Newton's second law

Newton’s second law is the quantitative heart of dynamics: while the first law tells us when motion changes (in the presence of a non-zero resultant force), the second tells us by how much it changes. It establishes the precise link between the cause — the force — and the effect — the acceleration.

Principle — Newton's second law

The sum of all forces acting on a body equals the body’s mass multiplied by its acceleration: $\ev{\sum \vv{F} = m\,\vv{a}}$ The reference frame must be inertial.

The equation is vectorial: force and acceleration have the same direction and the same sense, and are linked by the proportionality factor $m$, the mass. The greater the mass, the smaller the acceleration produced by a given force: mass measures inertia, that is, a body’s resistance to a change in its state of motion. Since it is an equality between vectors, it is equivalent to as many scalar equations as there are dimensions of space: in the plane, the single vector relation splits into two independent equations, one for each Cartesian axis.

Key formula

$\vv{F}_1 + \vv{F}_2 + \cdots = m\,\vv{a}$ In components: $\begin{cases} F_{1x} + F_{2x} + \cdots = m\,a_x \\ F_{1y} + F_{2y} + \cdots = m\,a_y \end{cases}$

This decomposition into components is what makes the second law a computational tool. In practice, its application follows a five-step procedure, always the same, which is worth learning by heart and carrying out in order:

1. Interaction diagram: draw all the objects involved and the squiggles representing their interactions.
2. Circle the object whose motion we want to study and count the forces acting on it.
3. Free-body diagram: draw the isolated object with all applied forces represented as vectors leaving the body.
4. Resolve each force into its components along the $x$ and $y$ axes.
5. Write the two scalar equations $\sum F_x = m\,a_x$ and $\sum F_y = m\,a_y$, and solve them.

Common mistakes

The second law holds only in an inertial reference frame. In an accelerating frame (a car braking, a rotating merry-go-round) the equality $\sum \vv{F} = m\,\vv{a}$ is no longer valid in its simple form, because apparent accelerations appear that are not associated with any real force.

Links

Topics: Dinamica Concepts: Seconda legge di Newton · Componenti di un vettore · Accelerazione Skills: Applicazione delle leggi di Newton · Diagramma di corpo libero Methods: Scomposizione in componenti cartesiane

Related exercises: Problema — Accelerazione da forza netta · Cassa con forza a tratti · Problema — Cassa spinta senza attrito