1.1 Force and Newton’s three laws

Mechanics is the science of motion under force. Its entire classical content rests on three laws stated by Newton in 1687, from which every problem — a thrown ball, an orbiting planet, a vibrating string, a slab of fluid — is assembled by applying the second law to the right body.

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: one in which the second law holds in the simple form above, without phantom forces. The second law is the operative one — every problem in classical mechanics is the assembly of $\mathbf F = d\mathbf p/dt$ for the right body or element. The third law is what closes the books: pairing every “force on A from B” with its reaction “force on B from A” is what lets internal forces cancel when a system is summed over.

For a body of constant mass $m$, the second law reduces to the familiar $\mathbf F = m\mathbf a$. The momentum form $\mathbf F = d\mathbf p/dt$ is the more fundamental one, holding also when the mass changes (a rocket shedding fuel) or when momentum is the natural variable (a stream of molecules striking a wall).

Systems of particles and the centre of mass

Sum the second law over all the particles of a system. By the third law the internal forces cancel in pairs, leaving only the external forces:

The total momentum moves as though the whole mass $M$ were concentrated at the centre of mass and the external force applied there. Internal forces — however violent — never accelerate the centre of mass. When $\mathbf F_\text{ext} = 0$ the total momentum is conserved; this is the formal basis for the collision analysis of the next lesson.

Free-body diagrams as a discipline

Turning a physical situation into equations is a procedure, not an inspiration. 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 $m\mathbf a$. Choose the body, choose the frame, list every contact and field force, and only then write equations. The discipline keeps two errors at bay: double-counting an internal force, and forgetting a constraint reaction (the normal force of a surface, the tension of a string). Every derivation in continuum mechanics — the force balance on a slab of fluid, on a segment of string, on an element of solid — is a free-body diagram drawn on an infinitesimal piece.