The Motion of a Pendulum - An Investigation Trevor Bland University Senior College Dr Richard Clark University of Adelaide Department of Applied Mathematics

“This material has been developed as a part of the Australian School Innovation in Science, Technology and Mathematics Project funded by the Australian Government Department of Education, Science and Training as a part of the Boosting Innovation in Science, Technology and Mathematics Teaching (BISTMT) Programme.” The Motion of a Pendulum - An Investigation

A simple pendulum is set up as shown A piece of rope about 1.5m long is tied to a shopping bag containing a basketball

A CBR (calculator based ranger) is clamped to a table so that the pendulum swings in a direct line with the sensor of the CBR

As the pendulum swings, the sensor of the CBR tracks the bob and collects, at varying times, its position, velocity, and acceleration

The CBR is attached to a TI graphics calculator

As the pendulum swings, the CBR sends the data to the TI which stores it in lists L1, L2, L3, and L4 Time in L1, position in L2, velocity in L3, and acceleration in L4

The TI can then be detached from the CBR Using the data in lists L1, L2, L3, and L4, the TI can be used to carry out a wide variety of analyses

Exploring Position vs Time draw a scatter plot of position vs time fit a sine function to the data

draw a scatter plot of velocity vs time Fit a sine function to the data Draw the position and velocity functions on the same axes Consider in what positions the bob’s velocity is equal to zero Exploring Velocity vs Time

draw a scatter plot of acceleration vs time Fit a sine function to the data Draw the position, velocity, and acceleration functions on the same axes Consider in what positions the bob’s acceleration is a maximum Exploring Acceleration vs Time

What might a scatter plot of velocity vs position look like? Use your TI to draw this scatter-plot Exploring Velocity vs Position

Simple Harmonic motion is oscillatory motion in a straight line about a mean position such that acceleration is always directed towards the mean position and is directly proportional to the displacement from the mean position. The motion of a pendulum approximates simple harmonic motion when the length of the rope is relatively long compared to the initial displacement Simple Harmonic Motion

The TI can be used to investigate whether the acceleration is directly proportional to the displacement from the mean position Draw a scatter-plot of acceleration vs displacement from the mean position – the plot should be a straight line passing through the origin Simple Harmonic Motion

The End