Theory
A compound pendulum of mass M, length between point of suspension and centre of mass h, and moment of inertia I, the period T is given by the equation [6]:
(Equation 1)
Therefore by measuring M, h, I and T we can calculate the value for g. The moment of inertia can be found by measuring the torsional oscillations of the pendulum, suspended by its centre of mass, compared to the torsional oscillations of a bar of known moment of inertia; with the axis’s through the centre of mass of the pendulum and bar parallel to each other. This is shown by the equation [7]:
(Equation 2)
It is important to note the error for each of the terms of the equation, as from the equation you can see that when determining the value for g to a 1 part in 1000 degree of accuracy, T needs to be measured to at least 1 part in 2000, whereas an accuracy of 1 part in 1000 will be adequate enough for M, I and h.
Experimental Method
The pendulum was set up as shown:
The pendulum was suspended from its knife edge, making sure that the edge was exactly horizontal, and the oscillations of the pendulum were taken.
For equation 1 to work, we needed to use an amplitude of less than ±20mm to ensure that the error in the approximation of equation 1 was not greater than 1 part in 1000.
The measurements for T were taken using a handheld stopwatch using a fixed reference point below the pendulum to time its oscillations as it passed this.
To measure T accurately to 1 part in 2000, first a series of measurements was taken of 5 periods of the pendulum. From these results an estimate of the error in T was found and to reduce this error we needed to time a larger amount of oscillations.
Before timing a very large amount of oscillations, we estimated an intermediary number of oscillations to measure, so as to check the value for T over a large amount of oscillations. This intermediary amount was estimated to be 36 oscillations. The estimate for T was then used when timing a large amount of oscillations, as it is hard to count for a long time, we could check the values with our better estimate. To reduce the error to 1 part in 2000 we approximated that we needed to time the oscillations over at least 200s, and to make our estimate safer we needed to measure a number of oscillations over 300-400s. We therefore estimated that we needed to time 180 oscillations, using our better estimate for T. After checking our best estimate from the 180 oscillations with our previous estimate of just 36 oscillations we could confirm that the amount of oscillations were counted correctly, and therefore improved our error in the measuring of T to 1 part in 2000.
The constants of M and h were measured. The mass of the pendulum was found by weighing it on a balance calibrated with 1 kg standard weights, therefore giving us an error in its mass of ± 0.0005kg. To find the centre of mass of the pendulum it was balanced on a knife edge, hence the length h could be measured to an accuracy of ±0.5mm.
To calculate the moment of inertia of the pendulum, we first had to measure the dimensions and mass of a bar that we could calculate its moment of inertia using the formula IBar = (MBar/12) (a2 + b2) [8] where a and b were the length and width of the bar and MBar was its mass. We then could calculate the moment of inertia of the pendulum by measuring both the pendulum’s and bar’s torsional oscillations with their axis’s of centre of mass parallel. The torsional oscillations of both the pendulum and the bar were measured by suspending them from a torsion wire and timing their oscillations about their centre of mass – where the pendulum’s centre of mass had already been measured by suspending it on a knife edge to an accuracy of 1 part in 1000, and the bar’s centre of mass also was measured to this accuracy.
From these measurements, the moment of inertia of the pendulum about its centre of mass was found using the equation IP = IBar (TP/TBar)2 [9] where TP and TBar were the torsional oscillations of the pendulum and bar respectively. From this we could then finally calculate the moment of inertia of he pendulum using equation 2.
Results
The constants
The compound pendulum was measured to have a mass of 2.092 ± 0.0005 kg, and a length of 0.6375 ± 0.0005m.
The steel bar that was used to find the moment of inertia of the pendulum was found to have a mass of 2.122 ± 0.0005 kg and the length and width of the bar were 0.74975 ± 0.00025m and 0.001908 ± 0.000005m respectively.
The time for one torsional oscillation of the steel bar suspended from the torsion wire was found to be 7.65 ± 0.1s, timing 5 oscillations 3 times. Likewise, the time taken for one torsional oscillation of the pendulum suspended by the same wire was measured to be 6.73 ± 0.1s, measuring 5 oscillations 3 times.
Therefore the moment of inertia of the pendulum about its centre of mass was calculated to be 0.08191 ± 0.00324 kg m2, and about the knife edge 0.932 ± 0.0035 kg m2.
The period of the pendulum
The period of the pendulum (measured 6 times over 5 periods) when suspended from its knife edge was 1.66 ± 0.0068 s. We then used this estimate to calculate that we needed to time the periods over 300 - 400s to have a sufficient accuracy in our value for the period. We measured an intermediary number of oscillations 3 times, which was worked out to be 36, to get a better estimate of 1.67 ± 0.00094s. Using this better estimate we worked out that we needed to time 180 oscillations of the pendulum, by dividing 300s by our better estimate. The time taken for the 180 swings of the pendulum was 300.66s of which we checked by dividing this by our better estimate to give exactly 180 swings. As the numbers were exact, we are confident that the number of oscillations measured was actually 180 and therefore the time for one period of T was 1.67 ± 0.00019s.
Using the equation:
Then combining it with equation 2, of:
We were able to rearrange the equations into a form that we could then calculate our best estimate of acceleration due to gravity, g:
From these equations g was found to be 9.89 ± 0.038 ms-2. The error found in g was calculated by combining the errors in the moment of inertia I, the mass M, the length h, and the time period T of the pendulum.
Discussion
The value that our experiment found for g (9.89 ± 0.0038 ms-2) was slightly higher than previously measured examples such as the value of 9.81 ms-2 quoted in Tipler and Mosca [10].
There are many reasons for this discrepancy. The acceleration due to gravity varies across the world from 9.779 ms-2 in Mexico City to 9.819 ms-2 in Oslo, Norway [11]. This is due to variances in the earths crust and therefore the actual distance to the centre of the Earth. Unfortunately, our measured value still exceeds the extremes of the values across the globe, and Bristol is not known to have a different value for g – the value in London is only 9.812ms-2 [11].
As we cannot contribute the discrepancy to the differences in the Earths crust, we have to examine our experimental method and causes for error that may have been overlooked. Firstly, we are positive that the measurements of the Pendulum’s mass and length could not have any overlooked error in them. Also in determining the period, T, of the pendulum, we were very thorough in our approach to minimise any errors so cannot contribute the discrepancy to an error here.
However, investigating the possible errors in the moment of inertia of the pendulum, there are a few points that could have cause to give a slight discrepancy in its readings, and therefore in the final value for g. Firstly our method for timing the oscillations was not as thorough as our method used to calculate T, hence allowing larger errors to be produced than what was recorded. As we timed 5 torsional oscillations of each of the pendulum and the standard steel bar, the oscillations became slower and did not move quite as far as the first period timed. To eliminate this in further experiments, we could only measure 3 oscillations or reduce the amount that the bar and pendulum are displaced by – but not both as this would increase the human error in stopping starting the stop watch.
If we assume that the moment of inertia of the pendulum was actually correctly measured, there are also some other factors that may have affected the value for g. There is the possibility that the oscillations of the pendulum were too large for the formulae used to derive the formulae in g; the approximations used in the derivation could only be made for small amplitudes. The derivation assumes that sin(x) ≈ x, and the amplitude in our experiment was 20mm. Fortunately, the amplitude used in the experiment shouldn’t cause much of an error from the approximation, as the amplitude was kept to a very small amount.
Another factor that could have resulted in a greater error in the experiment was that the oscillations of the pendulum were not entirely horizontal, and therefore some of its energy was lost due to movements away from the horizontal axis. This would cause the derivation in equation 1 to be incorrect, and hence provide a greater error in g than first was expected.
In evaluating these causes for possible error, we have come to the conclusion that the discrepancy in our value for g was due to the systematic error in calculating the moment of inertia of the pendulum. As well as this, the pendulum not rotating on a horizontal axis may have affected the final value slightly.
Conclusion
Our best estimate in measuring the acceleration due to gravity in the H. H. Wills Physics laboratory, Bristol University, UK is 9.89 ± 0.038 ms-2.
We believe that the difference in this final result and the accepted value for acceleration to gravity of 9.81 ms-2, were due to systematic errors in the experimental procedure, therefore further investigations must be carried out before we can take this value to replace the previously recorded measurements for the acceleration due to gravity.
References
[1] F H Newman and V H L Searle, “Gravitation”, Chapter 3 of The General Properties of Matter (Edward Arnold Publishers Ltd, 1957)
[2] A H Cook, “Gravity and the crust of the Earth”, Chapter 3 of Gravity and the Earth, (Wykeham Publications LTD, 1969)
[3] http://www.earthsci.unimelb.edu.au/ES304/MODULES/GRAV/NOTES/fallb.html
[4] http://www.earthsci.unimelb.edu.au/ES304/MODULES/GRAV/NOTES/spring.html
[5] http://en.wikipedia.org/wiki/Galileo_Galilei
[6] Tipler and Mosca, “Oscillations” Chapter 14 of Physics for Scientists and Engineers, (W H Freeman & Co, 6th Edition 2008) p 475
[7], [8], [9] Tipler and Mosca, “Rotation” Chapter 9 of Physics for Scientists and Engineers, (W H Freeman & Co, 6th Edition 2008) p 295, 297 & 298
[10] Tipler and Mosca, Appendix B of Physics for Scientists and Engineers, (W H Freeman & Co, 6th Edition 2008) p AP-3
[11] http://en.wikipedia.org/wiki/Gravity_(Earth)#Comparative_gravities_of_various_cities_around_the_world
