Physics 1 – Hooke's Law and Simple Harmonic Motion
Hooke's Law and Simple Harmonic Motion
Aims
 To study the oscillations of a mass suspended at the end of a spring.
 To study Hooke's law in the extension of a spiral spring.
 To analyse experimental data by graphical methods.
In this experimental tutorial you will first undertake a tutorial question to predict the
motion of and force on a mass oscillating on the end of the spring, and then perform
an experiment to measure these quantities.
Part 1: Tutorial Question (15 mins)
A mass is attached to the lower end of a spiral spring and oscillated vertically with
simple harmonic motion such that its displacement relative to is mean position, x , is
given by
 2
x  A sin 
 T

t

[1]
where A is the maximum displacement and T is the period.
1.1 Sketch the motion, clearly marking A and T .
The force, F , due to the extension of the spring from its mean position is given by
Hooke's Law:
F  kx
[2]
where k is the spring constant.
1.2 Hence, add a graph of force to your sketch.
The time period of the oscillations is given by
T  2
m
k
[3]
1.3 If the period is 0.5 s and the mass is 2 kg, what is the value of the spring constant?
1
Physics 1 – Hooke's Law and Simple Harmonic Motion
Part 2: Taking data (10 mins)
In this experiment the position of a mass suspended
on the end of a fixed spring is monitored used data
logging software on a PC. An ultrasound sensor1
monitors the position. A force sensor monitors the
force.
Force sensor
Spring
To start data taking:
Mass
 Click on EasySense software icon.
 Select EasyLog from New Experiment menu.
Ultrasound
 Check that the software has detected the force
sensor
and motion sensors – look for corresponding
Figure 1: Experimental set-up
boxes in top left of screen.
 Start the mass bouncing. Keep the motion as
vertical as possible – avoid swinging.
 Wait until the motion has become regular, and then click on "Start".
 Record data for approximately 20 seconds and then click on "Stop".
Part 3: Calculating period of oscillation and spring constant (5 mins)
 From the Display option on top toolbar select Show Table – this reveals the
columns of data of time, force and distance.
 Clicking on a row of the data brings up a marker on the graph. Using the up and
down arrow keys, move the marker until it is on a maximum of one of the sine
waves. Note down the corresponding time for this maximum. Now move along
10 maxima, and note the corresponding time.
3.1 Calculate the average period.
3.2 Hence calculate the spring constant.
Part 4: Calculating the spring constant from Hooke's Law (20 mins)
In this part you will analyse your data using Excel.
 In EasyLog, click on File – Transfer to Excel. This opens Excel with your data
clearly labelled in three columns.
4.1 Use Excel to plot (i) the force due to extension of the spring and (ii) distance
about mean position against time on a single graph.
Hints:
(a) To calculate the average of a column of data in Excel, use the
AVERAGE function.
1
This sends out a signal that is reflected from the base of the suspended mass. The reflected signal is
then detected. The time between emission and detection allows the distance between mass and detector
to be calculated.
2
Physics 1 – Hooke's Law and Simple Harmonic Motion
(b) When referencing a single, specific cell in an Excel equation, use $
signs, e.g. $E$2.
(c) Plot the distance in centimetres so that the scale is similar to that of the
force on your graph.
4.2 Compare this plot with the graph you sketched in Part 1.
4.3 From your data for force and distance, use Hooke's Law to calculate values for the
spring constant.
Hint:
(d) Consider your columns of data – discard any rogue points.
4.4 From your values, calculate a single, average value for k and compare this with
your value from Part 3.
Remember to save your work to disk and to add your spreadsheet to your lab book.
Part 5: Calculating velocity (Optional)
 Shut down the Easy Sense software, and then reopen it, this time selecting New
Experiment – Graph.
 When prompted de-select the force sensor as we are only interested in distance this
time.
 Set the data acquisition to run for 2 s, taking points every 20 ms. This will give
you 100 data points.
5.1 Plot this scaled velocity and distance on the same graph. What is the phase
difference between these two plots?
5.2 How can you explain this in terms of the motion and equation [1]?
Hints:
(a) As in Part 2 set the mass bouncing vertically again and let the motion
settle. Then start the data taking. This time you do not need to stop it
yourself.
(b) Transfer the data into Excel as before.
(c) Correct the distance data as before.
x  x2
(d) Calculate velocity using 1
where x1 , x2 are the corrected
t1  t 2
distances for consecutive data points and t1,t 2 their times.
(e) Scale the velocity so that its maximum value is similar to that of the
distance.
Further work
The following questions are related to the topic covered by this experimental tutorial.
 Exercise book questions D41 – D50.
3
Physics 1 – Hooke's Law and Simple Harmonic Motion
 Mastering Physics question
m.padgett@physics.gla.ac.uk
details
4
will
be
sent
by
Miles
Padgett
Physics 1 – Hooke's Law and Simple Harmonic Motion
Demonstrators' Answers, Hints, Marking Scheme and Equipment List.
Marking Scheme
Section
1.1
1.2
1.3
3.1
3.2
4.1
4.3
4.4
Discretionary mark
TOTAL
Mark
1
1
1
1
1
2
1
1
1
10
Answers
Part 1:
1.1 See the graph for Part 4
1.2 See the graph for Part 4
1.3 316 Nm-1
Part 3
3.1 Period ~0.585 s
3.2 Spring constant ~115 Nm-1
Part 4
4.1 See accompanying graph and data sheet
4.3 See Part 3. Can plot force against distance – should give a straight line, but very
fuzzy. Can be useful to spot outlying points which can significantly skew the
average.
Part 5
5.1 See graph. Can also do acceleration, but curve will not be smooth.
5
Physics 1 – Hooke's Law and Simple Harmonic Motion
Typical data and graph for Part 4
Time
s
Force
N
Distance
m
Distance
cm
0
0.03
0.06
0.09
0.15
0.18
0.21
0.24
0.27
0.3
0.33
0.36
0.39
0.45
0.48
0.51
0.54
0.57
0.6
0.63
0.66
0.69
0.75
0.78
0.81
0.84
33.8
33.7
34.2
34.7
35.9
36.6
36.9
37.2
37.2
37.1
36.9
36.5
35.8
34.6
34.1
33.7
33.6
33.5
33.7
33.9
34.5
35.1
36.2
36.8
36.9
37.3
0.233
0.232
0.23
0.224
0.213
0.21
0.205
0.203
0.201
0.201
0.204
0.207
0.213
0.224
0.227
0.232
0.234
0.235
0.234
0.231
0.228
0.222
0.211
0.208
0.203
0.201
23.3
23.2
23
22.4
21.3
21
20.5
20.3
20.1
20.1
20.4
20.7
21.3
22.4
22.7
23.2
23.4
23.5
23.4
23.1
22.8
22.2
21.1
20.8
20.3
20.1
Ave F
35.38154
Ave D
21.77538
Corrected F
Corrected
D
k(N/cm)
k(N/m)
-1.5815392
-1.6815376
-1.1815376
-0.6815376
0.51846313
1.21846008
1.51846313
1.81846237
1.81846237
1.71846008
1.51846313
1.11846161
0.41846085
-0.7815399
-1.2815399
-1.6815376
-1.7815399
-1.8815384
-1.6815376
-1.4815369
-0.8815384
-0.2815399
0.81846237
1.41846085
1.51846313
1.91846085
1.52461492
1.42461472
1.2246158
0.62461608
-0.4753847
-0.7753853
-1.2753848
-1.4753852
-1.6753841
-1.6753841
-1.375385
-1.0753844
-0.4753847
0.62461608
0.9246152
1.42461472
1.62461512
1.72461533
1.62461512
1.32461601
1.0246154
0.42461567
-0.6753851
-0.9753842
-1.4753852
-1.6753841
1.037337
1.180346
0.964823
1.09113
1.090618
1.571425
1.190592
1.232534
1.0854
1.025711
1.104028
1.040058
0.880257
1.251232
1.386025
1.180346
1.096592
1.09099
1.035038
1.118465
0.86036
0.663046
1.211846
1.454259
1.029198
1.145087
103.7337
118.0346
96.48231
109.113
109.0618
157.1425
119.0592
123.2534
108.54
102.5711
110.4028
104.0058
88.02574
125.1232
138.6025
118.0346
109.6592
109.099
103.5038
111.8465
86.03603
66.30465
121.1846
145.4259
102.9198
114.5087
6
Average k
(N/m)
120.9554591
Calculated
k
115.3581
Physics 1 – Hooke's Law and Simple Harmonic Motion
0.87
0.9
0.93
0.96
0.99
1.02
1.05
1.08
1.11
1.14
1.17
1.2
1.23
1.26
1.29
1.32
1.35
1.38
1.41
1.44
1.47
1.5
1.53
1.56
1.59
1.62
1.65
1.68
1.71
1.74
1.77
1.8
37.2
36.9
36.5
36.1
35.5
34.9
34.3
33.9
33.6
33.5
33.5
33.9
34.2
34.8
35.3
36.1
36.5
37
37.2
37.3
37.1
36.9
36.3
35.8
35.2
34.6
34
33.8
33.5
33.6
33.6
34.1
0.201
0.202
0.206
0.209
0.215
0.219
0.226
0.229
0.233
0.234
0.234
0.233
0.228
0.225
0.218
0.215
0.208
0.205
0.201
0.2
0.202
0.203
0.208
0.211
0.219
0.222
0.229
0.231
0.234
0.235
0.233
0.232
20.1
20.2
20.6
20.9
21.5
21.9
22.6
22.9
23.3
23.4
23.4
23.3
22.8
22.5
21.8
21.5
20.8
20.5
20.1
20
20.2
20.3
20.8
21.1
21.9
22.2
22.9
23.1
23.4
23.5
23.3
23.2
1.81846237
1.51846313
1.11846161
0.71846008
0.11846161
-0.4815369
-1.0815392
-1.4815369
-1.7815399
-1.8815384
-1.8815384
-1.4815369
-1.1815376
-0.5815392
-0.0815392
0.71846008
1.11846161
1.61846161
1.81846237
1.91846085
1.71846008
1.51846313
0.91846085
0.41846085
-0.1815376
-0.7815399
-1.3815384
-1.5815392
-1.8815384
-1.7815399
-1.7815399
-1.2815399
-1.6753841
-1.5753839
-1.1753846
-0.875384
-0.2753843
0.12461506
0.82461499
1.1246156
1.52461492
1.62461512
1.62461512
1.52461492
1.0246154
0.72461479
0.02461486
-0.2753843
-0.9753842
-1.2753848
-1.6753841
-1.7753843
-1.5753839
-1.4753852
-0.9753842
-0.6753851
0.12461506
0.42461567
1.1246156
1.32461601
1.62461512
1.72461533
1.52461492
1.42461472
7
1.0854
0.963869
0.951571
0.820737
0.430168
3.864195
1.311569
1.317372
1.168518
1.158144
1.158144
0.971745
1.153152
0.802549
3.312599
2.608937
1.146688
1.268999
1.0854
1.080589
1.09082
1.029198
0.94164
0.619589
1.456787
1.840582
1.228454
1.19396
1.158144
1.033007
1.168518
0.899569
108.54
96.38686
95.15708
82.07371
43.01684
386.4195
131.1569
131.7372
116.8518
115.8144
115.8144
97.1745
115.3152
80.25494
331.2599
260.8937
114.6688
126.8999
108.54
108.0589
109.082
102.9198
94.16401
61.95885
145.6787
184.0582
122.8454
119.396
115.8144
103.3007
116.8518
89.95695
Physics 1 – Hooke's Law and Simple Harmonic Motion
1.83
1.86
1.89
1.92
1.95
1.98
2.01
34.5
35.2
35.7
36.3
36.7
37.1
37.2
0.226
0.223
0.216
0.212
0.206
0.204
0.201
22.6
22.3
21.6
21.2
20.6
20.4
20.1
-0.8815384
-0.1815376
0.31846237
0.91846085
1.31846237
1.71846008
1.81846237
0.82461499
0.52461587
-0.1753841
-0.5753849
-1.1753846
-1.375385
-1.6753841
8
1.06903
0.346039
1.8158
1.596255
1.121728
1.249439
1.0854
106.903
34.60391
181.58
159.6255
112.1728
124.9439
108.54
Physics 1 – Hooke's Law and Simple Harmonic Motion
Typical data and graph for Part 5
Corrected
D
Velocity
Velocity/10
0.168
0.172
0.177
0.183
0.188
0.194
0.199
0.204
0.208
0.211
0.213
0.213
-0.01981
-0.01581
-0.01081
-0.00481
0.00018999
0.00619001
0.01119
0.01619
0.02019
0.02319
0.02519
0.02519
0.2
0.25
0.3
0.25
0.300001
0.25
0.25
0.2
0.15
0.1
0
0
0.26
0.213
0.02519
-0.1
0.28
0.3
0.211
0.208
0.02319
0.02019
-0.15
-0.2
0.32
0.204
0.01619
-0.2
0.34
0.2
0.01219
-0.25
0.36
0.38
0.4
0.195
0.189
0.183
0.00718999
0.00119
-0.00481
-0.3
-0.3
-0.25
0.42
0.178
-0.00981
-0.25
0.44
0.46
0.173
0.168
-0.01481
-0.01981
-0.25
-0.15
0.48
0.165
-0.02281
-0.15
0.5
0.52
0.54
0.56
0.58
0.6
0.62
0.64
0.66
0.68
0.7
0.72
0.74
0.76
0.78
0.8
0.82
0.162
0.161
0.161
0.161
0.163
0.167
0.171
0.176
0.181
0.187
0.192
0.198
0.203
0.207
0.21
0.212
0.213
-0.02581
-0.02681
-0.02681
-0.02681
-0.02481
-0.02081
-0.01681
-0.01181
-0.00681
-0.00081
0.00419
0.01019
0.01518999
0.01919
0.02218999
0.02419
0.02519
-0.05
0
0
0.1
0.199999
0.200001
0.25
0.249999
0.300001
0.25
0.299999
0.25
0.200001
0.15
0.1
0.05
-0.05
0.020000041
0.024999977
0.029999987
0.024999972
0.030000067
0.024999972
0.024999981
0.02000003
0.014999959
0.010000022
0
0
0.010000015
0.014999948
-0.02000006
0.019999955
0.025000037
0.030000015
-0.02999997
-0.025
0.025000037
0.024999963
-0.01499997
0.015000022
0.005000015
0
0
0.01000003
0.019999925
0.02000006
0.025
0.024999925
0.030000089
0.025
0.029999925
0.025
0.02000006
0.01499997
0.01
0.005000015
-
Time
s
Distance
m
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
Ave
distance
0.18781
9
Physics 1 – Hooke's Law and Simple Harmonic Motion
0.84
0.212
0.02419
-0.05
0.86
0.88
0.211
0.209
0.02319
0.02119001
-0.1
-0.2
0.9
0.205
0.01719
-0.25
0.92
0.94
0.2
0.195
0.01219
0.00718999
-0.25
-0.25
0.96
0.98
0.19
0.184
0.00219
-0.00381
-0.3
-0.25
1
1.02
0.179
0.174
-0.00881
-0.01381
-0.25
-0.25
1.04
0.169
-0.01881
-0.15
1.06
1.08
1.1
1.12
1.14
1.16
1.18
1.2
1.22
1.24
1.26
1.28
1.3
1.32
1.34
1.36
1.38
1.4
0.166
0.163
0.161
0.161
0.162
0.164
0.167
0.171
0.175
0.181
0.186
0.192
0.197
0.202
0.206
0.21
0.212
0.213
-0.02181
-0.02481
-0.02681
-0.02681
-0.02581
-0.02381
-0.02081
-0.01681
-0.01281
-0.00681
-0.00181
0.00419
0.00919
0.01419001
0.01819
0.02218999
0.02419
0.02519
-0.15
-0.1
0
0.05
0.1
0.15
0.199999
0.2
0.3
0.250001
0.3
0.25
0.249999
0.2
0.2
0.1
0.05
0
1.42
1.44
0.213
0.212
0.02519
0.02419
-0.05
-0.15
1.46
1.48
0.209
0.206
0.02119001
0.01819
-0.15
-0.25
1.5
1.52
0.201
0.196
0.01319001
0.00819
-0.25
-0.25
1.54
0.191
0.00319
-0.3
1.56
0.185
-0.00281
-0.25
1.58
1.6
1.62
1.64
1.66
1.68
0.18
0.175
0.17
0.166
0.163
0.161
-0.00781
-0.01281
-0.01781
-0.02181
-0.02481
-0.02681
-0.25
-0.25
-0.2
-0.15
-0.1
0
10
0.005000015
-0.005
0.009999955
-0.02000006
0.024999925
0.025000075
-0.025
0.029999925
-0.025
0.025000075
-0.025
0.015000045
0.014999881
-0.01000003
0
0.005000015
0.01000003
0.01499997
0.01999994
0.019999985
0.030000015
0.025000075
0.030000015
0.025
0.024999925
0.019999985
0.019999985
0.01000003
0.005000015
0
0.004999985
-0.01499997
0.015000045
-0.025
0.025000075
-0.025
0.030000015
0.024999851
0.025000075
-0.025
-0.02000006
-0.01499997
-0.01000003
0
Physics 1 – Hooke's Law and Simple Harmonic Motion
1.7
1.72
1.74
1.76
1.78
1.8
1.82
1.84
1.86
1.88
1.9
1.92
1.94
1.96
1.98
2
0.161
0.161
0.163
0.166
0.169
0.174
0.179
0.185
0.19
0.196
0.201
0.205
0.208
0.211
0.212
0.213
-0.02681
-0.02681
-0.02481
-0.02181
-0.01881
-0.01381
-0.00881
-0.00281
0.00219
0.00819
0.01319001
0.01719
0.02019
0.02319
0.02419
0.02519
11
0
0.1
0.15
0.15
0.25
0.249999
0.3
0.25
0.3
0.250001
0.2
0.15
0.15
0.05
0.05
0.012595
0
0.01000003
0.01499997
0.015000045
0.025
0.024999925
0.030000015
0.025
0.030000015
0.025000075
0.019999985
0.014999955
0.01499997
0.005000015
0.005000015
0.0012595
Physics 1 – Hooke's Law and Simple Harmonic Motion
Equipment List
1 meter stand
Boss head
Force sensor with cable
Ultrasound sensor with cable
Spring
Mass hanger with weights up to 1kg
Computer with EasySense software
12