Simple harmonic motion

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This article is about simple harmonic motion. For a detailed mathematical treatment of classical harmonic motion including damping and forcing, see harmonic oscillator.

Simple harmonic motion (SHM) is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The motion is periodic - as it repeats itself at standard intervals in a specific manner - and sinusoidal, with constant amplitude; the acceleration of a body executing SHM is directly proportional to the displacement of the body from the equilibrium position and is always directed towards the equilibrium position.

The motion is characterized by its amplitude (which is always positive), its period, the time for a single oscillation, its frequency, the reciprocal of the period (i.e. the number of cycles per unit time), and its phase, which determines the starting point on the sine wave. The period and frequency are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and velocity) of that system.

A general equation describing simple harmonic motion is $x(t) = A\cos \left( 2\,\pi \,ft+\phi\right)$ , where x is the displacement, A is the amplitude of oscillation, f is the frequency, t is the elapsed time, and $φ$ is the phase of oscillation. If there is no displacement at time t = 0, the phase $\phi= \frac{\pi}{2}$ .

Simple harmonic motion.

Simple harmonic motion can serve as a mathematical model of a variety of motions and provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.

[edit] Introduction

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 in order to align the two diagrams)

A typical example of a system that undergoes simple harmonic motion is a spring-mass system, which is a mass attached to a spring.

If the spring is unstretched, there is no net force on the mass - in other words, the system is in equilibrium. However, if the mass is displaced from equilibrium, the spring will exert a restoring force, which is a force that tends to restore it to the equilibrium position. In the case of the spring-mass system, this force is the elastic force, which is given by Hooke's Law, $F = - k x$ , where F is the restoring force, x is the displacement, and k is the spring constant.

Any system that undergoes simple harmonic motion exhibits two key features.

When the system is displaced from equilibrium there must exist a restoring force that tends to restore it to equilibrium.
The restoring force must be proportional to the displacement, or approximately so.

The spring-mass system satisfies both.

Once the mass is displaced it experiences a restoring force, accelerating it, causing it to start going back to the equilibrium position. As it gets closer to equilibrium the restoring force decreases; at the equilibrium position the restoring force is 0. However, at x=0, the mass has some momentum due to the impulse of the force that has acted on it; this causes the mass to shoot past the equilibrium position, in this case, compressing the spring. The restoring force then tends to slow it down, until the velocity reaches 0, whereby it will attempt to reach equilibrium position again.

As long as the system does not lose energy, the mass will continue to oscillate like so; thus, the motion is termed periodic motion. Further analysis will show that in the case of the spring-mass system the motion is simple harmonic.

[edit] Mathematics

[edit] Important terms

Amplitude: maximal displacement from the equilibrium.
Period: the time it takes the system to complete an oscillation cycle. Inverse of frequency.
Frequency: the number of cycles the system performs per unit time (usually measured in hertz = 1/s).
Angular frequency: $ω = 2π f$
Phase: how much of a cycle the system completed (system that begins is in phase zero, system which completed half a cycle is in phase $π$ ).
Initial conditions: the state of the system at t = 0, the beginning of oscillations.

[edit] Equation of motion

Simple harmonic motion is defined by the differential equation

$m\frac{d^2 x}{dt^2} = -kx$

where "k" is a positive constant, "m" is the mass of the body, and "x" is its displacement from the mean position. This is a second-order differential equation.

In the case of a spring, k is the spring constant. The equation can be derived from Newton's second law ( $F = d (m v) / d t$ ) as given by Hooke's law thus F=ma. If k is positive, the solution is a sinusoidal function (the exact form depends on the initial conditions); it can be easily verified that taking the second derivative of sin(x) gives -sin(x), thus satisfying the equation.

It can be shown, by differentiation, exactly how the acceleration varies with time. Using the angular frequency $ω$ , defined as

ω = 2π f = 2π / T,

the displacement is given by the function

$x(t) = A\cos \left( \omega t +\phi\right)$ .^[1]

Position, velocity and acceleration of an harmonic oscillator

Differentiating once gives an expression for the velocity at any time.

$v(t) = \frac{\mathrm{d}\,x(t)}{\mathrm{d}t} = - A\omega \sin \left( \omega t+\phi\right).$

And once again to get the acceleration at a given time.

$a(t) = \frac{\mathrm{d}^2\,x(t)}{\mathrm{d} t^2} = - A \omega^2 \cos \left( \omega t+\phi\right).$

These results can of course be simplified, giving us an expression for acceleration in terms of displacement.

$a = \frac{d^2x}{dt^2} = -\omega^2x,$

which is equivalent to:

$a(t) = -\left( 2\pi f \right)^2 x(t)$

or

$a = - ω 2 x$

Comparing to the definition, we may write

$ω 2 = k / m$

Now that the angular velocity $ω$ is known in terms of k and m, other things such as the period, velocity etc can be easily found.

[edit] Energy of simple harmonic motion

When and if total energy is constant and kinetic, the formula $E = \frac{kA^2}{2}$ applies for simple harmonic motion, where E is considered the total energy while all energy is in its kinetic form and A represents the mean displacement of the spring from its rest position in MKS units.

The kinetic energy is

$K = \frac{1}{2} m \left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2 = \frac{1}{2} k A^2 \sin^2(\omega_0 t + \phi)$ .

and the potential energy is

$U = \frac{1}{2} k x^2 = \frac{1}{2} k A^2 \cos^2(\omega_0 t + \phi)$

so the total energy of the system has the constant value

$E = \frac{1}{2} k A^2.$

[edit] Examples

An undamped spring-mass system undergoes simple harmonic motion.

Simple harmonic motion is exhibited in a variety of simple physical systems and below are some examples.

[edit] Mass on a spring

A mass M attached to a spring of spring constant k exhibits simple harmonic motion in space with

$\omega=2 \pi \ f = \sqrt{\frac{k}{M}}.\,$

Alternately, if the other factors are known and the period is to be found, this equation can be used:

$T= \frac{1}{f} = 2 \pi \sqrt{\frac{M}{k}}.$

This shows that the period of oscillation is independent of both the amplitude and gravitational acceleration.

The total energy is constant, and given by $E = \frac{kA^2}{2},$ where E is the total energy.

[edit] Uniform circular motion

Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion. If an object moves with angular frequency $ω$ around a circle of radius $R$ centered at the origin of the x-y plane, then its motion along the x and the y coordinates is simple harmonic with amplitude $R$ and angular speed $ω$ .

[edit] Mass on a simple pendulum

The motion of an undamped Pendulum aproximates to simple harmonic motion if the amplitude is very small.

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 string of length $\ell$ with gravitational acceleration $g$ is given by

$T= 2 \pi \sqrt{\frac{\ell}{g}}$

This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not the acceleration due to gravity (g), therefore a pendulum of the same length on the Moon would swing more slowly due to the Moon's lower gravitational acceleration.

This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to the sine of position:

$\ell m g \sin(\theta)=I \alpha$

where I is the moment of inertia; in this case $I = m\ell^2$ . When $θ$ is small, $\sin(\theta) \approx \theta$ and therefore the expression becomes

$\ell m g \theta=I \alpha$

which makes angular acceleration directly proportional to $θ$ , satisfying the definition of Simple Harmonic Motion.

For a solution not relying on a small-angle approximation, see pendulum (mathematics).

[edit] Useful Formulas

Given mass $M$ attached to a spring/pendulum with amplitude $A$ with acceleration $a$ :

$k = \frac{Ma}{A}$

$T_s = T_p = \frac{1}{f} = 2 \pi \sqrt{ \frac{M}{k}} = 2 \pi \sqrt{ \frac{\ell}{g}}.$

$E_{tot} = \frac{kA^2}{2} = \frac{MaA}{2}.$

Where:

k

is the spring constant.

M

is the mass (usually in kilograms)

a

is the acceleration.

A

is the amplitude.

f

is the frequency (usually in hertz).

T s

or

T p

is the time period of the spring or pendulum.

g

is the acceleration due to gravity (On Earth at sea level: 9.81 m/s²).

$\ell$ is the length of the pendulum.

E t o t

is the total energy.

[edit] See also

[edit] References

^ The choice of using a cosine in this equation is arbitrary in that $x(t) = A\sin \left( \omega t +\phi\right)$ is also a valid solution; the cos can be converted to a sin by subtracting π/2 and redefining the phase constant. Another valid form is $x(t) = A\sin\left(\omega t\right) + B\cos\left(\omega t \right).$

[edit] External links

easy java simulation on simple harmonic applet SHM by lookang inquiry learningeasy java simulation on simple harmonic applet SHM by lookang inquiry learning
Java simulation of spring-mass oscillator

[0] The choice of using a cosine in this equation is arbitrary in that $x(t) = A\sin \left( \omega t +\phi\right)$ is also a valid solution; the cos can be converted to a sin by subtracting π/2 and redefining the phase constant. Another valid form is $x(t) = A\sin\left(\omega t\right) + B\cos\left(\omega t \right).$

[1]

Simple harmonic motion

From Wikipedia, the free encyclopedia

Contents

[edit] Introduction

[edit] Mathematics

[edit] Important terms

[edit] Equation of motion

[edit] Energy of simple harmonic motion

[edit] Examples

[edit] Mass on a spring

[edit] Uniform circular motion

[edit] Mass on a simple pendulum

[edit] Useful Formulas

[edit] See also

[edit] References

[edit] External links

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