Editing Cavendish experiment (section)

==Derivation of ''G'' and the Earth's mass==
{{hatnote|For the definitions of terms, see the drawing below and the table at the end of this section.}}

The following is not the method Cavendish used, but describes how modern physicists would calculate the results from his experiment.<ref name=HarvLect>{{cite web|title=Cavendish Experiment, Harvard Lecture Demonstrations, Harvard Univ|url=http://sciencedemonstrations.fas.harvard.edu/icb/icb.do?keyword=k16940&pageid=icb.page80669&pageContentId=icb.pagecontent277503&state=maximize&view=view.do&viewParam_name=indepth.html#a_icb_pagecontent277503|accessdate=2013-12-30}}. '[the torsion balance was]...modified by Cavendish to measure ''G''.'</ref><ref>[https://books.google.com/books?id=dg0RAAAAIAAJ&pg=PA41 Poynting 1894], p. 41</ref><ref>Clotfelter 1987 p. 212 explains Cavendish's original method of calculation.</ref> From [[Hooke's law]], the [[torque]] on the torsion wire is proportional to the deflection angle <math>\theta</math> of the balance. The torque is <math>\kappa\theta</math> where <math>\kappa</math> is the [[torsion coefficient]] of the wire. However, a torque in the opposite direction is also generated by the gravitational pull of the masses. It can be written as a product of the attractive force of a large ball on a small ball and the distance L/2 to the suspension wire. Since there are two balls, each experiencing force ''F'' at a distance {{sfrac|''L''|2}} from the axis of the balance, the torque due to gravitational force is ''LF''. At equilibrium (when the balance has been stabilized at an angle <math>\theta</math>), the total amount of torque must be zero as these two sources of torque balance out. Thus, we can equate their magnitudes given by the formulas above, which gives the following:

:<math>\kappa\theta\ = LF \,</math>

For ''F'', [[Isaac Newton|Newton]]'s [[law of universal gravitation]] is used to express the attractive force between a large and small ball:

[[File:Cavendish Torsion Balance Diagram.svg|thumb|220px|Simplified diagram of torsion balance]]

:<math>F = \frac{G m M}{r^2}\,</math>

Substituting ''F'' into the first equation above gives

:<math>\kappa\theta\ = L\frac{GmM}{r^2} \qquad\qquad\qquad(1)\,</math>

To find the torsion coefficient (<math>\kappa</math>) of the wire, Cavendish measured the natural [[Resonance|resonant]] [[Torsion spring#Torsional harmonic oscillators|oscillation period]] ''T'' of the torsion balance:

:<math>T = 2\pi\sqrt{\frac{I}{\kappa}}</math>

Assuming the mass of the torsion beam itself is negligible, the [[moment of inertia]] of the balance is just due to the small balls. Treating them as point masses, each at L/2 from the axis, gives:

:<math>I = m\left (\frac{L}{2}\right )^2 + m\left (\frac{L}{2}\right )^2 = 2m\left (\frac{L}{2}\right )^2 = \frac{mL^2}{2}\,</math>,

and so:

:<math>T = 2\pi\sqrt{\frac{mL^2}{2\kappa}}\,</math>

Solving this for <math>\kappa</math>, substituting into (1), and rearranging for ''G'', the result is:

:<math>G = \frac{2 \pi^2 L r^2 \theta}{M T^2} \,</math>.

Once ''G'' has been found, the attraction of an object at the Earth's surface to the Earth itself can be used to calculate the [[Earth mass|Earth's mass]] and density:

:<math>mg = \frac{GmM_{\rm earth}}{R_{\rm earth}^2}\,</math>

:<math>M_{\rm earth} = \frac{gR_{\rm earth}^2}{G}\,</math>

:<math>\rho_{\rm earth} = \frac{M_{\rm earth}}{\tfrac{4}{3} \pi R_{\rm earth}^3} = \frac{3g}{4 \pi R_{\rm earth} G}\,</math>

===Definitions of terms===

{| class="wikitable"
!Symbol ||Unit ||Definition
|-
|<math>\theta</math>||[[radian]]s||Deflection of torsion balance beam from its rest position
|-
|''F''||[[newton (unit)|N]]||Gravitational force between masses ''M'' and ''m''
|-
|''G''||m{{sup|3}} kg{{sup|−1}} s{{sup|−2}}||Gravitational constant
|-
|''m''||kg||Mass of small lead ball
|-
|''M''||kg||Mass of large lead ball
|-
|''r''||m||Distance between centers of large and small balls when balance is deflected
|-
|''L''||m||Length of torsion balance beam between centers of small balls
|-
|<math>\kappa</math>||N m rad{{sup|−1}}||Torsion coefficient of suspending wire
|-
|{{math|''I''}}||kg m{{sup|2}}||Moment of inertia of torsion balance beam
|-
|''T''||s||Period of oscillation of torsion balance
|-
|''g''||m s{{sup|−2}}||Acceleration of gravity at the surface of the Earth
|-
|''M''{{sub|earth}}||kg||Mass of the Earth
|-
|''R''{{sub|earth}}||m||Radius of the Earth
|-
|<math>\rho</math>{{sub|earth}}||kg m{{sup|−3}}||Density of the Earth
|}