Kater's Pendulum Lecture Notes

Vocabulary terms and definitions related to the physics and application of Kater's Pendulum for measuring gravitational acceleration.

Aim of the Experiment

To find the gravitational acceleration in J-18 and to determine the latitude of the lab.

Oblate Sphere

The shape of the Earth as proved by Isaac Newton in 1687, meaning it is thicker at the equator due to the Earth's rotation.

Variation of $g$

The concept that the acceleration of gravity is not uniform across Earth due to its non-spherical shape and differences in altitude according to the law of universal gravitation.

Kater's Pendulum

An apparatus consisting of a long bar and two masses attached to the bar, which can oscillate on either of two knife-edge suspension points.

$a$ and $b$

The distances from the center of mass (cm) to the pivot points A and B, respectively.

Restoring Torque ($\tau$)

For oscillations about point A, it is defined by the equation $au = -Mg a ext{sin}( heta)$, where clockwise rotation is considered a negative torque.

Parallel Axis Theorem

A theorem used to express the moment of inertia about a pivot point ($I_a$) in terms of the moment about the center of mass ($I_{cm}$), given by $I_a = I_{cm} + M a^2 = M(k^2 + a^2)$.

$k$ (Radius of Gyration)

The value that defines the moment of inertia about the center of mass where $I_{cm} = Mk^2$.

Period of Swing ($T_a$)

The period of oscillation for small-amplitudes about point A, predicted by the formula $T_a = 2 ext{\pi} ext{\sqrt{\frac{k^2 + a^2}{ga}}}$.

Period of Swing ($T_b$)

The period of oscillation for small-amplitudes about point B, predicted by the formula $T_b = 2 ext{\pi} ext{\sqrt{\frac{k^2 + b^2}{gb}}}$.

Acceleration of Gravity Calculation

When $T_a = T_b$, the acceleration of gravity can be solved using the distance between knife suspension points ($a+b$) in the formula $T = 2 ext{\pi} ext{\sqrt{\frac{a+b}{g}}}$.

$a + b$ Value

The specific distance between the knife suspension points in the experiment, given as $100.06\,cm$.

Photo-gate

A device used to measure the oscillation period by detecting when the pendulum interrupts a light path, causing a voltage transition from $5\,V$ to $0\,V$.

Capstone

The specific software utilized on the computer for counting/timing the pendulum oscillations.