Theories

Theory

As we know, the exact value of gravitational attraction is relative to the distance between two masses. For Earth the force of gravity will depend on the distance between an object and the center of Earth's mass. If Earth were a perfect sphere we would be able to expect that a mass would feel the same gravitational force no matter where it was on the surface, because every point on the surface would be an equal distance to the center of Earth's mass. However due to Earth being more of an oblate spheroid the distance from the surface to the center of Earth's mass changes with latitude. The fact that gravity changes due to latitude brings us to the purpose of the experiment which is finding the gravity here at CNU.

Variable Definitions

$m\to$ weight of mass

$l\to$ length of pendulum

$g\to$ gravity

$T\to$ period of the pendulum

$\omega \to$ angle from straight down

$\mathbb{𝕀}\to$ moment of inertia of mass

Measuring Gravity

The simple pendulum theory can be looked at through its torque where:

$\tau =\mathbb{𝕀}\ddot{\theta }=r\times F$

The force that is perpendicular to $r$ is then plugged in and then converted to differential form:

$F=mg\sin \theta$

$m{l}^2\ddot{\theta }=-mgl\sin \theta$

$\ddot{\theta }+\frac{g}{l}\sin \theta =0$

Based off of this differential equation, the angular frequency is found to be:

$\omega =\sqrt{\frac{g}{l}}$

Period is then found to be:

$T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{l}{g}}$

Solved for gravity the equation becomes:

$g=\frac{4{\pi }^{2}l}{T^2}$

From this it is clear that the measurement of gravity depends only on the period of oscillation and the length of the pendulum.