IB Lab – Measuring ‘g’ with a Pendulum – Model Answer
Results
Data Processing
To find the uncertainty in period squared, there are (at least) two equally acceptable methods:
First Method
Double the relative uncertainty for period, using the equation:
e.g. for l = 0.100 ± 0.005 m, T = 0.69 ± 0.04 s:
∆(T2) = 0.48 x 2 x 0.04/0.69 = 0.055652 ≈ 0.06 (to 1 sig. fig.)
Second Method
Find the maximum and minimum values of period squared, take the difference and divide by two.
e.g. for l = 0.100 ± 0.005 m, T = 0.69 ± 0.04 s:
T2 = 0.692 = 0.4761
Tmax = (0.69 + 0.04)2 = 0.5329
Tmin = (0.69 – 0.04)2 = 0.4225
So ∆(T2) = (0.5329-0.4225)/2 = 0.0552 ≈ ...
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First Method
Double the relative uncertainty for period, using the equation:
e.g. for l = 0.100 ± 0.005 m, T = 0.69 ± 0.04 s:
∆(T2) = 0.48 x 2 x 0.04/0.69 = 0.055652 ≈ 0.06 (to 1 sig. fig.)
Second Method
Find the maximum and minimum values of period squared, take the difference and divide by two.
e.g. for l = 0.100 ± 0.005 m, T = 0.69 ± 0.04 s:
T2 = 0.692 = 0.4761
Tmax = (0.69 + 0.04)2 = 0.5329
Tmin = (0.69 – 0.04)2 = 0.4225
So ∆(T2) = (0.5329-0.4225)/2 = 0.0552 ≈ 0.06 (to 1 sig. fig.)
To find g, plot a graph of T2 vs. l.
Since, , so gradient, , so
mmax = 4.28229 s2m-1
m = 4.00792 s2m-1
mmin = 3.74975 s2m-1
ms-2
ms-2 ms-2
∆(g) = (gmax – gmin)/2 = (10.52828 – 9.21900)/2 = 0.65464 ≈ 0.7 ms-2
Conclusion
According to the graph, the square of the period of a pendulum is directly proportional to its length, since we have a straight line through the origin (within range of the worst lines). The gradient of the line gives the acceleration due to gravity as 9.9 ± 0.7 ms-2. The accepted value of 9.81 ms-2 (see IB Data Booklet, pg. 1) lies within this range.
As the length increases, the distance through which the bob moves also increases, so it makes sense that the period of oscillation should be greater.
Evaluation
The results all lie on or close to the line of best fit, so we can say that the results are reliable. There were no anomalous results. Possible sources of uncertainty in the measurements are:
- Resolution uncertainty of the instruments for measuring length and time
- Human reaction time for starting and stopping watch
- Perhaps the watch was not started/stopped exactly when beginning/ending an oscillation
- Perhaps the measurement of length was not to the centre of mass of the object
- The first few oscillations were quite erratic, which may have affected the period
Possible improvements to the procedure would be:
- Use data-logging apparatus (e.g. Acquire datalogger and motion sensor) to record the motion of the mass
- Use a larger range of lengths of string, to reduce the relative uncertainty in the length
- Measure the time for 50 oscillations instead of 10, to reduce the relative uncertainty in the period
- Mark clearly the centre of mass of the object
- Allow the mass to oscillate a few times before starting measurements
