrow\; a = \frac{0 - (2.8)^2}{2(1.0)} = -3.92\ \text{m/s}^2 \\ f &= ma = (0.45)(-3.92) \approx -1.8\ \text{N} \end{aligned}

So friction is $1.8\ \text{N}$ directed opposite the motion.

Summary

Dynamics studies the forces behind motion; the master equation is $\sum \vec F_{\text{ext}} = m\vec a$ , which holds only in an inertial reference frame.
Newton's zeroth law: forces add as vectors (superposition); first law: there exists a frame in which zero net force means constant velocity; third law: every force has an equal-and-opposite reaction on a different object.
An inertial frame is any frame moving at constant velocity relative to another inertial frame; accelerating frames are non-inertial.
The second law separates into independent $x$ , $y$ , $z$ component equations, and uses only the external forces acting on the chosen object.
Common forces — normal (perpendicular to the surface), friction (opposes sliding), tension, weight $w=mg$ , contact, and long-range — all trace back to gravity and electromagnetism.
Mass is intrinsic inertia and amount of matter; weight is the gravitational force $mg$ and is location-dependent. Force is measured in newtons, $1\ \text{N}=1\ \text{kg}\cdot\text{m/s}^2$ .