Editing Simple harmonic motion (section)

==Examples==
[[Image:Animated-mass-spring.gif|right|frame|An undamped [[spring–mass system]] undergoes simple harmonic motion.]]
The following physical systems are some examples of [[harmonic oscillator|simple harmonic oscillator]].

===Mass on a spring===
A mass {{math|''m''}} attached to a spring of spring constant {{math|''k''}} 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 motion===
Simple harmonic motion can be considered the one-dimensional [[projection (mathematics)|projection]] of [[uniform circular motion]]. If an object moves with angular speed {{math|''ω''}} around a circle of radius {{math|''r''}} centered at the [[origin (mathematics)|origin]] of the {{math|''xy''}}-plane, then its motion along each coordinate is simple harmonic motion with amplitude {{math|''r''}} and angular frequency {{math|''ω''}}.

=== Oscillatory motion ===
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 pendulum===
{{Infobox physical quantity
| image = ลูกตุ้มธรรมชาติ.gif
| caption = A [[pendulum]] making 25 complete [[oscillation]]s in 60 s, a frequency of 0.41{{overline|6}} [[Hertz]]
{{ubl
}}
}}
In the [[small-angle approximation]], the [[pendulum (mechanics)#Small-angle approximation|motion of a simple pendulum]] is approximated by simple harmonic motion. The period of a mass attached to a pendulum of length {{math|''l''}} 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]] {{math|''α''}} being proportional to the sine of the displacement angle:

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

where {{math|''I''}} is the [[moment of inertia]]. When {{math|''θ''}} is small, {{math|sin ''θ'' ≈ ''θ''}} and therefore the expression becomes

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

which makes angular acceleration directly proportional and opposite to {{math|''θ''}}, 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===
{{Main|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.
[[File:Scotch yoke animation.gif|thumb|200px|Scotch yoke animation]]