Physics 1051 Laboratory #2 Simple Harmonic Motion

Simple Harmonic Motion1

Physics 1051 Laboratory #2 Simple Harmonic Motion

PrelabWrite experiment title, your name and student number at top of the page.

Prelab 1: Write the objective of this experiment.

Prelab 2: Write the relevant theory of this experiment.

Prelab 3: List the apparatus and sketch the setup.

Have these ready to be checked by lab staff at the door on the day of your lab.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Introduction

The goal of this experiment is to familiarizeourselves with the properties of simpleharmonic motion and to collect data in whichwe can estimate the mass of an aluminiumcylinder and the spring constant of the spring.

To do so, we’ll collect data similarly to the lastexperiment, however we’ll also collect forceinformation of the tension in the spring.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

IntroductionRecall, any motion that repeats itself at regular intervals is calledperiodic or harmonic motion. Simple Harmonic Motion (SHM) satisfiesthe following properties:

§ Motion is periodic about an equilibrium position.§ The restoring force is proportional to and oppositely directed to a

displacement from the equilibrium position.

The displacement, velocity, and acceleration with respect to the equilibrium position x of a system undergoing SHM (as a function of time, t) can be described by

! " = % cos(*" + ,),/ " = −%* sin *" + , ,3 " = −%*4 cos *" + , .

Where % is the maximum displacement or amplitude of the motion, w isthe angular frequency of the motion, and ϕ is the phase constant orphase angle.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

IntroductionConsider a mass (m) hangning on a spring as our oscillating system.Hooke’s Law states that a spring restoring force acts the mass when it is displaced from its equilibrium position. This restoring force is described as

78 = −9Δ!⃗, where 9 is the spring constant. The negative sign in the equationindicates that force and displacement are opposite in direction. Δ!⃗ is thedisplacement of the object from equilibrium, meaning it is the amount ofstretch of the spring.

The force on an object can also be described by Newton’s second law.

Σ7⃗ = =3⃗.In our case, the measured force will be zeroed at equilibrium, which will negate the force of gravity on the system and allow us to measure the restoring force directly.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

IntroductionThe angular frequency * for a mass oscillating on a spring is defined in :

* = 2p> = 9/=� .The frequency is defined to be the number of oscillations that the system completes in one second, 9 the spring constant, and = the attached mass.The period A is the time taken for one complete oscillation, and can be expressed mathematically as

A = 1/>.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Apparatus and Setup

You have been provided with the following:

• Motion sensor• Force Probe• Aluminium cylinder• Metre stick• Spring• Support rods• Clamp

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Apparatus and Setup

The motion sensor is a device whichmeasures the distance to the closestobject.

It measures the distance to an object bysending out an ultrasonic tick, whichreflects off the object and returns to thesensor, allowing it to measure the time andextrapolate the distance to the object.

Connect the motion sensor to DIG/SONIC 1of the LabPro.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Force Probe

The force probe is a device which measures the force exerted on the hook at the bottom of the device.

To obtain accurate results, the force probe must first be calibrated and zeroed:

• Attach the force probe to a support rod.• Set the force probe is to "5 N" or �10 N� rather

than to "50 N�.• Plug the force probe into CH1 of the LabPro.

Make sure the LabPro is plugged into a power outlet and is connected to the computer.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Force Probe Calibration

Click the icon to launch Logger Pro.

Use a 200 g mass for calibration.

Click Experiment, Calibrate, then LabPro: 1 CH1: Dual Range Force.

Instructions continued on next slide…

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Physics 1051 Laboratory #2 Simple Harmonic Motion

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 9.81 newtons.

Click OK.

Force Probe Calibration

Remove the calibration weight, attach the spring to the force probe, and attach the large aluminium weight to the hook at the bottom of the spring.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Motion Sensor

Place the motion sensor on the floordirectly beneath the aluminium cylinder,as in the image to the right.

Position the motion sensor carefully -- thenarrow beam of ultrasound it emits caneasily miss the hanging mass altogether.

Remember, the motion sensor mustalways be between 15 cm and 100 cmbelow the mass for it to measure itsmotion reliably.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Zeroing the ProbesBefore collecting any data, the probesmust be zeroed. This tells the computerwhere the equilibrium point of theoscillator is.

Reduce the motion of the aluminiummass as much as possible, and thenselect Zero from the Experiment pull-down menu.

Select both Dual Range Force andMotion Detector and click OK.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Data CollectionStart your mass oscillating by lifting it a few centimetres and releasing it. Once the oscillation has stabilized, so that any wobble is minimized, hit the Collect button in Logger Pro. Logger Pro will display plots of the oscillating system for:

l Force vs time.l Position vs time.l Acceleration vs time.

Have an instructor check your graphs and initial your lab report.

Lab Report 1: Describe the phase relationship between the force vs time and the position vs time graphs. Do the crests of each plot align with one another in time? What is the approximate position when force is at a maximum?

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Using Logger ProYou may change the parameter plotted on the horizontal axis by clicking on the axis label as shown below, and selecting the new parameter from the popup menu that appears. For example, if your plot shows position vs time and you want to plot position vs acceleration, you click on the word �time�on the x axis and choose �acceleration�instead.

To obtain a linear fit to your data, click Analyze then Linear Fit. To find the uncertainties in the slope and intercept, double click on the box that appears and check Show Uncertainty.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Data Analysis

Logger Pro will display the coordinates of the plots if you click Analyze,

and Examine, or click the button.

The coordinates will be displayed in a pop-up box.

Use your data to determine the following:Lab Report 2:

a) As the mass oscillates, what are the relative values of position and acceleration? i.e. When position is max positive, what is the acceleration?

b) Explain how this mathematically makes sense based on the expressions of ! " and 3(").

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Analysis

Lab Report 3: Determine the period of oscillation and estimate the uncertainty from your data.

Lab Report 4: Measure and record the mass of the cylinder using a lab balance. Include uncertainty.

Lab Report 5: Calculate the spring constant and its uncertainty. Show all workings.

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Data Analysis

Working with your middle graph:

Lab Report 6: Using your data, make an appropriate plot to determine the mass of the cylinder. Describe your method. Include any equations and discuss how you decided what to plot. Record the mass with its uncertainty.

Lab Report 7: Compare the mass values found in LR4 and LR6 and comment on their agreement. Do they agree within uncertainty?

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Data Analysis

Working with your bottom graph:

Lab Report 8: Using your data, determine the spring constant. Describe your method. Include any equations and discuss how you decided what to plot. Record the spring constant with its uncertainty.

Lab Report 9: Compare your two values of spring constant in LR5 and LR8 and determine if they are in agreement. Agree within uncertainty?

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Data AnalysisRight click and choose Graph Optionsand include titles on each graphexplaining what each graph describes.

For the best printed graph:

Click File then Page Setup and choose the landscape orientation.

Click File then Print.

To select the only necessary page: Click Pages and choose Single.

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Include your printed graph with your reportP

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Physics 1051 Laboratory #2 Simple Harmonic Motion

Summary and Conclusions

Lab Report 10: Briefly summarize your experiment, in a paragraph or two.

Lab Report 11: List your experimental results and comment on how they agreed with the expected results.

Lab Report 12: List at least three sources of experimental uncertainty and classify them as random or systematic.

Include your prelab, any printed copies of your graph, and all data analysis with your report.

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