2.2: Newton's Laws of motion

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Newton defined a vector quantity called linear momentum \(\mathbf{p}\) which is the product of mass and velocity.

\[\label{eq:2.1} \mathbf{p} = m\dot{\mathbf{r}}\]

Since the mass \(m\) is a scalar quantity, then the velocity vector \(\dot{r}\) and the linear momentum vector \(\mathbf{p}\) are colinear.

Newton’s laws, expressed in terms of linear momentum, are:

Law of inertia: A body remains at rest or in uniform motion unless acted upon by a force.
Equation of motion: A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.\[\label{eq:2.2}\mathbf{F} = \frac{d\mathbf{p}}{dt}\]
Action and reaction: If two bodies exert forces on each other these forces are equal in magnitude and opposite in direction.

Newton’s second law contains the essential physics relating the force \(\mathbf{F}\) and the rate of change of linear momentum \(\mathbf{p}\).

Newton’s first law, the law of inertia, is a special case of Newton’s second law in that if

\[\label{eq:2.3}\mathbf{F}=\frac{d\mathbf{p}}{dt}=0\]

then \(\mathbf{p}\) is a constant of motion.

Newton’s third law also can be interpreted as a statement of the conservation of momentum, that is, for a two particle system with no external forces acting,

\[\label{eq:2.4} \mathbf{F}_{12} = -\mathbf{F}_{21}\]

If the forces acting on two bodies are their mutual action and reaction, then Equation \ref{eq:2.4} simplifies to

\[\label{eq:2.5} \mathbf{F}_{12}=-\mathbf{F}_{21}= \frac{d\mathbf{p_1}}{dt}+ \frac{d\mathbf{p_2}}{dt} = \frac{d}{dt}(\mathbf{p_1+p_2}) = 0\]

This implies that the total linear momentum \(\mathbf{P = p_1 + p_2}\) is a constant of motion. Combining Equations \ref{eq:2.1} and \ref{eq:2.2} leads to a second-order differential equation

\[\label{2.6} \mathbf{F}=\frac{d\mathbf{p}}{dt}=m\frac{d^2\mathbf{r}}{dt^2}=m\mathbf{\ddot{r}}\]

Note that the force on a body \(\mathbf{F}\), and the resultant acceleration \({\bf a = \ddot{r}}\) are colinear. Appendix \(19.3.2\) gives explicit expressions for the acceleration \({\bf a}\) in cartesian and curvilinear coordinate systems. The definition of force depends on the definition of the mass \(m\). Newton’s laws of motion are obeyed to a high precision for velocities much less than the velocity of light. For example, recent experiments have shown they are obeyed with an error in the acceleration of \(\Delta a \leq 5 \times 10^{-14}\mathit{m/s^2}\).