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Oscillating Pendulum, https://en.m.wikipedia.org/wiki/File:Pendulum-no-text.gif
The Mathematics of Pendulums
The Mathematics of Pendulums
Theoretical pendulums swing back and forth at a consistent rate, called a frequency ( f ) which is measured in Hertz, and they have a set time for the pendulum to swing and return to its original position before swinging again, called an oscillation time (T ), which is the period of a cosine wave.
Point-Mass Analysis
Point-Mass Analysis
Point mass analysis relates the distance from the pivot point to the center of mass and the force of gravity to the time it takes for the pendulum to oscillate. Interestingly, the mass of the pendulum is ignored; note the famous lead ball and feather drop experiment.
From measurements, r = 0.0526m
Plug the numbers into the formula, and T = 0.532 sec
Inertial Analysis
Inertial Analysis
By calculations, Itot = 5.375 g cm2
mr = 6.235
Thus, T = 0.5717
Definition of inertial oscillation time T in seconds (s), where:
Itot is the total inertia of the pendulum clock around the rotation point, including the pendulum body and the possible n screws/bolts included in your pendulum.
The value of Itot is computed as the sum of two main components:
The inertial effect of the pendulum body as the pendulum body mass m times the squared distance r between the rotation point and location of Center of Gravity (CoG) of the pendulum body
The inertial effect of the possible n screws/bolts as the screw/bolt mass m_k times the squared distance rk between the rotation point and location of Center of Gravity (CoG) of each screw/bolt
mr is the distance weighted mass of the pendulum clock around the rotation point, including the pendulum body and the possible n screws/bolts included in your pendulum.
The value of mr is also computed as the sum of two main components:
The distance weighted mass of the pendulum body as the pendulum body mass m times the distance r between the rotation point and location of Center of Gravity (CoG) of the pendulum body.
The distance weighted mass of the possible n screws/bolts as the screw/bolt mass mk times the distance rk between the rotation point and location of Center of Gravity (CoG) of each screw/bolt
Details for Point-Mass and Inertial (Rigid-Body) Analysis
Details for Point-Mass and Inertial (Rigid-Body) Analysis
Below you will find a table that contains numerous values and measurements that were used to determine the predictions for the pendulum's oscillation time. Apologies for the rather poor formatting and differing variable names.
Two-dimensional Simulation Analysis
Two-dimensional Simulation Analysis
Lastly, Working Model 2D was used to digitally simulate the oscillation of the pendulum and create a cosine graph displaying that. From the graph, WM2D predicted that the pendulum would oscillate every 0.66 seconds.
This value may have been more accurate had the escapement wheel been included in the model, however, as detailed in challenged, the escapement wheel caused other problems for the software
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