Trouble deriving the gravitational force of opposing mass in cavendish experiment

Question

I'm currently working on doing the cavendish experiment to determine G and I'm done with most derivations so have reached the point of corrections. One of the major corrections I have to make has to do with the gravitational force supplied by the other mass. Image for clarification:

Where $F_g$ is the main force working on the smaller masses, $F'_g$ is probably quite significant (I'm estimating aproximately 5%) so I'd like to calculate $F'_{g,||}$ but have yet been unable to derive a formula for it. Is anyone here able to help?

Answer 1

The distance $D$ between the left smaller mass and the right larger mass is given by Pythagoras:

$$D=\sqrt{d^2+L^2}$$

Thus for $F_g'$ we get:

$$F_g'=G\frac{mM}{D^2}$$

And:

$$F'_{g,||}=F_g'\sin \theta,$$

where:

$$\tan\theta=\frac{d}{L}$$

Or:

$$\sin\theta=\frac{d}{D}$$

Because $F_g=G\frac{mM}{d^2}$ we can even determine $\frac{F'_{g,||}}{F_g}$:

$$\frac{F'_{g,||}}{F_g}=\frac{d^3}{(d^2+L^2)^{\frac32}}$$