User:CosineKitty/MathPage

I am using this page to learn about mathematical markup.

Point mass ideal pendulum, small angle case:
 * $$t_c = 2 \pi \sqrt {\frac {L} {g_c}}$$

Distributed mass pendulum, with moment of inertia, small angle case:


 * $$t_c = 2 \pi \sqrt {\frac {I} {m g_i L}}$$

Equating the right hand sides of both of these equations, canceling like terms, and squaring both sides:


 * $$\frac {L} {g_c} = \frac {I} {m g_i L}$$

The ratio of moment-corrected gravity over experimentally determined gravity is:


 * $$\frac {g_i} {g_c} = \frac {I} {m L^2}$$

For a cylinder rotating about an axis passing through its diameter at its center of gravity:


 * $$I_x = \frac {m (3 r^2 + h^2)} {12}$$

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:


 * $$I = I_x + m L^2 = m \left( \frac {r^2} {4} + \frac {h^2} {12} + L^2 \right)$$

Substituting the rightmost expression for I into the gi/gc equation, we get:


 * $$\frac {g_i} {g_c} = \frac {\frac{r^2}{4} + \frac{h^2}{12} + L^2} {L^2}$$

Buoyancy of air requires a correction of:


 * $$\frac {g_a} {g_i} = \frac {m} {m - \rho V}$$

General formula for zero-angle limit of a conical pendulum consisting of a thin rod with an arbitrary linear density:


 * $$g = \left (\frac{2\pi} {t_c} \right )^2 \frac { \int_0^L s^2 \rho(s)\, ds } { \int_0^L s \rho(s)\, ds }$$