1. Simple Harmonic Motion
AH Physics Q&W

2. Wave motion
At N5 and Higher we studied wave characteristics such as v,f,λ and T but made no attempt to look at the motion of an individual particle in the wave motion itself. An example might be a water molecule in a transverse water wave.
In this section we look at this important motion which is called simple harmonic motion. (SHM)
Simple Harmonic motion is common in many areas of mechanics and engineering

3. Examples of SHM

4. SHM – the simple pendulum
Simple harmonic motion can be modelled using a swinging pendulum
The horizontal displacement of the pendulum “bob” varies directly with the restoring force acting on it.
This restoring force is the component of the weight of the bob perpendicular to the tension in the spring.
A simple pendulum can be used to determine a value for “g” on Earth
An equation for the period of a simple pendulum can be derived. It is given by:

5. Investigation –the simple pendulum
We are going to use the simple pendulum as an opportunity to practice some practical skills and introduce a more rigorous approach to the treatment of uncertainties in experimental measurement.
As part of the AH Physics course you have to complete a scientific report on a single experimental procedure. This is it!
This will also serve as practice for your Project!
The aim of this experiment is…
“To determine a value for Earth’s gravitational field strength using a simple pendulum”

6. Uncertainties
Before we do the pendulum experiment let’s have a look at how we should treat uncertainties in measurements at Advanced Higher level.
"Any measurement that you make without the knowledge of its uncertainty is completely meaningless." Professor Walter Lewin, MIT
Walter Lewin Last Lecture - MIT

7. Uncertainties – Raw results
1. Calibration Uncertainty in measuring apparatus. This is usually specified by the manufacturer of the equipment eg:
Wooden meter stick 0.5mm
Digital stopclock 0.5% of the reading plus 1 digit
If no calibration uncertainty is available it can be taken to be 0.5% and (hopefully) ignored.
2. Scale Reading Uncertainty in measuring equipment. This depends on the type of scale used. This is your uncertainty in reading the scale
Digital scale : ±1 of least significant digit.
Analogue scale: ±0.5 of the smallest division
3. Random uncertainty in the mean: same as in Higher Physics for repeated results (min 5)

8. Combining uncertainties
Multiple uncertainties may have to be combined. The simplest way to do this is to work out all uncertainties as percentages.
If any uncertainty is less than 1/3 of the others it can be ignored
If a quantity is squared in an equation, it’s percentage uncertainty is doubled.
(similar treatment is given if a quantity is raised to any power)
Uncertainties should be combined as follows:
The uncertainty in a final calculated value (Δw) is given by:
This doesn’t need to be done if you are using an Excel graphical method
Where x y and z are the percentage uncertainties in the relevant measured quantities

9. Combining uncertainties example:
A student drops a ball from a height of 2m and measures how long it takes to reach the floor. He repeats this 5 times. He measures the height with a tape measure that only has centimetre divisions on it.
He measures the time on a centi-second timer.
He uses his results to determine a value for acceleration due to gravity (g) using the equation of motion:
Results: Height 2.00
Time to fall 2m (s):
Determine the students value of the acceleration due to gravity (g) with the absolute uncertainty in this value.
Express your answer in the form: value ± uncertainty

10. Analysis of raw results:
Height : Calibration Uncertainty in tape (unknown) taken to be (0.5%) Scale reading Uncertainty is 0.5cm Scale Reading Uncertainty = 2.00 ± m (0.25%) Time : Calibration uncertainty in stopclock taken to be (0.5%) Scale reading uncertainty s in all raw time measurements. Scale Reading uncertainty in first time = ± (1.6%)
Most of the time you will not need to do this because the random uncertainty in the mean will be the dominant uncertainty and all the rest can be ignored…………………………………………………

11. Analysis of raw results continued:
Mean and random uncertainty in the mean.
Time to fall 2m (s):
Mean time t = 0.67± 0.05 s (7%)
Since this is more than 3 times bigger than all the other uncertainties, all the others can be ignored.
Therefore the uncertainty in t2 will be 14%

12. Calculation of ‘g’ Final answer: g = 8.9 ± 1.2 ms-2 (14%)
Ball is dropped from rest, therefore u=0 and a=g
The uncertainty in the time t is 7% therefore the uncertainty in t2 will be 14%
Therefore the uncertainty in g will be 14%
Final answer: g = 8.9 ± 1.2 ms-2 (14%)
Our calculated value of “g” is lower than expected but the actual value of 9.8 ms-2 lies within our uncertainty range (7.7 – 10.1)
In an evaluation of the procedure it should be identified that our value for the mean time is too big. This is probably due to air resistance or human reaction time i.e. a systematic uncertainty.
Results could be improved by eliminating reaction time from the procedure i.e. electronic timing / light gates……or by using a graphical method or using the simple pendulum procedure!

13. Simple Pendulum Procedure
Measure the length of the pendulum.
Suspend it from a clamp stand
Release it from a small angle (less than 10˚)
Measure the time for 10 swings of the pendulum
Repeat 5 times at each length
Repeat for different lengths over a range 20cm - 120cm
Plot a graph of period squared (T2) against length (l)
Gradient of the graph will be

14. Uncertainties in raw results
Record all your raw results in a table and consider the uncertainties in those raw results:
Length: calibration and scale reading uncertainty
Time: calibration and scale reading uncertainty
Random uncertainty in the mean period
Length l (m)
Mean period T2 (s2)
0.2 ± (0.4%)
0.79 ± (18%)
0.4 ± (0.2%)
1.61 ± (10%)
0.6 ± (0.1%)
2.40 ± (6%)
0.8 ± (0.1%)
3.24 ± (4%)
A sample of a summarised set of (my) results is shown.

15. Using a graphical method
Use Excel – it will make your life so much easier!
Enter your appropriate raw results on a spreadsheet then insert an x y scatter graph.
Get excel to draw the best fit line and display the equation of the line on the graph
This will show you the gradient and the y-intercept.
The y-intercept will show you if there is a systematic uncertainty in your procedure.
The uncertainty in the gradient can be determined using the LINEST function (see separate handout)

16. This can be tidied up a bit to look like this....

17. Graph of results
Gradient = 4.07

18. Using LINEST to find the uncertainty in the gradient
LINEST is a function built into Microsoft Excel that performs a statistical analysis of your data and automatically calculates the gradient of the best fit line and the uncertainty in the gradient.
To do this you need to firstly select 4 empty cells (2x2) in the Excel sheet and then type the following into the function bar:
Where the cell values should be your range of “y” vales and “x” values
=LINEST(B2:B5,A2:A5,TRUE,TRUE)
gradient
y- intercept
Then highlight the top left of the 4 empty cells and press “control + shift + enter” simultaneously.
The 4 cells will now have data in them….
4.07
-0.02
0.032
0.018
Uncertainty in gradient
Unc. In y-intercept

19. Uncertainty in the gradient
The uncertainty in the gradient is determined by using the LINEST function in Excel.
gradient
4.07
-0.02
0.032
0.018
Uncertainty in gradient
Therefore uncertainty in the gradient is (1%)
Final answer by graphical method: g = 9.7 ± 0.1 ms-2

20. Systematic uncertainty
Our graph doesn’t go through the origin.
From the LINEST function we can see there is a y-intercept of
This implies that our period measurement are all out by the same amount (due to human reaction times? – very unlikely)
It is more likely that we have a systematic uncertainty in our length measurement: We are not measuring the the centre of the bob from the point of suspension correctly. We have measured too long!
A graph that doesn’t go through the origin points to a systematic uncertainty.
(Note a systematic uncertainty doesn’t affect the gradient, this is why a graphical method is better.)

21. Report
Write up a report on the experiment using the “instructions for candidates” to guide you through the format of the report.
Basically:
Title
Aim
Apparatus
Procedure (written in impersonal past tense)
Results
Uncertainties
Conclusion
Evaluation
This has to be completed satisfactorily for the course award!