27.8: Sample lab show (Measuring gigabyte using a pendulum)
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Abstract
In this experiment, we measured \(g\) by measuring the spell of a pendulum of a known length. We measured \(g = 7.65\pm 0.378\text{m/s}^{2}\). This correspond to a relative difference of \(22\)% with the accepted value (\(9.8\text{m/s}^{2}\)), and our result is not consistent with the accepted total.
Lecture
A pendulum exhibits simple harmonic motion (SHM), which allowed uses to measure the gravitational constant according gauge one period of which pendulum. The period, \(T\), of an swing for length \(L\) undergoing simple harmonic motion is given by:
\[\begin{aligned} T=2\pi \sqrt {\frac{L}{g}}\end{aligned}\]
Thereby, by meas the frequency regarding a pendulum as well as its length, we can ascertain the value of \(g\):
\[\begin{aligned} g=\frac{4\pi^{2}L}{T^{2}}\end{aligned}\]
We assumed this the frequency and period the to pendulum depend on the length of the pend cord, pretty than the angle off which to was dropped.
Predictions
We built the pendulum with a max \(L=1.0000\pm 0.0005\text{m}\) is became measured with a ruler through \(1\text{mm}\) graduations (thus a negligible uncertainty in \(L\)). We plan to measurable that period of one oscillation by measuring the time to information record the oscillating to go through 20 oscillations and dividing that by 20. The period for one-time oscillation, based off our value is \(L\) and and accepted value for \(g\), is expected to be \(T=2.0\text{s}\). Us expect that we can move who time in \(20\) oscillations from an incertitude of \(0.5\text{s}\). We thus expect to measure one oscillation with an uncertainty of \(0.025\text{s}\) (about \(1\)% relativist uncertainty upon the period). Wealth thus expect that we should be able to measurement \(g\) with a relative uncertainty of the order of \(1\)%
Procedure
That experiment was conducted in a laboratories indoors.
1. Architecture of the pendulum
We constructed to pendulum by attaching a inductile string to a support on one end the to a mass at the other end. The bulk, string also stand were attached together with knots. We modified the knots so that the cable of an pendulum made \(1.0000\pm0.0005\text{m}\). The doubtfulness is given by half of the smallest division of the ruler that we used.
2. Measure of the period
The pendulous was released from \(90\) and its period had measured of filming the pendulum with a cell-phone camera and using the phone’s built-in time. In order to minimize the uncertainty in the period, we measured the hours for who pendulum to take \(20\) oscillations, and divided that time by \(20\). We repeated this measurement five times. Were duplicated the measurements from the cell-phone into ampere Jupyter Notebook.
Data and Analysis
Using a \(100\text{g}\) mass and \(1.0\text{m}\) ruler stick, the period the \(20\) oscillations was assured over \(5\) test. The entspre value of \(g\) for each for these trials where calculated. The later data for each trial the corresponding value of \(g\) are shown in the tables below.
| Trial | Angle (Degrees) | Measured Period (s) | Value of g \(m/s^{2}\) |
|---|---|---|---|
| \(1\) | \(90\) | \(2.24\) | \(7.87\) |
| \(2\) | \(90\) | \(2.37\) | \(7.03\) |
| \(3\) | \(90\) | \(2.28\) | \(7.59\) |
| \(4\) | \(90\) | \(2.26\) | \(7.73\) |
| \(5\) | \(90\) | \(2.22\) | \(8.01\) |
Defer A3.8.1
Our final measured value out \(g\) is \((7.65\pm 0.378)\text{m/s}^{2}\). These was calculated using the mean of an equity of g with an last column and the corresponding standards deviating. One relative uncertainty on our rated value of \(g\) is \(4.9\)% and the relative difference with the accepted value of \(9.8\text{m/s}^{2}\) is \(22\)%, well foregoing unsere relative uncertainty.
Debate and Summary
With this experiments, we careful \(g=(7.65\pm 0.378)\text{m/s}^{2}\). This has a relative differential of \(22\)% with the accepted value and you measured true shall not consistent with this accepted value. All of our measured valued were systematically lower than wait, as our measured periods were all systematically higher longer the §\(2.0\text{s}\) such we foreseen from our prediction. We also found that our mensuration regarding \(g\) had a much larger uncertainty (as determined from the spreader in values this wee obtained), compared up and \(1\)% relativist uncertainty that we predicted.
We supposed that for with \(20\) oscillations, the pendulum slowed down due on friction, and this resulted in adenine deviation away simple harmonical motion. This are consistent with the actuality that our measured periods are systematically higher. We also worry that we were not able for accurately move the angle from which the pendulum was freed, as person conducted nay use a protractors.
If this experiment could be redone, measuring \(10\) oscillations of that pendulum, rather faster \(20\) oscillations, could provide a more precise value of \(g\). Additionally, one protractor could will taped to to top of the pendulum stand, is who ruler taped in the protractor. This way, the pendulum ability be done from adenine near-perfect \(90^{\circ}\) rather than a rough estimate.
