# Simple Harmonic Motion Experiment Environmental Sciences

In this experiment we will look at simple harmonic motion, which is the oscillation around an equilibrium point. We will investigate angular frequency, frequency, period, and damped oscillation. Damped oscillation meaning that a frictional force is present which will serve to slow the velocity of the glider eventually to zero. Taking measurements of our glider's oscillation with varying amounts of damping force will allow us to calculate all the variables asked in this lab.

Investigation 1

Data and Analysis

The mass of the glider in kg was .376 kg.

Data from Oscillation

There are 300 data points from this experiment so here are the first 25.

Position, Run #1

**Â**

**Â**

Time ( s )

Position ( m )

Equil. Position x0

1.93E-03

0.6636

0.661146512

0.1019

0.6632

**Â**

0.2019

0.6632

**Â**

0.3019

0.6631

**Â**

0.4019

0.6631

**Â**

0.5019

0.6627

**Â**

0.6019

0.6624

**Â**

0.7019

0.662

**Â**

0.8019

0.6615

**Â**

0.9019

0.661

**Â**

1.0019

0.6605

**Â**

1.1019

0.66

**Â**

1.2019

0.6594

**Â**

1.3019

0.6591

**Â**

1.4019

0.6588

**Â**

1.5019

0.6586

**Â**

1.6019

0.6584

**Â**

1.7019

0.6584

**Â**

1.8019

0.6584

**Â**

1.9019

0.6584

**Â**

2.0019

0.6586

**Â**

2.1019

0.6589

**Â**

2.2019

0.6593

**Â**

2.3019

0.6596

**Â**

2.4019

0.66

**Â**

2.5019

0.6605

**Â**

This run proved that the glider did not vary for more than +- 1 cm. Using all 300 points of data we averaged them all together to get the equilibrium point shown above.

The oscillation of the glider was now tested with the glider starting at the 40 cm mark. The first 25 data points follow.

Position, Run #2

**Â**

Time ( s )

Position ( m ) x

4.34E-04

0.1493

0.1004

0.1398

0.2005

0.1846

0.3008

0.2807

0.4012

0.4052

0.5016

0.5392

0.602

0.6777

0.7024

0.8127

0.8027

0.936

0.903

1.0416

1.0033

1.1235

1.1034

1.1779

1.2035

1.2007

1.3035

1.1921

1.4034

1.1526

1.5032

1.0858

1.6029

0.9947

1.7026

0.8849

1.8022

0.7632

1.9019

0.6367

2.0015

0.5126

2.1012

0.3984

2.2009

0.3003

2.3006

0.2221

2.4005

0.1815

2.5005

0.1631

From this run we determined the following.

The rough estimate for the amplitude was .06 cm.

The rough estimate for the period was 2.6 s.

The rough estimate for the phase was -2.90 rad using the formula theta=-w*t0

The centered position was obtained by the equation (x-x0). The first 25 data points are shown.

Center

**Â**

Time (s)

Position (m)

4.34E-04

-0.51184651

0.1004

-0.52134651

0.2005

-0.47654651

0.3008

-0.38044651

0.4012

-0.25594651

0.5016

-0.12194651

0.602

0.016553488

0.7024

0.151553488

0.8027

0.274853488

0.903

0.380453488

1.0033

0.462353488

1.1034

0.516753488

1.2035

0.539553488

1.3035

0.530953488

1.4034

0.491453488

1.5032

0.424653488

1.6029

0.333553488

1.7026

0.223753488

1.8022

0.102053488

1.9019

-0.02444651

2.0015

-0.14854651

2.1012

-0.26274651

2.2009

-0.36084651

2.3006

-0.43904651

2.4005

-0.47964651

2.5005

-0.49804651

The oscillating glider data is graphed in Graph A at the end of this report. It is the blue data points.

The amplitude of the first positive peak was .540 m which is .054 cm. Compared to the rough estimate of .06 cm the rough estimate was only 11.1% off.

The graph of the first 6 positive peaks is shown in Graph B. The slope was 2.58 which represents the average period because it is the summation of the first six points of distance versus the time that it takes for each full oscillation. The period T is thus 2.58 s. The frequency was found using the equation f=1/T, Therefore f=.388 Hz. The angular frequency was obtained with the equation w=(2*pi)/T, therefore w=2.44 rad/s.

Phase is found with the equation theta=-w*tp. Therefore the phase for our data was -2.922.

To find the spring constant we used the equation k=w^2*m. Where m is the mass of the glider and in kg. Therefore k=1.11 which is only .909% error.

**Investigation 2**

Data and Analysis

Ring magnets:

Yellow 10.9 g & 10.8 g

Red 10.9 g & 11 g

Blue 10.9 g & 10.8 g

The first 25 data points for the oscillation with 2 magnets follows.

Position, Run #1

**Â**

Center

**Â**

Time ( s )

Position ( m )

Time (s)

Position (m)

1.76E-03

0.6042

1.11E-03

-0.05694651

0.1019

0.6651

0.1011

0.003953488

0.2021

0.7252

0.2012

0.064053488

0.3023

0.7809

0.3013

0.119753488

0.4024

0.8285

0.4014

0.167353488

0.5025

0.8658

0.5015

0.204653488

0.6026

0.8906

0.6017

0.229453488

0.7026

0.902

0.7018

0.240853488

0.8026

0.8992

0.802

0.238053488

0.9026

0.8829

0.9022

0.221753488

1.0025

0.854

1.0023

0.192853488

1.1024

0.8141

1.1024

0.152953488

1.2022

0.7656

1.2025

0.104453488

1.3021

0.7114

1.3026

0.050253488

1.4019

0.6548

1.4026

-0.00634651

1.5017

0.5989

1.5026

-0.06224651

1.6016

0.5468

1.6025

-0.11434651

1.7015

0.5014

1.7024

-0.15974651

1.8014

0.4653

1.8023

-0.19584651

1.9013

0.4401

1.9022

-0.22104651

2.0012

0.4278

2.002

-0.23334651

2.1012

0.4281

2.1019

-0.23304651

2.2013

0.4415

2.2017

-0.21964651

2.3014

0.4668

2.3016

-0.19434651

2.4015

0.5022

2.4015

-0.15894651

2.5016

0.5459

2.5014

-0.11524651

The first 25 data points for the oscillation with 4 magnets follows.

Position, Run #2

**Â**

Center

**Â**

Time ( s )

Position ( m )

Time (s)

Position (m)

1.11E-03

0.3832

1.11E-03

-0.27794651

0.1011

0.3867

0.1011

-0.27444651

0.2012

0.4032

0.2012

-0.25794651

0.3013

0.4317

0.3013

-0.22944651

0.4014

0.4713

0.4014

-0.18984651

0.5015

0.5194

0.5015

-0.14174651

0.6017

0.5738

0.6017

-0.08734651

0.7018

0.6312

0.7018

-0.02994651

0.802

0.689

0.802

0.02785349

0.9022

0.7437

0.9022

0.08255349

1.0023

0.7927

1.0023

0.13155349

1.1024

0.8333

1.1024

0.17215349

1.2025

0.8636

1.2025

0.20245349

1.3026

0.8818

1.3026

0.22065349

1.4026

0.8877

1.4026

0.22655349

1.5026

0.8813

1.5026

0.22015349

1.6025

0.8631

1.6025

0.20195349

1.7024

0.834

1.7024

0.17285349

1.8023

0.796

1.8023

0.13485349

1.9022

0.7511

1.9022

0.08995349

2.002

0.7021

2.002

0.04095349

2.1019

0.6514

2.1019

-0.00974651

2.2017

0.6018

2.2017

-0.05934651

2.3016

0.5561

2.3016

-0.10504651

2.4015

0.5165

2.4015

-0.14464651

2.5014

0.4849

2.5014

-0.17624651

The first 25 data points for the oscillation with 6 magnets follows.

Position, Run #3

**Â**

Center

**Â**

Time ( s )

Position ( m )

Time (s)

Position (m)

1.15E-03

0.3944

1.11E-03

-0.266746512

0.1012

0.4161

0.1011

-0.245046512

0.2013

0.4494

0.2012

-0.211746512

0.3014

0.4912

0.3013

-0.169946512

0.4016

0.5401

0.4014

-0.121046512

0.5017

0.5932

0.5015

-0.067946512

0.6019

0.6483

0.6017

-0.012846512

0.702

0.7023

0.7018

0.041153488

0.8022

0.7525

0.802

0.091353488

0.9023

0.7964

0.9022

0.135253488

1.0024

0.832

1.0023

0.170853488

1.1025

0.8576

1.1024

0.196453488

1.2025

0.8724

1.2025

0.211253488

1.3025

0.876

1.3026

0.214853488

1.4025

0.8684

1.4026

0.207253488

1.5025

0.8504

1.5026

0.189253488

1.6024

0.8228

1.6025

0.161653488

1.7023

0.7876

1.7024

0.126453488

1.8022

0.7467

1.8023

0.085553488

1.902

0.7019

1.9022

0.040753488

2.0019

0.6562

2.002

-0.004946512

2.1018

0.6116

2.1019

-0.049546512

2.2017

0.5707

2.2017

-0.090446512

2.3016

0.5351

2.3016

-0.126046512

2.4015

0.5065

2.4015

-0.154646512

2.5014

0.4864

2.5014

-0.174746512

This centered data was added to the graph created in step 20 of investigation 1. Graph A at the end of this report.

The following data is the time and the position coordinates for the first six positive peaks.

Peak Number

Time (s)

1

1.2

2

3.8

3

6.4

4

9

5

11.6

6

14.2

**Â**

**Â**

2 Magnets

**Â**

Peak Number

Time (s)

1

0.7

2

3.3

3

5.9

4

8.5

5

11.1

6

13.7

**Â**

**Â**

4 Magnets

**Â**

Peak Number

Time (s)

1

1.3

2

4

3

6.7

4

9.3

5

12

6

14.7

**Â**

**Â**

6 Magnets

**Â**

Peak Number

Time (s)

1

1.2

2

3.9

3

6.6

4

9.4

5

12.1

6

14.7

The period for the oscillation with no magnets was T = 2.6s.

The period for the oscillation with 2 magnets was T = 2.6s.

The period for the oscillation with 4 magnets was T = 2.674s.

The period for the oscillation with 6 magnets was T = 2.711s.

For each data set we then plotted the six peak amplitudes vs the peak times. These can be seen in the following graphs.

No magnets Graph F. Two magnets Graph G. Three magnets Graph H. Four magnets Graph I.

The decay constant a was found in each of the four plots. For no magnets a was .045. For 2 magnets a was .061. For 4 magnets a was .123. Fro 6 magnets a was .203.

The damping constant b was found using the equation b=2m*a. For no magnets b was .034. For 2 magnets b was .049. For 4 magnets b was .153. Fro 6 magnets b was .342.

Graph J shows b vs. number of magnets. My data does agree that as b increases the number of magnets also increases. The test with no magnets appears to have had damping forces acting upon it that could not be ignored.

Next we found w' by using the following equation w'=sqrt((k/m)-(b^2)/(4*m^2)).

w' for no magnets was .408 rad/s. For 2 magnets w' was .398 rad/s. For 4 magnets w' was .362 rad/s. For 6 magnets w' was .4 rad/s. T' for no magnets was 2.45s which is 5.77% error from T. T' for 2 magnets was 2.52 s which is 3.07% error from T. T' for 4 magnets was 2.76 s which is 8.84% error from T. T' for 6 magnets 2.5 s which is 7.78% error.

**Conclusion**

In this experiment we learned about Simple Harmonic Motion of a Glider on an Air Track. We graphed the oscillations and then analyzed the data in excel. The results seen above and the graphs seen below are the result of this experiment. The data and graphs show that because of the damping effect the oscillations will decrease exponentially as the damping effect increases. We learned how to find period, frequency, angular frequency, and phase using the experiment data and the formulas in the book.

**Questions**

1. w = sqrt( k/2m)

w = 1/sqrt(2) * sqrt(k/m)

w*sqrt(2) = sqrt(k/m)

T = 2pi/w

T = 2pi/sqrt(2)*w

T*sqrt(2) = 2pi/w

T goes up by a factor of 1.414.

For our data T would be 3.68 s for no magnets.

2.

Graph A:

Graph B: No mags

Graph C: 2 mags

Graph D: 4 mags

Graph E: 6 mags

Graph F: No mags

Graph G: 2 mags

Graph H: 4 mags

Graph I: 6 mags

Graph J:

Article name: **Simple Harmonic Motion Experiment Environmental Sciences essay, research paper, dissertation**