Ian Lin, Mo, and Caesar
2/27/17
This experiment is to investigate, examine, and determine the mathematical relationship between mass and period by using the power-law type equation.
Mass is how much matter there is, and is independent of gravitational strength. However, many of the measuring instruments that are used require the presence of gravity in order to function. An inertial balance is a device that's used to measure inertial mass by making measurements that don't depend on the strength of gravity. The period of oscillation of an inertial pendulum depends on how much mass is on it. It's reasonable to hypothesize some sort of proportional relationship between mass and period of oscillation. As mass increases, period of oscillation increases, and vice-versa. The starting mathematical model is given by T=A(m+Mtray)^N. There're three unknowns: A, N, and Mtray..
As we measure the mass and the oscillation period, the other three variables can be found by graphing the data and finding a line of best fit. That will give us the values of In A, N. The value of Mtray can be found through adjusting the value of the parameter Mtray until your graph gives you a beautiful straight line.
The lab first consists of setting up the inertial pendulum with a metal tray attached to two spring pieces of metal. When a force is applied to the tray, a vibration occurs. The oscillation period can then be measured. With the use of logger pro application and a photogate, the oscillation period for a range of masses from 0 to 800 grams can be recorded and graphed. The graph plots In T vs In ( m+Mtray) with the adjustment of the value of Mtray to give a correlation coefficient of around 0.9997 to 0.9999. Afterwards, a line of best fit is utilized and the values of n and In A is extrapolated from the computer.
Data Table
Mass in balance ( grams)
|
Period ( seconds)
|
Mass in balance ( grams)
|
Period ( seconds)
|
0
|
0.285
|
500
|
0.551
|
100
|
0.352
|
600
|
0.595
|
200
|
0.407
|
700
|
0.640
|
300
|
0.457
|
800
|
0.683
|
400
|
0.5091
|
Value of Mtray that gives max correlation of 0.9998
|
Slope ( value of n that goes with that value of Mtray)
|
Y-intercept ( value of n that goes with that value of Mtray)
|
230 ( grams)
|
0.5749
|
-4.383
|
250 ( grams)
|
0.6009
|
-4.570
|
270 ( grams)
|
0.6265
|
-4.756
|
Calculated mass of unknown 1
|
Calculated mass of unknown 2
|
|
Using your Mtray minimum value and its corresponding values for A,N
|
51.93 g
|
143.22 g
|
to g your Mtray intermediate value and its corresponding values for
A, n
|
51.96 g
|
144.92 g
|
Using your Mtray maximum value and its corresponding values for A, n
|
51.38 g
|
145.74 g
|
Mass from electronic balance:
91 g
|
Mass from electronic balance: 185 g
|
Sample Calculations
Conclusion
The experiment is to attempt to find a relationship between the mass of an object in question and the period of oscillation on the inertial pendulum through mathematical modeling. The given equation of T=A(m+Mtray) ^ n became in T= n In ( m+Mtray) + In A after taking the natural logarithm of the entire original equation. Then a graph of T vs In ( m+Mtray) should yield a line with slope n and y-intercept equal to In A with the correct value for the mass of the tray. Then we set about compiling a data table of the different periods of oscillation with respect to the different increasing masses placed on the inertial pendulum. The actual recording and detection of the period were done through the computer and sensor system. There aren't any significant sources of error regarding the actual measurement of the period due to it being computerized and not " by hand." Although air resistance was present, it can be disregarded due to the density of the mass placed on the balance and the time interval that was consumed. The biggest source of error is setting up the photo-gate and the inertial pendulum incorrectly so that the tape doesn't completely pass through the beam of the photo-gate when the balance is oscillating. The results of the data table does indeed verify the proportional relationship between time and mass. Afterwards, we had to adjust the value of the mass of the tray until a correlation coefficient of 0.9997 to 0.9999 was reached. In addition, there were uncertainties in the value of the mass of the tray, and so a range of values from minimum to maximum was also considered. We discovered that the values of the mass of the tray lie somewhere between 230 grams to 270 grams.
Nice introduction.
ReplyDeleteIt could be clearer why you are taking the ln of the equation.
You have most of these things in your description, but not quite all as clearly as they might be:
--Power law equation
--ln form
--what will be plotted on the y axis and on the x-axis
--what the slope and y-intercept of that graph will tell you
--how you are going to find the mass of the tray
You don't include any screen shots of your curve fits (this is essentially part of the "data" for the lab.
There is a large mismatch between the results from your curve fit and you experimental results for the unknown. This is very unusual for this lab. I would encourage you to check over your calculations.
The idea of a conclusion here is different that in an essay for an English class, where you would repeat and restate most of what you had already said earlier. Here it is about summarizing your results and looking at sources of error and uncertainty. It is not difficult to set up the photogate correctly, and you should have, as part of the required parts of the lab, checked how well it matches your results from a stopwatch.
When we set up our original equation all of the masses were cylinders centered in the tray.
Our unknown objects had different shapes and perhaps different placement in the tray. We didn't test separately to see if placement or shape made a difference. This is an assumption (that mass is the only variable) that maybe turns out not to be true