Simple harmonic motion

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Simple harmonic motion shown both in real space and phase space. The orbit is periodic. (Here the velocity and position axes have been reversed from the standard convention to align the two diagrams)

Template:Classical mechanics

In mechanics and physics, simple harmonic motion (sometimes abbreviated as Template:Abbr) is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an oscillation that is described by a sinusoid which continues indefinitely (if uninhibited by friction or any other dissipation of energy).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Simple harmonic motion can serve as a mathematical model for a variety of motions, but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. The motion is sinusoidal in time and demonstrates a single resonant frequency. Other phenomena can be modeled by simple harmonic motion, including the motion of a simple pendulum, although for it to be an accurate model, the net 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; see small-angle approximation). Simple harmonic motion can also be used to model molecular vibration.

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

IntroductionEdit

The motion of a particle moving along a straight line with an acceleration whose direction is always toward a fixed point on the line and whose magnitude is proportional to the displacement from the fixed point is called simple harmonic motion.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In the diagram, a simple 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 the equilibrium position then there is no net force acting on the mass. However, if the mass is displaced from the equilibrium position, the spring exerts a restoring elastic force that obeys Hooke's law.

Mathematically, <math display="block"> \mathbf{F}=-k\mathbf{x}, </math> where Template:Math is the restoring elastic force exerted by the spring (in SI units: N), Template:Math is the spring constant (N·m⁻¹), and Template:Math is the displacement from the equilibrium position (in metres).

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, it accelerates 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, at Template:Math, the mass has momentum 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 its velocity reaches zero, whereupon it is accelerated back to the equilibrium position again.

As long as the system has no energy loss, the mass continues to oscillate. Thus simple harmonic motion is a type of periodic motion. If energy is lost in the system, then the mass exhibits damped oscillation.

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

DynamicsEdit

In Newtonian mechanics, for one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, can be obtained by means of Newton's second law and Hooke's law for a mass on a spring.

<math display="block"> F_\mathrm{net} = m\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -kx,</math> where Template:Mvar is the inertial mass of the oscillating body, Template:Mvar is its displacement from the equilibrium (or mean) position, and Template:Math is a constant (the spring constant for a mass on a spring).

Therefore, <math display="block"> \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -\frac{k}{m}x</math>

Solving the differential equation above produces a solution that is a sinusoidal function: <math> x(t) = c_1\cos\left(\omega t\right) + c_2\sin\left(\omega t\right),</math> where <math display="inline"> {\omega} = \sqrt{{k}/{m}}.</math> The meaning of the constants <math> c_1</math> and <math> c_2</math> can be easily found: setting <math> t=0</math> on the equation above we see that <math> x(0) = c_1</math>, so that <math> c_1</math> is the initial position of the particle, <math> c_1=x_0</math>; taking the derivative of that equation and evaluating at zero we get that <math> \dot{x}(0) = \omega c_2</math>, so that <math> c_2</math> is the initial speed of the particle divided by the angular frequency, <math> c_2 = \frac{v_0}{\omega}</math>. Thus we can write: <math display="block"> 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).</math>

This equation can also be written in the form: <math display="block"> x(t) = A\cos\left(\omega t - \varphi\right),</math> where

<math> A = \sqrt{{c_1}^2 + {c_2}^2} </math>
<math>\tan \varphi = \frac{c_2}{c_1}, </math>
<math>\sin \varphi = \frac{c_2}{A}, \; \cos \varphi = \frac{c_1}{A} </math>

or equivalently

<math> A = |c_1 + c_2i|, </math>
<math>\varphi = \arg(c_1 + c_2i) </math>

In the solution, Template:Math and Template:Math are two constants determined by the initial conditions (specifically, the initial position at time Template:Math is Template:Math, while the initial velocity is Template:Math), and the origin is set to be the equilibrium position.Template:Cref2 Each of these constants carries a physical meaning of the motion: Template:Math is the amplitude (maximum displacement from the equilibrium position), Template:Math is the angular frequency, and Template:Math is the initial phase.Template:Cref2

Using the techniques of calculus, the velocity and acceleration as a function of time can be found: <math display="block"> v(t) = \frac{\mathrm{d} x}{\mathrm{d} t} = - A\omega \sin(\omega t-\varphi),</math>

Speed: <math> {\omega} \sqrt {A^2 - x^2} </math>
Maximum speed: Template:Math (at equilibrium point)

<math display="block"> a(t) = \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = - A \omega^2 \cos( \omega t-\varphi).</math>

Maximum acceleration: Template:Math (at extreme points)

By definition, if a mass Template:Math is under SHM its acceleration is directly proportional to displacement. <math display="block"> a(x) = -\omega^2 x.</math> where <math display="block"> \omega^2=\frac{k}{m}</math>

Since Template:Math, <math display="block">f = \frac{1}{2\pi}\sqrt{\frac{k}{m}},</math> and, since Template:Math where Template:Math is the time period, <math display="block">T = 2\pi \sqrt{\frac{m}{k}}.</math>

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

EnergyEdit

Substituting Template:Math with Template:Math, the kinetic energy Template:Math of the system at time Template:Math is <math display="block"> K(t) = \tfrac12 mv^2(t) = \tfrac12 m\omega^2A^2\sin^2(\omega t - \varphi) = \tfrac12 kA^2 \sin^2(\omega t - \varphi),</math> and the potential energy is <math display="block">U(t) = \tfrac12 k x^2(t) = \tfrac12 k A^2 \cos^2(\omega t - \varphi).</math> In the absence of friction and other energy loss, the total mechanical energy has a constant value <math display="block">E = K + U = \tfrac12 k A^2.</math>

ExamplesEdit

File:Animated-mass-spring.gif

An undamped spring–mass system undergoes simple harmonic motion.

The following physical systems are some examples of simple harmonic oscillator.

Mass on a springEdit

A mass Template:Math attached to a spring of spring constant Template:Math exhibits simple harmonic motion in closed space. The equation for describing the period: <math display="block"> T= 2 \pi\sqrt\frac{m}{k}</math> 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 motionEdit

Simple harmonic motion can be considered the one-dimensional projection of uniform circular motion. If an object moves with angular speed Template:Math around a circle of radius Template:Math centered at the origin of the Template:Math-plane, then its motion along each coordinate is simple harmonic motion with amplitude Template:Math and angular frequency Template:Math.

Oscillatory motionEdit

The motion of a body in which it moves to and from a definite point is also called oscillatory motion or vibratory motion. The time period is able to be calculated by <math display="block"> T= 2 \pi\sqrt\frac{l}{g}</math> where l is the distance from rotation to the object's center of mass undergoing SHM and g is gravitational acceleration. This is analogous to the mass-spring system.

Mass of a simple pendulumEdit

Template:Infobox physical quantity In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a pendulum of length Template:Math with gravitational acceleration <math>g</math> is given by <math display="block"> T = 2 \pi \sqrt\frac{l}{g}</math>

This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not of the acceleration due to gravity, <math>g</math>, 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 <math>g</math> 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 for angular acceleration Template:Math being proportional to the sine of the displacement angle:

<math display="block">-mgl \sin\theta =I\alpha,</math>

where Template:Math is the moment of inertia. When Template:Math is small, Template:Math and therefore the expression becomes

<math display="block">-mgl \theta =I\alpha</math>

which makes angular acceleration directly proportional and opposite to Template:Math, 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 yokeEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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.

File:Scotch yoke animation.gif

Scotch yoke animation