The Motion of a Pendulum An Investigation
Trevor Bland
Dr Richard Clark
University Senior College
University of Adelaide
Department of Applied Mathematics
The Motion of a Pendulum An Investigation
“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 x(t )  a sin(bt  c)  d
to the data
Position v Time
Position (m)
2.5
2
1.5
1
0.5
0
time (sec)
Time (sec)
Exploring Velocity vs Time
• draw a scatter plot of velocity vs time
• Fit a sine functionv(t )  a sin(bt  c)  d
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
Position and Velocity
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
Position (m)
Velocity (m/s)
Time (secs)
Exploring Acceleration vs
Time vs time
draw a scatter plot of acceleration
•
• Fit a sine functiona(t )  a sin(bt  c)  d
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
Position, Velocity & Acceleration
6
4
2
Position (m)
0
Velocity (m/s)
-2 1
acceleration (m/s/s)
-4
-6
Time (sec)
Exploring Velocity vs
Position
• What might a scatter plot of velocity vs position
look like?
• Use your TI to draw this scatter-plot
Velocity v Position
Velocity (m/s)
2
1.5
1
0.5
0
-0.5 0
-1
0.5
1
1.5
-1.5
-2
Position (m)
2
2.5
Simple Harmonic Motion
• 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
Acceleration vs displacement
6
Acceleration
4
2
0
-1
-0.8
-0.6
-0.4
-0.2
-2 0
-4
-6
Displacement
0.2
0.4
0.6
0.8
1
The End