DescriptionAPHY106 Lab 6

Harmonic Motion

HARMONIC MOTION

SUMMARY

The displacement and oscillation of a spring is used to calculate the spring constant and the mass

attached to it. An experiment using a simple pendulum is performed to compare measured and

theoretical results.

THEORY

Spring:

Hooke’s Law states that a displaced spring exerts a restoring force equal to

! = −

Where x is the displacement of the spring (the amount it is stretched or compressed) and k a physical

characteristic of the spring called the spring constant. The greater the value of k, the harder it is to

stretch or compress a spring. The negative sign in Hooke’s Law indicates that the restoring force

points opposite the force that displaces the spring.

If a mass (m) is hung at rest then the net force on the mass is zero and

∑ = ! − = 0 →

=

(Equation 1)

If the mass is then pulled down a further amount A and allowed to oscillate the displacement of the

spring (measured with respect to the equilibrium position) is

= ( )

(Equation 2)

where w is called the angular frequency and is equal to

”

$%

= .# = &

(Equation 3)

where T is the period of oscillation, which is the time it takes to complete one cycle.

Not all springs obey Hooke’s Law, but those that do are said to be ‘ideal’ springs.

Simple pendulum:

A simple pendulum is a mass (m) attached to a frictionless pivot via a

massless cable of length L. If the mass is displaced by a small angle and

released it will oscillate with angular frequency

‘

$%

= .( = &

(Equation 4)

Since the period can (relatively) easily be observed and measured,

pendulums are historically useful tools for timekeeping or measuring

heights.

REFERENCE

Harmonic Motion is covered in Cutnell & Johnson, Chapter 10.

1

APHY106 Lab 6

Harmonic Motion

EQUIPMENT DESCRIPTION

The equipment for this lab is somewhat simple, being just a spring suspended

from above with a mass attached to it as shown. The masses are placed on a

mass holder which is lightweight enough to displace the spring only a

negligible amount when it is empty. Displacements are measured using a

computerized sensor placed underneath the spring. For the pendulum portion

of the lab the period is calculated via a rotational sensor connected to Logger

Pro. The pendulums themselves are masses clamped onto the ends of

lightweight wooden sticks.

COLOR-KEY FOR ONLINE LABS

These online versions of APHY106 labs are based on the in-person versions

and retain most of the actual lab instructions even though you won’t be

doing the experiment.

Green text highlights commentary to help you understand the lab or how to

write your report.

Blue text highlights key steps of the lab process that you should read carefully.

Red text highlights steps important to your Excel data sheet.

ONLINE VS IN-PERSON

There is a considerable amount of data analysis required when doing this lab in person, including

sinusoidal curve-fitting and analysis of results to extract values of interest (such as the period). Much

of that has been done for you here. The post-lab analysis is mostly the same.

PROCEDURE

In this online version of this lab the uncertainties of the Logger Pro sensor measurements and of the

lab masses will not be considered. You will however include (and use) an uncertainty in time when

you calculate the value of the unknown mass in section 1.3.

1. MASS ON A SPRING

Set the motion sensor underneath the mass holder, pointed up. This sensor will measure the distance

between itself and the mass. The difference between the distances measured by the sensor for an

empty holder and a holder containing a mass equals the displacement of the spring. Click on the

‘Harmonic Motion’ icon to start Logger Pro and verify it is measuring the distances correctly.

1.1 DETERMINING THE SPRING CONSTANT

Before putting any mass on the holder zero the sensor. Put the first mass (m) on the mass holder.

The spring will likely oscillate slightly as a result of the disturbance. Wait until the oscillation stops (ie.

the spring is at equilibrium) and record the distance (x) from the sensor. Record the mass and distance

values in Excel. (All data for this lab is in a separate data file on Brightspace.) Repeat the procedure

for the other three masses (so you have four distinct displacement measurements). In Excel, use an

x/y scatter plot to plot the displacements (x) on the x-axis against the weights (mg) on the y-axis. Add

a trendline (setting the y-intercept to 0) to your plot to get the slope, which is the spring constant, k.

Be sure the equation is displayed on your plot. (You are graphically solving Equation 1 here.)

1.2 PERIODIC OSCILLATION

Attach the unknown mass to the holder and let it hang until it is at equilibrium. From equilibrium,

carefully pull the mass down several centimeters and release it so it oscillates up and down with no

sideways motion. Click on collect in Logger Pro and record a few seconds of displacement versus

time data. Your data should be sinusoidal and periodic (ie. repeats itself over time). If it isn’t, check

2

APHY106 Lab 6

Harmonic Motion

with your TA that you are collecting data correctly. Select a portion of your data that contains at least

one complete cycle and copy it into your Excel file.

1.3 ESTIMATING THE PERIOD

Plot the displacement versus time data using an x/y scatter plot and examine the plotted data to

estimate the period of oscillation. (See the bottom of this page for a visual illustration of the period of

a sinusoidal plot.) Since the data is saved in 0.05 second increments, your absolute uncertainty will

be 0.05 seconds. Calculate the percent uncertainty of your estimated period.

1.4 CALCULATING THE UNKNOWN MASS

Using the spring constant (you calculated in 1.1) and your estimated period (from 1.3), use Equation

3 (from the theory of this lab) to calculate the value of the unknown mass. Also calculate the absolute

uncertainty of your estimate using the same percent uncertainty as your period.

2. SIMPLE PENDULUM

The pendulum in this lab is a wooden stick attached at the top to a rotary motion sensor which acts as

a pivot. A mass is attached to the end of the stick. The mass of the stick is very small compared to

the mass attached at the end, so this functions as an adequate approximation of a simple pendulum.

Since the equations for the periods of a simple pendulum uses the small angle approximation sinq »

q, you should only swing the pendulum through angles of no greater than 10 degrees

2.1 MEASURE THE PENDULUM

Carefully measure the length of the pendulum (L) from the pivot point to the location of the attached

mass. Record the value in Excel.

2.2 CALCULATE THE PERIOD

With the pendulum length you just measured use Equation 4 to calculate the expected period of the

pendulum.

2.3 DETERMINING THE PERIOD OF THE SIMPLE PENDULUM

Restart Logger Pro using the ‘Pendulum’ shortcut and zero the rotational sensor. Displace the

pendulum by about 10 degrees and release it so it swings back and forth smoothly. Click on collect in

Logger Pro and record a few seconds of angle versus time data. Click on the plot and use the sine fit

option to calculate the period. Record the period in your Excel file. (You will compare the two values

in your report.)

Estimating the period

The image at right shows a typical

sinusoidal curve, which your plotted data

will resemble. A is the amplitude, which

is the maximum displacement from

equilibrium, and T is the period, which is

the time it takes the oscillation to

complete one cycle. In your plotted data,

you can estimate the period by

calculating the time difference between

subsequent peaks.

3

APHY106 Lab 6

Harmonic Motion

Before you upload your lab report, carefully read the grading comments on

Brightspace for your previous reports.

LAB REPORT QUESTIONS (Answer each in a separate paragraph in your results section.)

This lab report must be 100% your own work, and labs are automatically scanned for

plagiarism when submitted. If you collaborate with any other person, including sharing work

with them, you may fail the course and be reported to student discipline.

1. Based on the four mass displacements, what was the spring constant of your spring? Was this an

ideal spring? Explain why or why not. (Read the lab theory if you aren’t sure how to answer this.)

2. When the unknown mass oscillated on the spring, what was your estimate for the period of

oscillation? Including the uncertainty, what value did you calculate for the unknown mass?

3. For the simple pendulum experiment, what was the percent difference between your expected

and measured period of oscillation? List (with explanation) some possible reasons why the

measured period was less than the expected value.

To calculate the percent difference use the percent error formula with the expected value assumed

to be correct. To answer the second part of this question, think about how the pendulum used in the

lab may differ from the (theoretical) simple pendulum, as well as what sorts of uncertainties may

have affected the results that you did not consider.

The data for this lab is in a separate Excel file on Brightspace. Be sure to use the data

corresponding to your TA; if you use any other data your lab will not be graded.

Lab reports (including Excel data sheets) are submitted via Brightspace and have due dates

listed on Brightspace. Late labs may not be graded.

4

Purchase answer to see full

attachment