(From, Henry Cavendish: the man end the measurement by Isobel Falconer, page, 475. (1999) Article can be found here.)

1. A more modern analysis of the method is as follows. Let \(2a\) be the length of the torsion rod, \(m\) the mass of the ball, \(M\) the mass of a large sphere and \(d\) the distance between the centers of the ball and sphere, supposed the same on each side.
2. Then, when the spheres are moved from one side of the ball to the other, the rod moves round through an angle \(\theta\), given by,

\begin{equation*} \mu \; \theta = \frac{4 \;G \;M\; m\; a}{d^2} \end{equation*}

where \(\mu\) is the couple required to twist the rod through 1 radian. \(\mu\) can be found from the period of vibration of the torsion system, \(T\), and its calculated moment of inertia, \(I\):

\begin{equation*} \mu = \frac{4\pi^2\,I}{T^2} \end{equation*}

3. Then, using the relation,

\begin{equation*} G = \frac{g\;r^2}{M_e} \end{equation*}

where \(g\) is the acceleration due to gravity, \(r\) is the radius of the earth and \(M_e\) is the mass of the earth, and defining \(g\) by the length, \(L\), of a seconds pendulum, the density of the earth

\begin{equation*} \Delta = \frac{\left [ 3/(4\pi\,r) \right ] \; L\,M\,m\,T^2}{d^2\,I\,\theta} \end{equation*}