DI – The Periodic Motion of a Pendulum – Solutions for Sample Data
In this directed investigation you will analyse the periodic motion of a pendulum. The motion a pendulum approximates straight line motion even though the bob hanging at the end of the string moves in a curve. This is because if the string is relatively long compared to the initial displacement, the curve made by the bob is close enough to a straight line.
A CBR (calculator based ranger) is attached to a TI graphics calculator. As the pendulum swings, the sonic sensor of the CBR tracks the bob of the pendulum and collects, at varying times , values of position , velocity , and acceleration . These values are sent to the TI graphics calculator and stored in lists L1, L2, L3 and L4 respectively. The calculator can then be used to draw scatter-plots of position vs time, velocity vs time, acceleration vs time, velocity vs position and acceleration vs displacement from the mean position. The TI graphics calculator can be used to fit functions to these scatter-plots.
Introduction
If a CBR is not available, an excel file of sample data, data.xls, can be used to do this investigation. Unfortunately excel does not do a sine regression. Instructions for transferring this excel file from a PC to a graphics calculator are available from the Home screen. Students using this sample data should omit activities 1 and 2.
Mathematical Investigation and Analysis
Activity 1 – Prepare the CBR and TI Graphics Calculator for data collection
Transfer the RANGER program to your TI graphics calculator
Your teacher will have the RANGER program on his/her TI and will arrange for this program to be transferred to all students’ TI graphics calculators. The RANGER program is available free of charge from the internet, http://education.ti.com/
On your TI adjust the RANGER program settings
Press , Choose RANGER, Press and then again. The opening screen is displayed.
Press . The MAIN MENU is displayed.
MAIN MENU
1: SETUP / SAMPLE
2: SET DEFAULTS
3: APPLICATIONS
4: PLOT MENU
5: TOOLS
6: QUIT
Choose SETUP / SAMPLE and press . The current settings are shown. The location of the cursor is shown by . Use the cursor keys, , , to move to the setting you want to change. Press to cycle through the available options for that setting. When the option is correct, use the cursor keys to move to the next setting. To change TIME, enter 1 or 2 digits, and then use the cursor keys to move to the next setting.
For this activity, the options for each setting should be:
REALTIME: NO
TIME (S): 5 SECONDS
DISPLAY: DISTANCE
BEGIN ON: [ENTER]
SMOOTHING: LIGHT
UNITS: METRES
Once the options are set correctly move the cursor so that it sits just to the left of START NOW
ie START NOW
Activity 2: Set up a simple pendulum and collect some data
Form a group of four students. Set up a simple pendulum as follows. Put a basketball in a shopping bag. This will serve as the bob of your pendulum. Tie a piece of rope that is approximately 1.5m long to the shopping bag. One student will stand on a chair and hold the rope, a second student will set the pendulum in motion and a third student will connect the TI graphics calculator to the CBR. The CBR is mounted on a table as shown. Align the pendulum so it swings in a direct line with the CBR. Position the CBR at least 0.5 metres from the closest approach of the basketball.
Make a final check that the TI is ready to receive the data.
Check that the following options are set correctly:
REALTIME: NO
TIME (S): 5 SECONDS
DISPLAY: DISTANCE
BEGIN ON: [ENTER]
SMOOTHING: LIGHT
UNITS: METRES
Ensure that the cursor sits just to the left of START NOW
ie START NOW
press
A message POINT CBR AT TARGET is displayed.
Set the pendulum in motion and when it has settled down press . You will hear a clicking sound as the data is collected, and the message TRANSFERRING... is displayed on the calculator.
When the data collection is complete, the calculator automatically displays a position-time scatter-plot of the collected data points. If you want to repeat the sample press and select REPEAT SAMPLE
Important information: Exiting the RANGER program:
Always exit the RANGER program using the QUIT option from the MAIN MENU.
The RANGER program performs a proper shutdown of CBR when you choose QUIT. This ensures that CBR is properly initialized for the next time you use it.
Activity 3 Exploring Position vs Time
Detach your calculator from the CBR. Recall that the values of time , position , velocity , and acceleration are stored in lists L1, L2, L3 and L4 respectively.
Use your graphics calculator to draw a scatter-plot of position against time. Fit a sine curve to your data. Draw the graph of the sine function over your scatter-plot. State the equation of the position function, , giving values of , , and correct to three significant figures.
Use any suitable graphing software to sketch the graph of the function for . Print your graph and paste a copy below.
Recall that the period of the function is . Use this formula to calculate the period of the function to the nearest tenth of a second.
Period seconds (nearest tenth)
What is the amplitude of the function to the nearest cm.
Amplitude metres (nearest cm)
Which of the following distances best describes the amplitude: AB, MB or BC?
MB best describes the amplitude.
Activity 4 Exploring Velocity vs Time
Use your graphics calculator to draw a scatter-plot of velocity against time. Fit a sine curve to your data. Draw the graph of the sine function over your scatter-plot. State the equation of the velocity function, , giving values of , , and correct to three significant figures.
Calculate the period of the function to the nearest tenth of a second.
Period seconds (nearest tenth)
What is the amplitude of the function . Give your answer correct to three significant figures.
Amplitude
Use any suitable graphing software to sketch the graphs of and , on the same axes. Print your graph and paste a copy below.
Label clearly a point on your graph of where the velocity is a maximum. Draw a vertical line through to meet the graph of at . State the position of the bob when its velocity is a maximum eg “the bob is at ” or “the bob is at ” or etc.
The bob is at M
The slope of a position vs time graph at any given point in time gives the velocity of the body at that point in time. Label clearly a point on your graph of where the slope of the curve is zero. Draw a vertical line through and mark the point where this line meets the graph of .
Activity 5 Exploring Acceleration vs Time
Use your graphics calculator to draw a scatter-plot of acceleration against time. Fit a sine curve to your data. Draw the graph of the sine function over your scatter-plot. State the equation of the acceleration function, , giving values of , , and correct to 3 significant figures.
Calculate the period of the function to the nearest tenth of a second.
Period seconds (nearest tenth)
What is the amplitude of the function . Give your answer correct to three significant figures.
Amplitude
Use any suitable graphing software to sketch the graphs of , and , on the same axes. Print your graph and paste a copy below.
Label clearly a point on your graph of where the acceleration is a maximum. Draw a vertical line through to meet the graph of at . State the position of the bob when its acceleration is a maximum.
The bob is at B when its acceleration is a maximum
The slope of the graph of at any given point in time gives the acceleration of the body at that time. Label clearly a point on your graph of where the slope is zero. Draw a vertical line through and mark the point where this line meets the graph of . State the position of the bob at this time.
The bob is at M, the mean position
Activity 6 – Exploring Velocity vs Position
Use your graphics calculator to draw a scatter-plot of velocity against position.
Sketch the scatter-plot below. Label your axes and show scales.
Label clearly a point on your graph where the velocity of the bob is a maximum. State the position of the bob when its velocity is a maximum.
The bob is at M, its mean position, when the velocity is a maximum.
Label clearly a point on your graph where the bob is closest to the CBR. What is the velocity of the bob at this time.
The velocity of the bob is zero when the bob is closest to the CBR.
Comment on the velocity of the bob when it is in the mean position, M.
The velocity is either a maximum or a minimum when the bob is in its mean position.
Comment on the speed of the bob when it is in the mean position, M.
The speed is a maximum when the bob is in its mean position.
Activity 7 – Exploring Acceleration vs Distance from Mean 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 a pendulum approximates straight line motion even though the bob hanging at the end of the string moves in a curve. This is because if the string is relatively long compared to the initial displacement, the curve made by the bob is close enough to a straight line.
Use your graphics calculator to draw a scatter-plot of acceleration, against , where = displacement from the mean position. Fit a straight line to your data. Draw the graph of the line over your scatter-plot. State the equation of the function giving values of and correct to three significant figures.
Use any suitable graphing software to sketch the graph of . Print your graph and paste a copy below.
Does the motion of a pendulum approximate simple harmonic motion? Explain your answer briefly
Yes, the motion of a pendulum does approximate simple harmonic motion. The graph of a against d is approximately a straight line passing through the origin, implying that a is directly proportional to d. Furthermore the direction of a is always towards the mean position.
Conclusion
Students should write a brief summary of their findings.
U:\07_Assists Grant\Maths\Pendulum\CD\DIpend_sol.doc