Newtonian mechanics in one page

Force, momentum, energy, torque, free-body diagrams, continuum limit.

Every dynamical statement in the bookshelf — Euler’s equation for a fluid, the basilar membrane’s damped oscillator, the Rayleigh–Plesset balance on a spherical bubble — reduces, at root, to Newton’s second law applied to the right element. This chapter is the apparatus, with one interactive per consequential equation so the algebra can be driven and seen.

The three laws

In an inertial frame:

1. A body free of external force moves with constant velocity.
2. The rate of change of momentum equals the applied force, $$\mathbf{F} \;=\; \frac{d\mathbf{p}}{dt}, \qquad \mathbf{p} \;=\; m\mathbf{v}.$$
3. Forces between two bodies are equal and opposite: $\mathbf{F}_{12} = -\mathbf{F}_{21}$.

The first law fixes what counts as an inertial frame: any frame in which the second law has the simple form above without phantom forces. The second is the operative one — every problem in classical mechanics is the assembly of $\mathbf{F} = d\mathbf{p}/dt$ for the right body or fluid element. The third law is what closes the books: it is what lets us write Euler’s equation for a slab of air by pairing each “pressure force from the right” with a reaction force on the right slab.

For a point mass with constant $m$, Newton’s second law reduces to $\mathbf{F} = m\mathbf{a}$. For a system of particles, summing over the system and using the third law to cancel internal forces gives

so internal forces never accelerate the centre of mass. This is the formal statement of momentum conservation when $\mathbf{F}_\text{ext} = 0$.

Impulse: the integrated form of Newton’s second law

Integrate $\mathbf{F} = d\mathbf{p}/dt$ over a finite time interval:

The left side is the impulse. For brief, high-force events — a molecule rebounding off a wall, a hammer strike, an audio click — the integrated impulse is what matters, not the detailed time-course of the force.

Slide the pulse amplitude $F_\text{peak}$ and duration $\tau$. The shaded area under $F(t)$ is the impulse $J = F_\text{peak}\,\tau / 2$, and the resulting velocity change is $\Delta v = J/m$. This bookkeeping is the heart of kinetic theory — pressure on a wall is the rate at which the wall receives impulse per unit area, as the kinetic-theory chapter develops.

Collisions, momentum, and the energy ledger

Momentum is conserved in every collision (no external horizontal force). Kinetic energy is not — only when the collision is elastic does the kinetic energy after equal that before. The single parameter that controls this is the coefficient of restitution $e$:

Combined with momentum conservation $m_1 v_1^i + m_2 v_2^i = m_1 v_1^f + m_2 v_2^f$, this determines the post-collision velocities completely.

Set the two masses, their initial velocities, and the restitution. The momentum readout is always conserved; the kinetic-energy readout shows the lost-to-heat-and-deformation fraction at $e < 1$. At $e = 0$ the bodies coalesce (perfectly inelastic); at $e = 1$ all kinetic energy is recovered (perfectly elastic). Most physical collisions sit in between; for the molecular collisions of kinetic theory and the wall-bouncing of a contained gas, we idealise $e = 1$ — and the entire kinetic-theory derivation of pressure depends on that idealisation.

Work, energy, and conservation

Define work done by $\mathbf{F}$ along a path as $W = \int \mathbf{F}\cdot d\mathbf{r}$. Newton’s second law then implies the work–energy theorem:

A force is conservative if the work it does around any closed path is zero, equivalently if $\mathbf{F} = -\nabla U$ for some scalar potential $U$. For such forces, $T + U$ is conserved along trajectories. Gravity, the electrostatic force, and the elastic restoring force of a spring are conservative; friction and viscous drag are not.

Toggle between a conservative radial spring force and a non-conservative drag force; toggle the path between a direct line and an arc over the top. The conservative work depends only on the endpoints — it is the difference in potential $U(A) - U(B)$ for any path. The drag work grows with path length — it costs more to take the long way. This is what conservative means operationally, and it is the basis for the energy bookkeeping in the acoustic energy density of a sound wave (where the kinetic-plus-potential split holds because the restoring pressure is conservative).

Torque, angular momentum, the lever principle

For a body with position $\mathbf{r}$ relative to some origin and momentum $\mathbf{p}$, the angular momentum about the origin is $\mathbf{L} = \mathbf{r}\times\mathbf{p}$ and the torque of a force is $\boldsymbol{\tau} = \mathbf{r}\times\mathbf{F}$. Newton’s second law extends in form:

For a rigid body in static equilibrium, both the net force and the net torque about every point must vanish. The lever principle is the simplest application: two forces $F_L$ and $F_R$ acting at distances $L_L$ and $L_R$ on opposite sides of a pivot balance when

A small force at a long arm matches a large force at a short arm.

Slide the pivot and the two masses to see the torque-balance condition come into and out of balance. The mechanical advantage $L_L/L_R$ rises sharply as the pivot approaches the short-arm mass — which is why a 1.5× lever can be biologically valuable when combined with other amplifications (see the middle-ear-ossicle example in the key-examples sub-page).

Free-body diagrams as a discipline

A free-body diagram isolates one body (or one differential element of a continuum), draws every external force acting on it, and sets the vector sum equal to $m\mathbf{a}$. The practice — choose the body, choose the frame, list every contact and field force, and only then write equations — keeps two errors at bay: double-counting internal forces, and forgetting a constraint reaction.

The continuum limit

Newton’s laws were stated for a point mass, but every governing equation in the rest of this bookshelf is for a continuum — a fluid, an elastic solid, a membrane. The continuum form is not new physics: it is Newton’s laws applied to a differential element, together with a constitutive law saying how that element responds to stress. The standard sandbox for seeing this happen is a 1-D chain of masses connected by springs.

For a chain of $N$ identical masses $m$ at positions $x_j = j a$, each connected to its neighbours by springs of constant $\kappa$, the equation of motion for mass $j$ is

The right-hand side is a discrete second difference. Expanding $u_{j\pm 1} = u(x_j \pm a, t) \approx u(x_j, t) \pm a\, u_x + \tfrac12 a^2 u_{xx} + \ldots$, the discrete second difference becomes $a^2\, u_{xx}(x_j) + O(a^4)$, and in the limit $a \to 0$ with $\kappa a^2 / m \to c^2$ fixed, the chain equation collapses to the 1-D wave equation

A pulse on a chain of $N$ masses propagates as a wave. For small $N$ the chain is dispersive — short wavelengths travel slower than long ones — and the pulse spreads and ripples. As $N$ grows, the chain’s spectrum fills out smoothly and approaches the continuum wave equation. The dashed reference is the continuum solution; the blue dots are the chain. This is how Newton’s particle-mechanics becomes the field theory the fluid mechanics and elasticity chapters take as their starting point.

The same logic applies in three dimensions. A fluid element of volume $dV$ has mass $\rho\, dV$; the net surface force on it is the divergence of the stress tensor times $dV$; Newton’s second law per unit volume is $\rho\, D\mathbf{u}/Dt = \nabla\cdot\boldsymbol{\sigma}$ — the master equation of continuum mechanics, of which Euler’s equation and the Navier–Stokes equation are special cases.

A preview of Lagrangian mechanics

Newton’s second law has a deep reformulation as a variational principle. Among all kinematically admissible trajectories $\mathbf{r}(t)$ between two endpoints, the actual trajectory of a particle is the one that makes the action

stationary. The Euler–Lagrange equation for $S$ stationary recovers $\mathbf{F} = m\mathbf{a}$ exactly. The Lagrangian formulation is equivalent to Newtonian mechanics for point particles, but it generalises cleanly to constrained systems (where eliminating constraint forces is awkward in Newtonian terms) and to fields (where it gives the natural variational route to the wave equation, used in the Sound book’s energy-route lesson).

The full Lagrangian / Hamiltonian apparatus is beyond the scope of this chapter. The takeaway: every equation in this bookshelf can be derived either by free-body-diagram Newtonian arguments or by writing down a Lagrangian and applying $\delta S = 0$. The two routes give the same equations but very different intuitions about which terms are fundamental.

For the cross-book applications — middle-ear lever, kinetic-theory pressure, Euler’s equation for a slab, Rayleigh–Plesset bubble dynamics, and acoustic energy bookkeeping — see the key examples sub-page.