I am using this page to learn about mathematical markup.
Point mass ideal pendulum, small angle case:
![{\displaystyle t_{c}=2\pi {\sqrt {\frac {L}{g_{c}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/953bfb7cf9320519520dcb6750f296a8d7db6ecc)
Distributed mass pendulum, with moment of inertia, small angle case:
![{\displaystyle t_{c}=2\pi {\sqrt {\frac {I}{mg_{i}L}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c73913cd87fe298f4a5c8811093100cf23b42c2)
Equating the right hand sides of both of these equations, canceling like terms, and squaring both sides:
![{\displaystyle {\frac {L}{g_{c}}}={\frac {I}{mg_{i}L}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5cea3fc134e4d421a91905b6d82d512216f0ad)
The ratio of moment-corrected gravity over experimentally determined gravity is:
![{\displaystyle {\frac {g_{i}}{g_{c}}}={\frac {I}{mL^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4c07564d2d72940b8e34b3635b78285bd7391f)
For a cylinder rotating about an axis passing through its diameter at its center of gravity:
![{\displaystyle I_{x}={\frac {m(3r^{2}+h^{2})}{12}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84efea3053c59ae07675b6f7b3df30ffbce22bdb)
Using the parallel axis theorem, I move the rotation axis from center of gravity to the pivot point a distance L above the center of gravity:
![{\displaystyle I=I_{x}+mL^{2}=m\left({\frac {r^{2}}{4}}+{\frac {h^{2}}{12}}+L^{2}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/944d3db4566aaf7c7c3f2a4f44e268e0abc66ee7)
Substituting the rightmost expression for I into the gi/gc equation, we get:
![{\displaystyle {\frac {g_{i}}{g_{c}}}={\frac {{\frac {r^{2}}{4}}+{\frac {h^{2}}{12}}+L^{2}}{L^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32d7094574628e6b43e001c68e94caf8c8b1c62d)
Buoyancy of air requires a correction of:
![{\displaystyle {\frac {g_{a}}{g_{i}}}={\frac {m}{m-\rho V}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32cb4c03c05bfd035dbfd5fd6636c926217bada5)
General formula for zero-angle limit of a conical pendulum consisting of a thin rod with an arbitrary linear density:
![{\displaystyle g=\left({\frac {2\pi }{t_{c}}}\right)^{2}{\frac {\int _{0}^{L}s^{2}\rho (s)\,ds}{\int _{0}^{L}s\rho (s)\,ds}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/776466bcdaf7e34523db2ed55745086559027ee0)