The Simple Pendulum
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The Simple Pendulum
Introduction
The aim of this experiment was to experimentally find the time period of a simple pendulum, and use to compute the local acceleration due to gravity. These values were compared with the theoretical predictions based on the equation, =2πLg , where T is the time period, L is the effective length of the simple pendulum (measured quantity) and g is the local acceleration due to gravity. In order to compute the total error between the predicted and actual values, the effects of all random errors in each measured quantity were taken into account and analyzed using the method of error propagation.
Data
Average diameter of the bob = 1.9 cm => Average radius (ρ) = 0.95 cm
Average length of string (λ) = 80.27 cm
Average effective length of pendulum (L) = 81.22 cm
Time period of N swings as calculated using a stop clock (N = 10)
Trial # Time for 10 swings (s) Time Period (s)
- 18.03 1.803
- 18.01 1.801
- 18.01 1.801
- 18.06 1.806
- 18.03 1.803
- 18.05 1.805
- 18.1 1.810
- 18.08 1.808
- 18.06 1.806
- 18.02 1.802
Average time period (T) = 1.805 seconds
Data Analysis
The experimental value of ‘g’ is computed using the equation g=4π2LT2
Sample Calculation – For the first set, T = 1.803 seconds. L is the average effective length = 81.22 cm = 0.8122 m.
Therefore g=4π2(0.8122)(1.803)2 = 9.853 m/s2
Similarly, g for the all sets were computed and are as tabulated below:
Trial # g (m/s2) d = |ge - gth|| (m/s2)
Average ‘g’ = 9.837 m/s2
The actual value of ‘g’ is taken to be 9.792 m/s2 = gth
Average d = 0.048
Average error in length measurement: σρ = 0.1; σλ = 0.5 cm
Therefore σL = 0.6 cm
Percentage error in L = 0.63 %
Percentage error in T = 0.16%
Using the method of error propagation:
g=4π2LT2
% σg = % σL + 2(% σT)
Therefore % σg = 0.63 + (2 x 0.16) = 0.95%
Implies σg = 0.0095
Average d = 0.048
σd = σge + σgth = 0.0095 + 0.005 = 0.0145 m/s2
Discussion
Average d = 0.048 m/s2, where d is the difference between the theoretical and actual values of ‘g’.
σd = 0.0145 m/s2
The permitted error in the experimental value is 1% of the theoretical value, which is 1% of 9.792 = 0.0972 m/s2. The obtained error is 0.048 m/s2 which is less than the permitted value. Therefore the experimental results are within the permitted range.
Propagation of error
The error in a quantity is the combination of the errors that it is a function of.
In general,
1) If C is a sum of two quantities, then
The absolute error in C is the sum of absolute errors of the independent quantities that C is a function of. This is also true for differences.
2) If C is a product of 2 quantities, then
The relative error in C is the sum of the relative errors of the independent quantities that C is a function of. This is also true if there is a division involved.
3) If C is dependent on a quantity that is raised to the power n, then the relative error in C is equal to n times the relative error in that quantity.
Using these rules for error propagation,
1) The absolute error in L is the sum of absolute errors in the measurement of the radius and length of the string.
2) The relative and percentage errors in ‘g’ is the sum of relative error in L and (2 x relative error in T)
Average absolute error in the measurement of ‘g’ is given by average d = 0.045 m/s2.
Therefore the relative error in the measurement = 0.045/9.792 = 0.00459 m/s2.
Comparing this with σd = σge + σgth = 0.0095 + 0.005 = 0.0145 m/s2, it is evident that the d < σd and hence the results lie within the permitted experimental error range.
Another way to check the validity of the experiments perhaps is by analyzing backwards. Given the actual value of ‘g’, the time period is computed as , T=2πLg = 1.808 seconds.
This is compared with the experimental values of T as shown in the table below:
Measured T (seconds) Absolute Error (seconds)
The average absolute error = 0.0039 seconds.
Therefore the average relative error = 0.0039/1.808 = 0.0021 seconds or 0.2 %.
Since the error is much less than 0.5%, the validity of the experimental results is again verified.
Conclusion
The theoretical value of ‘g’ is taken to be 9.792 m/s2. The experimental value of ‘g’ has been computed using the equation g=4π2LT2, where L and T are the length and time period of the simple pendulum, and are measured quantities. The error in the measured quantities is propagated to ‘g’ and is computed. The overall error in the experimental results is 0.45% which is well within the permitted range. In other words, d < σd. The following are the possible errors in measurement:
1) The length of the string is measured using a meter scale, which is accurate only up to 0.1 cm, while the bob diameter is measured using a Vernier Caliper, which is accurate up to 0.01 cm. Though this does not cause significant problems, while estimating of ‘g’ up to two or three decimal places, the addition of the two lengths measured may result in error in the last two decimal places. This is a measurement error.
2) Parallax error while taking measurements – this must be avoided since it it is the fault of the person doing the experiment.
3) Error while taking time measurements: These can be minimized by taking more number of trials.
4) Random error due to external disturbances like speed of fan nearby (It is best to switch off all fans while doing the experiment).
5) Though the assumptions that the bob is a point mass and that the angle of deflection of the bob is less, are valid, they may contribute to the overall error. It is prudent to make sure that the angle of deflection is indeed small, which was ensured in this case.
