Newton's Laws of Motion/Second Law

Physical Law

Newton's Second Law of Motion is one of three physical laws that forms the basis for classical mechanics.

Statement of Law

The total force applied on a body is equal to the derivative with respect to time of the linear momentum of the body:

$\ds \mathbf F$

$=$

$\ds \dfrac {\d \mathbf p} {\d t}$

where $p$ denotes linear momentum

$\ds $

$=$

$\ds \map {\dfrac \d {\d t} } {m \mathbf v}$

where $m$ denotes mass and $\mathbf v$ denotes velocity

Also presented as

Newton's Second Law of Motion is also seen presented in the form:

$\ds \mathbf F$

$=$

$\ds m \dfrac {\d \mathbf v} {\d t}$

$\ds $

$=$

$\ds m \mathbf a$

where $\mathbf a$ denotes acceleration

which is not its most general form, as it assumes constant mass.

Indeed, as Isaac Newton himself put it:

The acceleration produced by a particular force acting on a body is directly proportional to the magnitude of the force and inversely proportional to the mass of the body.

Also known as

Newton's Second Law of Motion is also often referred to as just Newton's Second Law.

Some refer to it as just Newton's Law, on the grounds that it is the most significant of all of Newton's Laws of Motion, but there is more than one of those.

Also see

At velocities near the speed of light, see Einstein's Law of Motion.

Source of Name

This entry was named for Isaac Newton.

Sources

1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VI}$: On the Seashore
1965: J.W. Leech: Classical Mechanics (2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Introduction: $(2)$
1977: A.J.M. Spencer: Engineering Mathematics: Volume $\text { I }$ ... (previous) ... (next): Chapter $1$ Ordinary Differential Equations: $1.1$ Introduction
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.7$: A Simple Approach to $E = M c^2$
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.8$: Rocket Propulsion in Outer Space
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): force
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): momentum (linear momentum) (plural momenta)
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Newton's laws of motion
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): force
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): momentum (linear momentum) (plural momenta)
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Newton's laws of motion
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Newton's laws of motion