Measuring g - Discourses on Two New Sciences - Galileo Provided a Nifty Argument Regarding Large and Small Objects - Sciences Assignment Help

Assignment Task:

Introduction
One of the earliest mechanics observations must have been “things fall.” Aristotle maintained that objects fall with a speed proportional to their weight. This seems reasonable when comparing the behavior of dandelion seeds and kernels of corn. In a book published in 1638, abbreviated title of Discourses on Two New Sciences, Galileo provided a nifty argument regarding large and small objects tied together with very fine threads that challenged this belief.1 In addition he pointed out air has an effect that cannot be ignored for very light objects, like dandelion seeds, but becomes more and more ignorable as the density of the falling object grows.

There are also shape factors that must be considered. To demonstrate this, take a sheet of paper and hold it so the flat surface is horizontal, then drop it, observing the fall. Using the exact same sheet of paper, hold it so the flat surface is vertical. Comment on the observed difference.

Galileo was motivated by the recognition that a ball rolling down a slope moves faster and faster, and a ball rolling up a slope moves slower and slower. From this he reasoned if the ball rolled along a level surface it should not change speed at all. This is counter to our everyday observations. If you roll a ball on a level surface, the ball eventually stops. Aristotelian physics claimed this happens because the ball’s “natural state” is a state of rest. For the ball, motion is “unnatural.” Galileo maintained both motion and rest are natural. We observe rest more often because of friction. Thechoice to use hard bronze balls rolled along a smooth, inclined groove was made to minimize the effects of friction. He did not simply drop the objects so they would fall straight down because he could not measure the time accurately enough over any reasonable height. Rolling was his way around the limitations of his technology.

Rolling introduces a bit of complication, though, that would have prevented Galileo from obtaining an accurate measure of the acceleration due to gravity (the name we give to the acceleration observed when an object falls with negligible air resistance). We can exploit technology to perform a more direct measurement of the acceleration due to gravity. In this experiment you will test Galileo’s challenge to Aristotelian physics – objects fall with an acceleration that is independent of their weight. Analysis of your results will be via fit to an appropriate plot.

Experiment

2.1 Equipment
washer(s), tape measure, masking tape (included in kit) or push pins (from your own supplies), hard surface item used as a target Acoustic stopwatch in PhyPhox

2.2 Procedure
From playing with the Acoustic Stopwatch it is probably apparent that you need a bit of quiet to do this experiment. It does not need to be silent, but you should pay attention for excessive background noise or short, loud bursts of sound that may trigger the sensor. Be sure to set up the appropriate conditions.

1. Find a section of wall where you can attach your tape measure. Attach the tape to the wall such that it is held vertically and it has no slack in it. The bottom of the tape measure should touch the floor.

2. Place your “hard surface item” on the floor against the tape measure. This could be a hardcover book, an candy tin (like what Altoids come in), or a flat baking sheet. You could even try using the box your equipment kit was shipped in. The purpose of this is two-fold. First it provides a flat surface for the washer to hit producing a sound that will trigger the acoustic stopwatch. Second, it raises the “floor” to compensate for any systematic uncertainty associated with aligning the end of the tape measure against the floor. As long as your raise the target surface by greater than about 2 cm above the floor, you should be able to determine its location along the tape measure.

3. Start the Acoustic Stopwatch in PhyPhox. Set the phone near where the washer will hit the target. Your goal is to have the phone close enough that it will respond to that sound, but not so close that the washer will bounce and land on it. Test the setup several times to make sure everything works – the stopwatch triggers on a reproducible starting sound and it triggers to stop only when the washer hits the target. Find the lowest height you can drop the washer and have the stopwatch trigger when the washer hits the target. What restriction does this place on your experiment?

4. Choose some large initial height close to the 2 m mark on your tape measure. Hold a washer at that height and drop it onto the target. Should the bottom, middle, or top of the washer be held at this height? Why? As you release the washer make some kind of noise that will start the stopwatch. Verify that when the washer hits the target, the stopwatch stops. If you’ve done things carefully the time displayed represents the free fall time.

5. Repeat the free fall time measurement 5-7 times. Be sure to record each measurement in a well-structured data table. You will be collecting many data points, so it may be a good idea to record directly in your chosen spreadsheet program. This way you won’t have to write the numbers, then later enter them in the computer. If you choose to save time this way be sure to write in your logbook that you have done so and where the file is saved. If it is saved on a computer, which computer and what is the file path? If it is saved online, what service and what is the file path?

6. Once you have multiple measurements for one height, choose a second height and repeat the experiment. Continue choosing new initial heights and measuring the free fall time for the washer. You should aim for at least 5-7 di↵erent starting heights.

7. Now that you have your data for one washer, tape 3-5 washers together so you have larger mass. Repeat the entire experiment using this new mass. Should you use the same heights as before or should you choose di↵erent heights? What are the advantages of either method? What are the disadvantages of either method? Justify your choice.

8. At the end of all your data collection you should have 50-100 trials. Identify which data represent identical conditions, and can be averaged. Only average those data for which you have a good physical argument allowing you to expect they should be the same.

You will need to plot your data to determine g. Your data should consist of two sets of 5-7 pairs of numbers. For each set you chose the starting height, so this is your independent variable. You measured the free fall time, so this is your dependent variable. Your goal is to write the data in such a way that a plot predicts a straight line. Then, you will be able to use your chosen spreadsheet program to obtain a linear fit The equation will be in slope-intercept form, so the model needs to be in the same form. Pay attention to the process we follow as we determine how to plot the data.

If we compare Eqs.3.4 and 3.6 we can identify d $ x, 0 $ b, and 2/g $ m provided we say t2 $ y. Taking the square of the times transforms the collected data into something that when plotted are expected to fall on a straight line with slope 2/g and t2-intercept (when d = 0) of 0. Note that we have not changed the information provided by the data. Squaring all of the measured times changes each of the collected numbers in precisely the same way giving us new numbers while retaining the values.

Construct a plot of t2 (on the y or dependent axis) vs. d (on the x or independent axis) for your two experiments. You may put them both on the same plot, since they represent two di↵erent trials of the same experiment. Doing so will make comparing the results much easier. If you do this, though, you must use di↵erent data markers for the two runs. Since you are testing for a possible e↵ect of mass on acceleration due to gravity, you have to present your data in a way that distinguishes the two configurations.

Once you have your data plotted, fit trendlines to both sets. Be sure to display your equation on the graph. Appropriately label your axes and create a reasonable legend. Each instructor may have slightly di↵erent formatting expectations, so be sure to follow whatever your instructor requires.

3 Questions & Comments

• Describe any difficulties you had executing the procedure. Was it easy to work with PhyPhox? Which steps were difficult to interpret? What might you do di↵erently if you were to repeat this experiment?
• Did you choose to use the same heights for both masses or di↵erent heights? Why did you make the choice you made?
• What value of g do you measure? Was there a di↵erence in results between the two masses used?
• If your instructor requires it, report your measured value(s) of g with appropriate uncertainty.
• Students often balk at analyzing the data via plots. Many would much rather calculate g for each trial and then average those results. This exercise will demonstrate the benefit of the plotting method.

1. Since your data are already in a spreadsheet, use Eq.3.1 to calculate g for each run.
2. Now, imagine there was some systematic error in your measured distance, and the number you recorded was actually 10 cm too large. To simulate this, add 10 cm to each of your heights. It’s best to do this in a separate column in yoru spreadsheet.
3. Use your modified height data and your recorded times to calculate g for each point. Note any di↵erences between these values and those you calculated above.
4. Use your modified height data and your recorded times to create a t2 vs. dmod plot. Determine g from your slope and compare this to the other slope value obtained. How do the intercepts of the two fits compare?
5. Explain why the plotting method is the method you will most often be expected to use. 

• Address any other requirements from your lab instructor.
• Remember, “human error” is not an acceptable source of uncertainty or di↵erence from expectations.

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