42.1: Introduction to Simple Harmonic Motion
- Page ID
- 99832
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The small-angle approximation of the simple plane pendulum is an example of what is called simple harmonic motion. Simple harmonic motion is the motion that a particle exhibits when under the influence of a force of the form given by Hooke's law (named for the 17th century English scientist Robert Hooke):
A force of this form describes, for example, the force on a mass attached to a spring with spring constant \(k\), where \(k\) is a measure of the stiffness of the spring. In this case \(F\) is the force exerted by the spring, and \(x\) is the distance of the mass from its equilibrium position - that is, the "resting" position at which the mass can be left where it will not oscillate.
Substituting Hooke's law as the force in Newton's second law \(F=m a\) (and recalling the acceleration \(a=d^{2} x / d t^{2}\) ) gives the equation
\[-k x=m \frac{d^{2} x}{d t^{2}}\]
This is a second-order linear differential equation with constant coefficients, and can be solved for \(x(t)\) using standard methods from the theory of differential equations. We won't go into the theory of differential equations here, but just present the result. The solution is
\[x(t)=A \cos (\omega t+\delta) .\]
Here \(\omega\) is called the angular frequency of the motion, and measures how fast the particle oscillates back and forth. The constant \(A\) is called the amplitude of the motion, and is the maximum distance the particle travels from its equilibrium position, \(x=0\). The constant \(\delta\) called the phase constant, and determines where in its cycle the particle is at time \(t=0\). A plot of \(x(t)\) is shown in Fig. \(\PageIndex{1}\).
Since the sine and cosine function differ only by a phase shift ( \(\sin \theta \equiv \cos (\theta-\pi / 2)\) ), we could replace the cosine function in Eq. \(\PageIndex{3}\) with a sine by simply adding an extra \(\pi / 2\) to the phase constant \(\delta\). So either the sine or the cosine can be used equally well to describe simple harmonic motion; here we will choose to use the cosine function.
The calculus may also be used to find the velocity of the particle at any time \(t\); the result is
\[v(t)=-A \omega \sin (\omega t+\delta)\]
so that the maximum speed of the simple harmonic oscillator is
\[\left|v_{\max }\right|=A \omega\]
Further, it can be shown that the acceleration at any time \(t\) is
\[a(t) =-A \omega^{2} \cos (\omega t+\delta) \]
\[ =-\omega^{2} x(t).\]
Multiplying Eq. \(\PageIndex{7}\) by the particle mass \(m\), we find
\[m a(t)=F(t)=-m \omega^{2} x(t) .\]
Comparing this with Eq. \(\PageIndex{1}\) we see that
\[k=m \omega^{2},\]
or
\[\omega=\sqrt{\frac{k}{m}} .\]
In Eq. \(\PageIndex{3}\), the amplitude \(A\) depends on how far the particle was displaced from equilibrium before being released; the phase constant \(\delta\) just depends on when we choose time \(t=0\); but the angular frequency \(\omega\) depends on the physical parameters of the system: the stiffness of the spring \(k\) and the mass of the particle \(m\).