2.2: Newton's Laws of motion
- Page ID
- 9566
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(\Delta a \leq 5 \times 10^{-14}\mathit{m/s^2}\).