Safety Considerations
To ensure the experiment is carried out safely, I will have to make sure:
- The string is secured to the cork and the cork to the clamp,
- The clamp stand is stable (using weights) and therefore cannot topple over causing damaging to equipment and others around,
- The pendulum is oscillation at a reasonable pace so as not to cause an imbalance,
- And that I am always aware of my surroundings
Reasons for Procedures
From the preliminary experiment, the following percentage errors were calculated for the measurement of time:
It is evident to see that as the number of oscillations timed increases, the percentage error for that time measurement decreases. Therefore to ensure the results of the experiment are as reliable as possible, it would be wise to record the time for twenty oscillations.
I am going to use a maximum length of 1.100m. As I have calculated the approximate length of the grandfather clock to be 1.000m, recording the time period for 1.100m will enable me to acquire the length using a graphical method. I am having a minimum length of 0.300m. For smaller lengths, the pendulum would oscillate more rapidly there would therefore be a smaller time period. This would subsequently be harder to observe and record meaning more human errors would be expected.
Beginning at 0.300m, I will be taking readings of length increasing it 0.100m at a time. During this, I will obtain nine values for the length which can be plotted onto the graph and allow a line of best fit to be drawn.
I will be taking the reading for the length twice and taking three measurements of time. This will help to reduce the errors that may occur in the experiment
Justification for Design
I am going to measure the length and the time period, as this will enable me to plot a graph that will enable me to calculate a value for the gravitational field strength.
It is very difficult to measure the true length, L, of the pendulum to its centre of mass. I am therefore going to measure the length of the string l but will have to add the end correction (x) to the centre of mass to find the genuine value for L.
If I plot a graph of T2 against L, the graph will be a straight line but will not pass through the origin, as the intercept is not equal to zero.
The y intercept (c) is equal to: and will have units = s2
I predict the graph will look like the one found below:
T2 (s2)
l (m)
From the graph, I will be able to calculate the gradient:
And will have units = s2m-1
And then from this, I can work out the value for the gravitational field strength and from the y intercept on the axis I can calculate a value for the end correction (x). This will help me calculate the true length L of my grandfather clock.
The value of g that I will obtain will be the value of freefall (units ms-2) but as g can also be thought of as the gravitational field strength the actual units will be Nkg-1. I therefore hope to obtain a value for the gravitational field strength equal to 9.81Nkg-1, as this is the value quoted both in the textbook and on the datasheet.
Aspects of Plan Based on Procedures
As I know the time period for the grandfather clock is 2 seconds, I can make a calculation for an approximate length of the grandfather clock’s pendulum as seen next.
T2 (s2) I intend to use my graph
as illustrated to find a value
4 for the length of the
Grandfather clock.
L l (m)
To find the length of the grandfather clock pendulum, I will use the time axis and at the point where T2 is equal to 4s2, will draw a line across to the line of best fit and then downward to the length axis.
As I have predicted that the length of the grandfather clock is approximately 1.000m, in order to be able to read a value for the length from my graph, I will have to record the time for lengths greater than 1.0m. However it would not be wise to measure lengths far exceeding 1.0m as the time period would be too long. I will not be measuring extremely small values for the length, as the time period will be very small also making it hard to record.
Use of Preliminary Work or Secondary Sources
From my preliminary experiment, I hope to determine the minimum ad maximum length I should measure, the number of complete oscillations to be counted and an overall grasp of how I will do the main experiment.
As the number of oscillations increases, the percentage error decreases and I will therefore use twenty oscillations for the main experiment. However as I am plotting T2, the time percentage error will be doubled.
Identification of Significant Sources of Error
The reaction time error can be discounted due to the idea that if the time is started late initially and stopped late at the end, the time interval would be the same as when the timer was started correctly on both occasions. The most significant error is that due to the length measurement.
Action Proposed to Minimise Errors
To ensure the experiment is done accurately and to minimise errors I will follow certain procedures. These are:
- Make certain the pendulum undergoes a complete oscillation,
- Make sure that I continually time the same number of oscillations,
- Take all measurements at eye level,
- If the pendulum oscillation is conical, stop timing and start again,
- Use a countdown method to assure the timing is accurate done,
- Repeat the reading for length twice to ensure there is no
change in length
- Repeat the reading for time measured three times and find an
average
- Time multiple oscillations therefore reducing the percentage
error
Using a split cork makes it easy to pull the string through and change its length and it made sure that as the string oscillated, the length has the same size at each end of swing as illustrated below.
Check on Inconsistent Readings
The line of best fit crossed every point bar one. This point however is fairly accurate as it lies close to the line of best fit. The repeated reading is circled on the graph
Calculations
Support or Contradiction of Prediction
In my plan I stated that for a pendulum to undergo simple harmonic motion,
But the length here is equal to the length of the inextensible string in addition to the length of the end correction.
I predicted that the graph would be a straight line with a small intercept =
The shape of my graph confirms this theory.
- Value of g
The official value given from the gravitational field strength from the Accessible Physics Textbook is 9.81Nkg-1. I obtained a value of 9.70Nkg-1. In comparison with the official value, there is a 1.12% difference.
My estimation for g is close to the official value, hence, my results support my prediction.
- Length
Having been informed that the time period of the grandfather clock is two seconds, I used a calculation, which enabled me to predict that the length would be roughly 1m (0.994m). I acquired a value from the graph of 0.960m and this agrees with my original prediction, as there is only a 0.91% error.
Possible sources of error
Pendulum not undergoing a complete oscillation
- Pendulum undergoing a conical oscillation
- Countdown method done inaccurately
Time measured:
Length measured:
Micrometer screw gauge:
Reaction time:
Most Error Sensitive Measurements
My most error sensitive measurements were due to my metre rule and tape measure. These errors were very small therefore the equipment was reliable.
Categorising Errors – systematic and random
Systematic errors (have an effect on every reading) for example:
- Having a faulty stopwatch with a zero order
- Reaction time
- Counting deficit or excess of timed oscillations throughout
- Having an unstable clampstand
However I think I eliminated these systematic errors through my precautions
Random errors (do not have an effect on every reading) for example:
- Having the angle greater than 10˚ on one/two occasions
- The string slipping and therefore changing the length during a
reading
- Counting 19 instead of twenty oscillations once
Estimates of Error on All Measurements
Comment on Data Discrepancy or Anomalous Results
From my graph it is apparent that I have one slight anomalous point. This point is probably due to one of the random errors listed earlier perhaps the pendulum having an angle of oscillation greater than 10˚.
Variation of Repeats or Uncertainty of Data
In the variation of length there is no variation. However, when measuring time an increased amount of variation is apparent. The worst examples are tabulated below,
The worst percentage error in the variation of repeats is:
Comment on Suitability of Techniques Used
My techniques were suitable as they were able to verify my prediction and I found little difficulty throughout.
Reliability of Conclusions
I believe my conclusion is reliable as:
- My points are close to the lie of best fit,
- My value of g is close to the official value,
- I timed twenty oscillations, therefore reducing the errors
incurred,
- I repeated my readings for length twice and time three times,
- The value for the length obtained is similar to my predicted
value,
- I used a countdown method,
- And I began timing when the pendulum was at the centre of its
oscillation and moving fastest.
Reliability of Experiment
- Compare calc value of L in prelim to L found how reliable?
- % error from alt gradient high or low?
- Compare value of g to official value % error from diff. if small % the experiment is reliable
- Compare calculated x with measured x % diff how reliable
- Comment on size of errors on instruments – very small. % error smaller for T than L reaction time ( fairly large but ca be discounted)
Proposals for Improvements
- More precise equipment
- More than one person
- Length measurement better
