Physics 1051
Laboratory #1
Simple Harmonic Motion
Introduction to
Simple Harmonic Motion
Physics 1051
Laboratory #1
Simple Harmonic Motion
Contents
Part I: Setup
Apparatus and Setup
Setup
Estimating the Period and Frequency
Launching LoggerPro
Calibrating the Force Probe
Zeroing the Force Probe
Data Acquisition
Part II: Oscillation Parameters
Oscillation Parameters
Period and Phase
Oscillation and Circular Frequency
Part III: Dynamics
Changing the Plot Axes
Force vs. Acceleration
Force vs. Displacement
Relating m, k, and ω
Wrap it Up!
Physics 1051
Laboratory #1
Part I: Setup
Apparatus and Setup
•
This is a photograph of the experimental
set-up you will be using in this module.
•
When the spring is stretched a little bit
away from its rest position and then
released, the mass will oscillate up and
down.
•
You will measure its position using the
ultrasonic motion detector, and the force
using the force probe.
Simple Harmonic Motion
Physics 1051
Laboratory #1
Simple Harmonic Motion
Setup
• Attach a force probe to a stand or support as shown above. Make sure
that the force probe is set to "5 N" or “10 N” rather than to "50 N“.
•The force probe should be plugged into CH1 of the LabPro.
• Hang a spring from the force probe.
• Attach the large aluminum mass provided to the bottom of the spring.
• Place
the motion sensor directly beneath the mass. Connect the motion
sensor to DIG1 of the LabPro.
Position the motion sensor carefully -- the narrow beam of ultrasound it emits
can easily miss the hanging mass altogether.
Remember, the motion sensor must always be between 45 cm and 100 cm
below the mass for it to measure its motion reliably.
Physics 1051
Laboratory #1
Simple Harmonic Motion
Estimating the Period & Frequency
Pull the mass down a few cm and release it. Release it carefully, so that sideways motion is
minimised.
Observe the motion carefully.
Use a stop watch to measure the time for 10 oscillations and record it in your
Activity Log. Record the uncertainty in this value.
Calculate the period from the time for 10 oscillations. What is the uncertainty in the
period? Record these in your Activity Log.
QUESTION 1: Using your period in Table 1, calculate the frequency and circular
frequency of your oscillator and record them in Table 3 of your Activity Log.
The support is not perfectly rigid and you will tend to introduce some
sideways motion even if you are very careful.
Physics 1051
Laboratory #1
Simple Harmonic Motion
Launching LoggerPro
• Click on the icon below to launch LoggerPro.
LoggerPro should display three graphs (distance against time,
acceleration against time, and force against time) each with the same
maximum time. The maximum time should be 5 sec. If you don't see
these graphs, or if the time axes do not have the same maximum time,
consult an instructor now.
Physics 1051
Laboratory #1
Simple Harmonic Motion
Calibrating
the Force Probe
•
You will have been provided with a mass of about 200 g to use for
calibration.
•
Click the
•
Click the force probe icon and select Calibrate.
icon above the graph.
Continued…
Physics 1051
Laboratory #1
Simple Harmonic Motion
Calibrating the Force Probe
•
With nothing attached to the force probe
click the Calibrate Now button.
•
The Reading 1 value is 0 N. Click Keep.
•
Now hang the 200 g mass from the force
probe and enter the force in the Reading 2
cell. Click Keep.
Remember that a 1 kg mass weighs
9.8 Newtons.
Physics 1051
Laboratory #1
Simple Harmonic Motion
Zeroing the Force Probe
•
Remove the calibration weight, attach the spring to the force probe, and
attach the large aluminum weight to the hook at the bottom of the spring.
•
Reduce the mass's motion as much as possible, and then select Zero from
the Experiment pull-down menu.
Physics 1051
Data Acquisition
Laboratory #1
Simple Harmonic Motion
• Set the mass to oscillating, being as careful as possible to keep sideways motion to a
minimum.
• Click on the Collect button. LoggerPro will collect force and motion data for the
prescribed time, and then stop automatically.
• To discard the data, and restart, just click on the Collect button again.
• Repeat the above steps until you have collected
a good data set.
• If, for a particular graph, a data trace runs outside
the graph boundaries, adjust the graph limits by
going into Analyze and then selecting
Autoscale Graph > Autoscale.
Physics 1051
Laboratory #1
Simple Harmonic Motion
Part II: Oscillation Parameters
Now we can use LoggerPro to analyze your data and find the parameters
which describe the oscillation.
Click on the
button.
With the help of the vertical line which
should then appear in the graph window,
determine the times, t1 and t2, of the first and
the second maximum of the displacement.
Also record the height of the first maximum
and the first minimum.
Enter these values, and their uncertainties,
into your Activity Log. Uncertainty in any
data point is found by selecting the point
and then looking at the point immediately
before or after it. The uncertainty is then the
difference between the two sucessive
points.
t1
t2
Physics 1051
Laboratory #1
Simple Harmonic Motion
Calculate the Period and Phase
Calculate the oscillation period from the two times in Table 2. Use the error
formulae to calculate the uncertainty. Record the value in your Activity Log.
If the value which you have just calculated for the period is not within 10%
of your value from timing the oscillations, consult an instructor now.
Now calculate the phase constant ϕ.
• For an oscillation described by x(t) = A cos(ωt + ϕ), ϕ is given by
ϕ = 2π(1 - t1/T)
where t1 is the time of the first maximum.
Physics 1051
Laboratory #1
Simple Harmonic Motion
Oscillation and Circular Frequency
Using the period from Table 3, calculate the oscillation frequency and
circular frequency, and enter the results in your Activity Log.
The amplitude, A, of the oscillations and the mean displacement can be
estimated from the maximum and minimum displacement in your data.
Estimate A and the mean displacement from the heights you recorded in
Table 2, and enter the result in your Activity Log.
QUESTION 2: Compare the estimated period from Table 1 and the
measured result from Table 3. If the two values do not agree
within the uncertainties explain why.
Physics 1051
Laboratory #1
Simple Harmonic Motion
Part III: Dynamics
Changing Plot Axes to F vs. a
• Still working with the same
graph, change the quantity
plotted on the horizontal axis
from time to acceleration. Do
this by clicking on the axis label
as shown below, and selecting
the quantity from the popup
menu that appears.
• The points should lie along a
nearly straight line or a narrow
ellipse. Why?
Physics 1051
Laboratory #1
Simple Harmonic Motion
Force and Acceleration
• Calculate and display the regression line for this
data set. To do so, pull down the Analyze menu
and select Linear Fit. Then double click on the
box that appears and in the Standard Deviations
section check both the Slope and Intercept.
Make sure your graph is labelled (axes and title) properly. Print the graph and attach it to
your Activity Log.
Enter the fitted values of the slope and intercept of the regression line, and
their associated uncertainties, into the appropriate locations in your Activity Log.
QUESTION 3: What physical quantity does the slope of the F vs. a graph
represent?
QUESTION 4: Using the above information give the mass, m (with appropriate units and
uncertainty estimate), for the mass used in your experiment. Compare this
value obtained by weighing the mass. If the values do not agree, explain
why.
Physics 1051
Laboratory #1
Simple Harmonic Motion
Force and Displacement
• Still working with the same graph, change the quantity plotted on the horizontal
axis from acceleration to position.
• The points should lie along a nearly straight line or a very narrow ellipse. Why?
• As you did with the force vs. acceleration graph calculate and display the
regression line for this data set.
Make sure your graph is labelled properly. Print the graph and attach it to
your Activity Log.
Enter the fitted values of the slope and intercept of the regression line,
and their associated uncertainties, into the appropriate locations in your
Activity Log.
QUESTION 5: Using the above information give the spring constant, k (with
appropriate units and uncertainty estimate), for the spring used in
your experiment.
QUESTION 6: Why is the slope of the F vs d plot negative? What does thexintercept represent physically?
Physics 1051
Laboratory #1
Simple Harmonic Motion
Relation Between m, k and the
Frequency
QUESTION 7: Using the values you have just obtained in questions 4 and 5 for
the mass and spring constant, calculate the circular frequency,
ω , (with appropriate units and uncertainty estimate), of the
oscillating mass in your experiment. Compare this result and the
value obtained from your previous calculations in Table 3.
Physics 1051
Laboratory #1
Simple Harmonic Motion
Summary Questions
QUESTION 8: Possible sources of error in this experiment include
A. Air resistance
B. Sideways motion of the mass
C. Size of time steps in the data collection
D. Movement of support stand.
Identify which of these are systematic errors and which are random
errors. Which do you think is the DOMINANT source of error in
this experiment. Why?
QUESTION 9: By doing this experiment, did you show that Equation (3) of the
introduction was true, i.e. that force and acceleration are linearly
related? Did you show that a sinusoidal solution for x(t) represents
periodic motion?
QUESTION 10: Why do we need to introduce a non-zero phase constant ϕ in the
equation x(t) = A cos(ωt + ϕ)?
Physics 1051
Laboratory #1
Simple Harmonic Motion
Wrap it up!
Check that you have completed all the Tables of your Activity Log.
Make sure that you have answered all the Questions completely.
Attached to your Activity Log should be the following graphs:
–
–
Force vs. Acceleration
Force vs. Displacement