Laboratory Report
EF0EFS Foundation Engineering Fundamentals, Pendulum Laboratory
Name Inzamaam Khaliq
Convener Peter Marshall
Module Title Foundation Engineering Fundamentals
Module Code EF0EFS
Hand in date 26/11/18
Laboratory performed on 19/11/18
Contents
Introduction & Aim Page 3
Background Theory Pages 3 – 4
Apparatus List Page 5
Risk Assessment Pages 5 – 6
Method Pages 6 – 7
Results For Part 1 Pages 8 – 9
Results For Part 2 Pages 9 – 10
Graph For Result 2 Page 11
Graph For Result 2 (Proportional) Pages 12 – 13
Accuracy & Comparison Page 13
Conclusion Page 14
References Page 15
Introduction & Aim
In this laboratory report I will be investigating the effects of changing the angle of initial release and the length on the period of oscillation of a simple pendulum. This lab report will be broken down into various sections and sub sections. Some of the main sections being:
Hypothesis
Background Theory
Results from both experiments and uncertainties
Graphs and deducing the relationship
Accuracy
Conclusion
This is all the main criteria that will be covered in this laboratory report.
Background Theory
Pendulums have been around for years and have been for many applications, one of them being an old fashioned clock. The word simple pendulum is defined as an object that has a small mass which is then suspended from a string as show on the right hand side. The person who famously first discussed and performed experiments on the simple pendulum was called an Italian scientist named Galileo Galilei. Galileo first began to study the pendulum around the time of 1602 but however he had his interest sparked in1582 by the swinging the motion of a chandelier in a cathedral. Eventually Galileo managed to find out that the period is independent of the mass of the bob and proportional to the square root of the length of the pendulum.
Expanding on the simple pendulum you will now be able to view the conditions which make it perform a simple harmonic motion and from this we can derive an expression for its period. Now we define the displacement to be the arc length. From looking at the diagram we can now see that the force on the bob is a tangent to the arc and Is therefore equal to -mg sin . Tension in the string exactly cancels the component mg cos parallel to the string. This leaves a net force towards the equilibrium position at = 0.
Now we can show the restoring force is directionally proportional to the displacement. Then we have a harmonic oscillator. Therefore, making the equation:
F = -mg
The displacement is the proportional to . This is only when theta is radians, the arc length in a circle is related to its radius by the equation of:
s=L
so, rearrange the formula to get
= s/L
For small angles the expression for the restoring force is
F = -mg/L * s
After inspection this you will now be able to determine that this equation can be simplified in the form of
F = -kx
Using this equation we can now find the period of a pendulum for amplitudes that are about 15. Therefore the final equation being [1]:
T = 2 * √L/g
Apparatus list
The equipment below is the equipment that you will need to acquire to carry out this experiment successfully and effectively:
Stopwatch
Clamp Stand Set
1 G Clamp
150 grams mass (Bob)
1 Metre rule
2 Wooden Block
1.5 Meters of String
1 Protractor
After acquiring all your equipment make sure to setup as shown on the right.
Risk Assessment
In this experiment there are multiple risks that can cause harm to you or any bystanding people. Some of these risks being:
The apparatus could fall onto your body parts if not correctly set up or tightened.
The Ball could get loose overtime and could become a projectile.
Entrapment of fingers when tightening the clamp.
These are some of the risks that could potentially happen however these risks won’t happen if you follow some certain safety procedure such as:
Make sure that everything is visually observed before using and of not you should notify whoever is in charge and replace the faulty piece of equipment.
Keep checking to monitor the equipment throughout the experiment and make sure you are wearing safety glasses at all times.
Finally make sure you are performing your experiment out of harm’s way and that you are wearing closed shoes so if in an event of an accident your feet are protected.
These are all the risks and safety procedures that should be followed when preforming this experiment.
Method
For both tests one and two you will need to setup the apparatus as shown under the section of apparatus list. Remember to bare in mind that you are following all the safety procedures and that the experiment is carried out in a safe environment away from any people that may be in harm’s way.
Part 1
Create a table which has the rows columns:
Length (m) [0.03-0.12]
Time (s) [1-3]
Average Time for 20 Oscillations (s)
Mean Time for One Oscillation (s)
Set the length of the pendulum to a suitable 1.3m. To measure the length of the pendulum you will have to measure from the free point of the wire down to centre of the mass which is called the Bob.
Next move the pendulum side to side (laterally) from its equilibrium position until the centre of the Bob is 3 cm away from its equilibrium position.
Now record the time for 20 oscillations
Repeat the step above three times so you can produce an average time for 20 oscillations.
Calculate the mean time for one oscillation this is done by diving the mean time by 20.
Repeat Steps 3 to 6 for the 3 times using different lengths which are:
6 cm
9 cm
12 cm
Finally calculate the angle between the string and the horizontal for each starting position.
Part 2
Create a table which has the rows columns:
Length (m) [1.3 – 1]
Time (s) [1-3]
Average Time for 20 Oscillations (s)
Mean Time for One Oscillation (s)
Move the pendulum laterally until the string makes a 5 degrees angle from the vertical string and release the Bob.
Now record the time for 20 oscillations
Repeat the step above three times so you can produce an average time for 20 oscillations.
Calculate the mean time for one oscillation this is done by diving the mean time by 20.
Decrease the length of the pendulum by 5cm and repeat steps 2 – 5 until your last reading is 1 metre.
Part 3
Plot a graph of period against length.
Analyse the shape of the graph and the trend, from this piece of information deduce the nature of the relationship between the period of oscillations and the length of the pendulum.
Plot a new graph showing the new relationship
Write down the equation of the line in terms of length, Period and the constant of proportionality.
Add error bars to the graph and determine the line of best fit and worst fit
Using the gradient now relate the constant of proportionality of your line of best fit to constant and to G (the acceleration of free fall) and finally the complete equation the linking the period to the length of a pendulum.
Results For Part 1
Length (m) Time (s) Average Time for 20 oscillations (s) Mean time for one oscillation mean (s)
0.03 43.09 43.44 43.30 43.28 2.16
0.06 45.55 45.39 43.30 44.72 2.24
0.09 45.60 45.45 45.44 45.50 2.27
0.12 45.58 45.60 45.55 45.58 2.28
Working Out the angle
I will now demonstrate how to find the angle between the string and the horizontal for each starting position.
Length (m) Working Out Angle (°)
0.03 cos−1 (0.03/1.3) 88.678
0.06 cos−1 (0.06/1.3) 87.355
0.09 cos−1 (0.09/1.3) 86.030
0.12 cos−1 (0.12/1.3) 84.704
Absolute Uncertainty (Part 1)
Absolute uncertainty for 20 oscillations is 0.5 s. Therefore for 1 oscillation the absolutely uncertainty is 0.5/20 = 0.025 s. Now you need to minus and add the absolute uncertainty value to the mean time for a single oscillation:
Uncertainty for Rule is 0.0005
Average Oscillation For 20 Oscillations
0.03 m – 0.5 s = 43.78 and 42.78
0.06 m – 0.5 s = 45.22 and 44.22
0.09 m – 0.5 s = 46.00 and 45.00
0.12 m – 0.5 s = 46.08 and 45.08
Mean Time For One Oscillation
0.03 m – 0.025 s = 2.185 and 2.135
0.06 m – 0.025 s = 2.215 and 2.265
0.09 m – 0.025 s = 2.295 and 2.245
0.12 m – 0.025 s = 2.305 and 2.255
Results For Part 2
Length (m) Time (s) Average Time for 20 oscillations (s) Mean time for one oscillation mean (s)
1.30 45.51 45.53 45.32 45.39 2.27
1.25 44.66 44.50 44.70 44.62 2.23
1.20 43.96 44.10 43.80 43.95 2.20
1.15 43.32 43.50 43.10 43.31 2.17
1.10 42.19 42.30 42.05 42.18 2.11
1.05 41.61 41.51 41.78 41.68 2.08
1.00 40.26 40.39 40.45 40.36 2.02
The angle used in this set of results was 5°
Now we have to square the mean time for one oscillation mean.
Length (m)
Mean Time for one oscillation mean squared (s)
1.30 5.15
1.25 4.97
1.20 4.84
1.15 4.71
1.10 4.45
1.05 4.33
1.00 4.08
Absolute Uncertainty (Part 2)
Absolute uncertainty for 20 oscillations is 0.5 s. Therefore for 1 oscillation the absolutely uncertainty is 0.5/20 = 0.025 s. Now you need to minus and add the absolute uncertainty value to the mean time for a single oscillation:
Uncertainty for Protractor is 0.5°
Uncertainty for Rule is 0.0005
Average Oscillation For 20 Oscillations
1.30 m – 0.5 s = 45.89 and 44.89
1.25 m – 0.5 s = 45.12 and 44.62
1.20 m – 0.5 s = 44.45 and 43.45
1.15 m – 0.5 s = 43.81 and 42.81
1.10 m – 0.5 s = 42.68 and 41.68
1.05 m – 0.5 s = 42.18 and 41.18
1.00 m – 0.5 s = 40.86 and 39.86
Mean Time For One Oscillation
1.30 m – 0.025 s = 2.295 and 2.245
1.25 m – 0.025 s = 2.255 and 2.205
1.20 m – 0.025 s = 2.225 and 2.175
1.15 m – 0.025 s = 2.195 and 2.145
1.10 m – 0.025 s = 2.135 and 2.085
1.05 m – 0.025 s = 2.105 and 2.055
1.00 m – 0.025 s = 2.045 and 1.995
Mean Time For One Oscillation Squared
1.30 m – 0.05 s = 5.10 and 5.12
1.25 m – 0.05 s = 5.02 and 4.92
1.20 m – 0.05 s = 4.89 and 4.79
1.15 m – 0.05 s = 4.76 and 4.66
1.10 m – 0.05 s = 4.50 and 4.40
1.05 m – 0.05 s = 4.38 and 4.28
1.00 m – 0.05 s = 4.13 and 4.03
Graph For Result 2
As you can see the length isn’t proportional to the mean for one oscillation as the graph above is displayed as a curve. To achieve a proportional relationship, you will have to determine a way to ensure the line of best fit is linear rather exponential. Furthermore, whilst taking a look at the graph the data I have calculated is quite accurate as it is very close to the line of best fit and there are no visible outliers.
Graph for Result 2 (Proportional)
Absolute uncertainty = Line of best fit – Line of worst max
-0.554522 = 3.461538 – 4.01606
Gradient = line of best fit Absolute uncertainty
To achieve a proportional relationship for this graph I squared the mean for one oscillation so that linear line of best fit can be drawn. Furthermore, whilst taking a look at the graph the data I have calculated is quite accurate as it is very close to the line of best fit and there are no visible outliers. Additionally, all error bars are in range of the best and I have drawn a minimum and maximum line of worst fit to calculate the percentage uncertainty of the gradient.
Accuracy and Comparison of Results Against Theory
Checking the accuracy of my results by substituting the numbers into the equation
T = 2 * √L/g
Trying the length 1.3m
T = 2* √(1.3/9.81)
= 2.287 and the results acquired without the use of the formula was 2.27
Trying the length 1.0m
T = 2* √(1.0/9.81)
= 2.01 and the results acquired without the use of the formula was 2.02
As you can see above my results for this laboratory are quite accurate as there is only a small margin of error and the value, I managed to acquire without the use of the formula is nearly identical. However, there is no way you would be able to obtain the actual outcome because there is always some form of human error which could have been the error of measuring the string, stopping the clock or just a simple rounding error. In conclusion I am pleased with the accuracy of my set of results as there is only a small area of error but to minimise this error I will use better equipment and to five results rather than three to minimise any errors.
Conclusion
Overall, I believe this experiment was success as I manged to meet my aim which was to investigate the effects of changing the angle of initial release and the length on the period of oscillation of a simple pendulum. Now that I have performed this experiment, I can confidently say that as you decrease the length the time taken decreases therefore implying the speed of the pendulum is increasing. Another reason why I believe this experiment was performed successfully because the theoretical result that I plugged into the equation T = 2 * √L/g (which I found using some background theory) were very similar to the results I found from the practical. One thing I would change next time is to make sure that I use a machine that will record the oscillations rather than just manually counting the oscillations and pressing the stop button on the stop clock. The reason why is because I could have made a counting error and I will be minimising my reaction time If it is done via a machine which therefore decreases the uncertainty and provides more accurate results. But all in all I believe this laboratory was a success.
References
[1]"The Simple Pendulum – College Physics", Opentextbc.ca, 2018. [Online]. Available: https://opentextbc.ca/physicstestbook2/chapter/the-simple-pendulum/. [Accessed: 25- Nov- 2018].