In the non scientific world, the words mass and weight are often used interchangeably. However, to a physicist (or physics student) the difference between the actual things these two values measure is important. Mass is a measure of an objects inertia, a quantity that does not change depending on its proximity to other matter. Weight, or gravitational mass is a force acting on an object that varies in strength according to its mass and the gravitational field in which its weight is being measured. Of course. You know that. Because you're a physics professor. But I needed to type that out to prove that I knew that, probably. If not sorry for wasting your time. And more of time just now. And now.
Back to the point, spring scales and balances measure mass by way of gravity, in a sense. They are calibrated to account for gravity in their measurements, anyways. But in order to see that mass is not dependent on gravity, we must find a way to measure mass without using gravity as a catalyst for our observations. This is, probably, the overreaching point of this lab, along with the more functional purpose of familiarizing the class with the lab equipment and the scientific process. All of that fingerflubbering aside, the real goal of this lab was to establish a mathematical relationship between the mass of an object on a horizontal pendulum and the period of that pendulum.
THE SETUP
Me and my peers set up the lab as follows: An inertial balance device was c-clamped to the table with a piece of tape on its free end as shown:
Next, we used a stand to fasten a photogate, a device used to measure the period. It operates by sending a continuous stream of light to a sensor, when an object breaks this beam, such as piece of tape on the end of inertial balance device (hint hint), it starts a count. A program then records the time between each break of the light beam it measures the period, the photogate setup was as shown:
The photogate was plugged into a lab computer and a LabPro file was opened to record and manipulate the data. Finally, we got a set of 100 gram weights to add to the device in order to measure the period with different masses on it.
THE PROCEDURE
After setting up the necessary equipment, we began collecting data. We pulled back the inertial balance device with increasing amounts of weight on it, starting with just the mass of the tray and ending with 800 grams, going up 100 grams for each measurement. In order to keep some accuracy, we pulled the device back the same distance each time and letting the photogate record the period for 5 seconds each time. The periods were recorded on the following chart:
This was then plotted on a graph with mass as the x component and period as the y component. I do not have the original file, so I replicated the result on Microsoft excel:
This graph is not perfectly linear, and appears to follow a slight downward facing curve. This indicates that the relationship between period and mass is dictated by a power law of the form:
Where T is the period, m is mass, Mtray is the mass of the tray, n is an exponent, and A is some constant that converts kg to seconds. In this equation we have 3 unknowns, A, Mtray and n. In order to figure out the values of these three variables, I linearized the equation by taking the logarithm of both sides like so:
This new equation has a the a form close to y=mx+b, so we expect that if we graph this function with the correct value for Mtray, it will form a line with slope n and y intercept lnA. Using the LabPro tool, we are able to quickly subsitute many values for Mtray, and gauge their accuracy with the measure of linear fit provided by the program. Doing this, we were able to produce this graph:
Although it was recomended to get the linear fit to 0.9999, with our data we found it impossible to do so, the closest we could achieve was 0.9998. This indicates some issue with our data collection process. The range of values for Mtray which yielded this value for linear fit was 263-333 grams. I created 3 different possible equations, one from the lower limit, one from the upper, and one from the average. These are as follows:
EQ1-Lower limit: Mtray = 263 grams
T=0.6578(m+0.263)^0.623
EQ2-Upper limit: Mtray = 333 grams
T=0.6307(m+0.333)^0.7168
EQ3-Average: Mtray = 298 grams
T=0.645(m+0.298)^0.6749
In order to test the accuracy of our equation, we used objects that we did not have exact weights for, put them on the inertial balance machine, and measured the periods. In our case, we used a set of keys and a wooden block. The keys yielded a period of 0.348 and the wooden block 0.431. In order to make use of these quantites we rearranged the equations above to solve for mass as shown:
Plugging each of these into our 3 equations and solving for mass, we produced these quantities for mass. (note: EQ(n) stands for the equations listed in order above and mk and mb represent the mass of the keys and the wooden block, respectively)
EQ1- mk=102.1g mb=249g
EQ2- mk=103g mb=255g
EQ3- mk=102.8g mb=252g
Afterwards, we measured the actual weight of the two objects on a scale, the results were:
mk= 102g
mb=256g
CONCLUSIONS
Although equation 1 gets closer to the mass of the keys than the other two equations, equation 2 gets the closest to the mass of the block, and equation 3 is less accurate that equation 1 for the mass of the keys and equation 2 for the mass of the block, but is more accurate than equation 1 for the mass of the block and likewise for equation 2 with respect to the mass of the keys. This indicates that our none of our equations are perfect, but equation 1 may be more accurate for low masses and equation 2 for larger ones. With more accurate data collection, and more data points a future experiment following the method laid out in this lab could yield a single, more accurate equation. These results are close enough, however, to say with reasonable confidence that mass is a quantity that is innate to an object and not dependent on gravity that is related to inertia and that the period and mass of an object in this system is dictated by an equation resembling those posted above.
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