Editing Pendulum (section)

=== Compound pendulum ===

Any swinging [[rigid body]] free to rotate about a fixed horizontal axis is called a '''compound pendulum''' or '''physical pendulum'''. A compound pendulum has the same period as a simple gravity pendulum of length <math>\ell^\mathrm{eq}</math>, called the ''equivalent length'' or ''radius of oscillation'', equal to the distance from the pivot to a point called the ''[[center of percussion|center of oscillation]]''.<ref name="HuygensCompound">{{cite web
 | first = Christian | last = Huygens
 | translator-first=Ian | translator-last=Bruce
 | title = Horologium Oscillatorium
 | website = 17centurymaths
 | publisher = 17thcenturymaths.com
 | year = 1673
 | url = http://www.17centurymaths.com/contents/huygenscontents.html
 | access-date = 2009-03-01
}}, Part 4, Proposition 5</ref> This point is located under the [[center of mass]] of the pendulum, at a distance which depends on the mass distribution of the pendulum. If most of the mass is concentrated in a relatively small bob compared to the pendulum length, the center of oscillation is close to the center of mass.<ref>{{cite book
 | last=Glasgow | first=David
 | title=Watch and Clock Making
 | year=1885
 | publisher=Cassel & Co.
 | location=London
 | page = [https://archive.org/details/watchandclockma00glasgoog/page/n264 278]
 | url=https://archive.org/details/watchandclockma00glasgoog
}}</ref>

The radius of oscillation or equivalent length <math>\ell^\mathrm{eq}</math> of any physical pendulum can be shown to be
<math display="block">\ell^\mathrm{eq} = \frac{I_O}{mr_\mathrm{CM}}</math> 

where <math>I_O</math> is the [[moment of inertia]] of the pendulum about the pivot point <math>O</math>, <math>m</math> is the total mass of the pendulum, and <math>r_\mathrm{CM}</math> is the distance between the pivot point and the [[center of mass]].
Substituting this expression in (1) above, the period <math>T</math> of a compound pendulum is given by
<math display="block">T = 2\pi \sqrt\frac{I_O}{mgr_\mathrm{CM}}</math>
for sufficiently small oscillations.<ref>{{cite book
 | last=Fowles
 | first=Grant R
 | title=Analytical Mechanics, 4th Ed
 | year=1986
 | publisher=Saunders
 | location=NY, NY
 | pages = 202 ff
}}</ref>

For example, a rigid uniform rod of length <math>\ell</math> pivoted about one end has moment of inertia <math display="inline">I_O = \frac{1}{3}m\ell^2</math>. The center of mass is located at the center of the rod, so <math display="inline">r_\mathrm{CM} = \frac{1}{2}\ell</math> Substituting these values into the above equation gives <math display="inline">T = 2\pi\sqrt{\frac{\frac{2}{3}\ell}{g}}</math>. This shows that a rigid rod pendulum has the same period as a simple pendulum of two-thirds its length.

[[Christiaan Huygens]] proved in 1673 that the pivot point and the center of oscillation are interchangeable.<ref name="HuygensReciprocity">[http://www.17centurymaths.com/contents/huygenscontents.html Huygens (1673) Horologium Oscillatorium], Part 4, Proposition 20</ref>  This means if any pendulum is turned upside down and swung from a pivot located at its previous center of oscillation, it will have the same period as before and the new center of oscillation will be at the old pivot point. In 1817 [[Henry Kater]] used this idea to produce a type of reversible pendulum, now known as a [[Kater pendulum]], for improved measurements of the acceleration due to gravity.