Movatterモバイル変換

[0]ホーム

Jump to content

Simple harmonic motion

Edit links

From Wikipedia, the free encyclopedia

To-and-fro periodic motion in science and engineering

Simple harmonic motion shown both in real space andphase space. Theorbit isperiodic. (Here thevelocity andposition axes have been reversed from the standard convention to align the two diagrams)

Classical mechanics
Part of a series on
${\textbf {F}}={\frac {d\mathbf {p} }{dt}}$ Second law of motion
History Timeline Textbooks
Branches Applied Celestial Continuum Dynamics Field theory Kinematics Kinetics Statics Statistical mechanics
Fundamentals Acceleration Angular momentum Couple D'Alembert's principle Energy kinetic potential Force Frame of reference Inertial frame of reference Impulse Inertia / Moment of inertia Mass Mechanical power Mechanical work Moment Momentum Space Speed Time Torque Velocity Virtual work
Formulations Newton's laws of motion Analytical mechanics Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton–Jacobi equation Appell's equation of motion Koopman–von Neumann mechanics
Core topics Damping Displacement Equations of motion Euler's laws of motion Fictitious force Friction Harmonic oscillator Inertial / Non-inertial reference frame Motion (linear) Newton's law of universal gravitation Newton's laws of motion Relative velocity Rigid body dynamics Euler's equations Simple harmonic motion Vibration
Rotation Circular motion Rotating reference frame Centripetal force Centrifugal force reactive Coriolis force Pendulum Tangential speed Rotational frequency Angular acceleration / displacement / frequency / velocity
Scientists Kepler Galileo Huygens Newton Horrocks Halley Maupertuis Daniel Bernoulli Johann Bernoulli Euler d'Alembert Clairaut Lagrange Laplace Poisson Hamilton Jacobi Cauchy Routh Liouville Appell Gibbs Koopman von Neumann
Physics portal Category
v t e

Inmechanics andphysics,simple harmonic motion (sometimes abbreviated asSHM) is a special type of periodic motion an object experiences by means of arestoring force whose magnitude is directlyproportional to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in anoscillation that is described by asinusoid which continues indefinitely (if uninhibited byfriction or any otherdissipation ofenergy).^[1]

Simple harmonic motion can serve as amathematical model for a variety of motions, but is typified by the oscillation of amass on aspring when it is subject to the linearelastic restoring force given byHooke's law. The motion issinusoidal in time and demonstrates a singleresonant frequency. Other phenomena can be modeled by simple harmonic motion, including the motion of asimple pendulum, although for it to be an accurate model, thenet force on the object at the end of the pendulum must be proportional to the displacement (and even so, it is only a good approximation when the angle of the swing is small; seesmall-angle approximation). Simple harmonic motion can also be used to modelmolecular vibration. A mass-spring system is a classic example of simple harmonic motion.

Simple harmonic motion provides a basis for the characterization of more complicated periodic motion through the techniques ofFourier analysis.

Introduction

[edit]

The motion of aparticle moving along a straight line with anacceleration whose direction is always toward afixed point on the line and whose magnitude is proportional to the displacement from the fixed point is called simple harmonic motion.^[2]

In the diagram, asimple harmonic oscillator, consisting of a weight attached to one end of a spring, is shown. The other end of the spring is connected to a rigid support such as a wall. If the system is left at rest at theequilibrium position then there is no netforce acting on the mass. However, if the mass is displaced from the equilibrium position, the springexerts a restoringelastic force that obeysHooke's law.

Mathematically, $\mathbf {F} =-k\mathbf {x} ,$ whereF is the restoring elastic force exerted by the spring (inSI units:N),k is thespring constant (N·m⁻¹), andx is thedisplacement from the equilibrium position (inmetres).

For any simple mechanical harmonic oscillator:

When the system is displaced from its equilibrium position, a restoring force that obeys Hooke's law tends to restore the system to equilibrium.

Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, itaccelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, atx = 0, the mass hasmomentum because of the acceleration that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then slows it down until itsvelocity reaches zero, whereupon it is accelerated back to the equilibrium position again.

As long as the system has noenergy loss, the mass continues to oscillate. Thus simple harmonic motion is a type ofperiodic motion. If energy is lost in the system, then the mass exhibitsdamped oscillation.

Note if the real space andphase space plot are not co-linear, the phase space motion becomes elliptical. The area enclosed depends on the amplitude and the maximum momentum.

Dynamics

[edit]

InNewtonian mechanics, for one-dimensional simple harmonic motion, the equation of motion, which is a second-order linearordinary differential equation withconstant coefficients, can be obtained by means ofNewton's second law andHooke's law for amass on aspring.

$F_{\mathrm {net} }=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-kx,$ wherem is theinertial mass of the oscillating body,x is itsdisplacement from theequilibrium (or mean) position, andk is a constant (thespring constant for a mass on a spring).

Therefore, ${\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-{\frac {k}{m}}x$

Solving thedifferential equation above produces a solution that is asinusoidal function: $x(t)=c_{1}\cos \left(\omega t\right)+c_{2}\sin \left(\omega t\right),$ where ${\textstyle {\omega }={\sqrt {{k}/{m}}}.}$ The meaning of the constants $c_{1}$ and $c_{2}$ can be easily found: setting $t=0$ on the equation above we see that $x(0)=c_{1}$ , so that $c_{1}$ is the initial position of the particle, $c_{1}=x_{0}$ ; taking the derivative of that equation and evaluating at zero we get that ${\dot {x}}(0)=\omega c_{2}$ , so that $c_{2}$ is the initial speed of the particle divided by the angular frequency, $c_{2}={\frac {v_{0}}{\omega }}$ . Thus we can write: $x(t)=x_{0}\cos \left({\sqrt {\frac {k}{m}}}t\right)+{\frac {v_{0}}{\sqrt {\frac {k}{m}}}}\sin \left({\sqrt {\frac {k}{m}}}t\right).$

This equation can also be written in the form: $x(t)=A\cos \left(\omega t-\varphi \right),$ where

$A={\sqrt {{c_{1}}^{2}+{c_{2}}^{2}}}$
$\tan \varphi ={\frac {c_{2}}{c_{1}}},$
$\sin \varphi ={\frac {c_{2}}{A}},\;\cos \varphi ={\frac {c_{1}}{A}}$

or equivalently

$A=|c_{1}+c_{2}i|,$
$\varphi =\arg(c_{1}+c_{2}i)$

In the solution,c₁ andc₂ are two constants determined by the initial conditions (specifically, the initial position at timet = 0 isc₁, while the initial velocity isc₂ω), and the origin is set to be the equilibrium position.^[A] Each of these constants carries a physical meaning of the motion:A is theamplitude (maximum displacement from the equilibrium position),ω = 2πf is theangular frequency, andφ is the initialphase.^[B]

Using the techniques ofcalculus, thevelocity andacceleration as a function of time can be found: $v(t)={\frac {\mathrm {d} x}{\mathrm {d} t}}=-A\omega \sin(\omega t-\varphi ),$

Speed: ${\omega }{\sqrt {A^{2}-x^{2}}}$
Maximum speed:v =ωA (at equilibrium point)

$a(t)={\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-A\omega ^{2}\cos(\omega t-\varphi ).$

Maximum acceleration:Aω² (at extreme points)

By definition, if a massm is under SHM its acceleration is directly proportional to displacement. $a(x)=-\omega ^{2}x.$ where $\omega ^{2}={\frac {k}{m}}$

Sinceω = 2πf, $f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}},$ and, sinceT =⁠1/f⁠ whereT is the time period, $T=2\pi {\sqrt {\frac {m}{k}}}.$

These equations demonstrate that the simple harmonic motion isisochronous (the period and frequency are independent of the amplitude and the initial phase of the motion).

Energy

[edit]

Substitutingω² with⁠k/m⁠, thekinetic energyK of the system at timet is $K(t)={\tfrac {1}{2}}mv^{2}(t)={\tfrac {1}{2}}m\omega ^{2}A^{2}\sin ^{2}(\omega t-\varphi )={\tfrac {1}{2}}kA^{2}\sin ^{2}(\omega t-\varphi ),$ and thepotential energy is $U(t)={\tfrac {1}{2}}kx^{2}(t)={\tfrac {1}{2}}kA^{2}\cos ^{2}(\omega t-\varphi ).$ In the absence of friction and other energy loss, the totalmechanical energy has a constant value $E=K+U={\tfrac {1}{2}}kA^{2}.$

Examples

[edit]

An undampedspring–mass system undergoes simple harmonic motion.

The following physical systems are some examples ofsimple harmonic oscillator.

Mass on a spring

[edit]

A massm attached to a spring of spring constantk exhibits simple harmonic motion inclosed space. The equation for describing the period: $T=2\pi {\sqrt {\frac {m}{k}}}$ shows the period of oscillation is independent of the amplitude, though in practice the amplitude should be small. The above equation is also valid in the case when an additional constant force is being applied on the mass, i.e. the additional constant force cannot change the period of oscillation.

Uniform circular motion

[edit]

Simple harmonic motion can be considered the one-dimensionalprojection ofuniform circular motion. If an object moves with angular speedω around a circle of radiusr centered at theorigin of thexy-plane, then its motion along each coordinate is simple harmonic motion with amplituder and angular frequencyω.

Oscillatory motion

[edit]

The motion of a body in which it moves to and from a definite point is also calledoscillatory motion or vibratory motion. The time period is able to be calculated by $T=2\pi {\sqrt {\frac {l}{g}}}$ where l is the distance from rotation to the object's center of mass undergoing SHM and g is acceleration due to gravity. This is analogous to the mass-spring system.

Mass of a simple pendulum

[edit]

Apendulum making 25 completeoscillations in 60 s, a frequency of 0.416Hertz

In thesmall-angle approximation, themotion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a pendulum of lengthl with gravitational acceleration $g {\displaystyle g}$ is given by $T=2\pi {\sqrt {\frac {l}{g}}}$

This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not of the acceleration due togravity, $g {\displaystyle g}$ , therefore a pendulum of the same length on the Moon would swing more slowly due to the Moon's lower gravitational field strength. Because the value of $g {\displaystyle g}$ varies slightly over the surface of the earth, the time period will vary slightly from place to place and will also vary with height above sea level.

This approximation is accurate only for small angles because of the expression forangular accelerationα being proportional to the sine of the displacement angle:

$-mgl\sin \theta =I\alpha ,$

whereI is themoment of inertia. Whenθ is small,sin θ ≈θ and therefore the expression becomes

$-mgl\theta =I\alpha$

which makes angular acceleration directly proportional and opposite toθ, satisfying the definition of simple harmonic motion (that net force is directly proportional to the displacement from the mean position and is directed towards the mean position).

Scotch yoke

[edit]

Main article:Scotch yoke

A Scotch yoke mechanism can be used to convert between rotational motion and linear reciprocating motion. The linear motion can take various forms depending on the shape of the slot, but the basic yoke with a constant rotation speed produces a linear motion that is simple harmonic in form.

Notes

[edit]

^
The choice of using a cosine in this equation is a convention. Other valid formulations are:
$x(t)=A\sin \left(\omega t+\varphi '\right),$ where $\tan \varphi '={\frac {c_{1}}{c_{2}}},$
sincecosθ = sin(⁠π/2⁠ −θ).
^
The maximum displacement (that is, the amplitude),x_max, occurs whencos(ωt ±φ) = 1, and thus whenx_max =A.

References

[edit]

^"Simple harmonic motion | Formula, Examples, & Facts | Britannica".britannica.com. 2024-09-30. Retrieved2024-10-11.
^"Simple Harmonic Motion – Concepts".

Fowles, Grant R.; Cassiday, George L. (2005).Analytical Mechanics (7th ed.). Thomson Brooks/Cole.ISBN 0-534-49492-7.
Taylor, John R. (2005).Classical Mechanics. University Science Books.ISBN 1-891389-22-X.
Thornton, Stephen T.; Marion, Jerry B. (2003).Classical Dynamics of Particles and Systems (5th ed.). Brooks Cole.ISBN 0-534-40896-6.
Walker, Jearl (2011).Principles of Physics (9th ed.). Hoboken, New Jersey: Wiley.ISBN 978-0-470-56158-4.