phys lab 1 000

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Physics

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Feb 20, 2024

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Physics 207 Lab 1: Measurements Section GH3 Physics 207 February 16, 2024 Introduction: Everything we do in the fields of science and mathematics involves measurements. Relationships between these measurements are examined and studied in order to help us better

Physics 207 πunderstand the physical world. In physics we assign a different combination of units to measurements called SI units allowing easier comparison between measurements along with consistency. In this lab we learn to make certain measurements, calculate the uncertainty of our measurements and at the end graph our measurements to see a correlation in heartbeats and head size. In this experiment we use the formulas C = 2 πr, the circumference formula for a circle and ρ = m / V the density formula for objects. Specifically in this experiment we derive π from the circumference formula by dividing C by the diameter which is 2r. Furthermore, when we were calculating the density in this lab we recognized that the volume was not given to us therefore we had to calculate volume using V= , and then plug this value into ρ = m / V. From this we can ?𝑤ℎ derive the equation ρ = m/ . ?𝑤ℎ Procedure: Before our first exercise we made sure we had all materials in front of us along with correct measuring tools with the correct units. We then proceeded to measure Tyrone’s head with the rope and then marked on the rope where to stop and then measured the rope to the stop point. After this we then proceeded to record the time between heartbeats by counting up to 60 heartbeats and then dividing the number of beats by the time on the stopwatch. At this instant there could possibly have been considerable room for error as we may have been off by seconds, nanoseconds, or so on when stopping the timer, which impacts the data. For the second exercise we estimated the value of π through multiple mediums. ● At first we measured the diameter of three circular objects around us and recorded the data in an excel sheet. We then represented the data in a graph making our x-axis the

Physics 207 diameter and the y-axis the circumference. The slope indicated in the equation was the rate of change between diameter in circumference approaching the value of π. ● We then created a circle using toothpicks and used more toothpicks to form the diameter and solved for π, dividing the circumference by the diameter. ● Our last estimation of π involved using google maps and finding a circular object using the maps interface. We took the diameter and circumference of a circular top of a building in epcot. We then solved for π, dividing the circumference by the diameter. For our next exercise we measured the length of a 2D printed fish. In order to find the area of uncertainty in our measurement. We concluded that the smallest spacing on the ruler was .25 centimeters. We divided .25 by 2 and had an uncertainty of .05 cm. We then did an exercise where we calculated the uncertainty of a wooden block. Using a ruler, we found the length, width and height of the block, measuring all three dimensions of the object, which gives us the volume. We then weighed our block on a scale to obtain the mass. Using the formula ρ = m / V ( Density= mass/volume), we found the density of the wooden block. For our last exercise we looked at a basic pendulum. We measured the time it took to swing back and forth at different lengths of rope. In other words we measured the period of the oscillation. Our rope was a little more than a meter in length and less than 1.5 meters. We took a stop watch and measured the time for ten oscillations. After every trail we shortened the length of the rope for 5 trials. We calculated the period of oscillations by dividing the time in seconds by ten. We did this to minimize calculation and human error. Data tables, Calculations and Graphs:

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Physics 207 Head Size(cm) Time between heartbeats (s) 56 0.64 61 0.83 54 0.68 57 0.79 55.5 0.84 54 0.63 55 0.48 60 0.81 58 0.75 58 0.63 54 1.2 57 1.1 57 1.3 57 0.48 55 0.77 58.8 1.16

Physics 207 53 0.68 57 1.05 56.3 1.1 61 0.84 60 1 64 0.92 55 0.83 61 1.14 60 0.94 54 2.3 58 0.74 61 0.88

Physics 207 Circle Measurements for π : Object number diameter (cm) circumference (cm) 1 6.5 20.4 2 4 12.6 3 4.5 14.1

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Physics 207 Estimate of π : 3.1286 Measuring the circumference/diameter with toothpicks: Circumference: 8 toothpicks Diameter: 2.5 sticks Formula derived and used: π = 𝐶 2𝑟 π estimate: (8) / (2.5)= 3.2 sticks Measuring circumference/diameter of a circle via google maps Circumference of epcot circle: 306.4 m Diameter of epcot circle: 95.7072m Formula derived and used: π = 𝐶 2𝑟 π estimate: (306.4) / (95.7072)= 3.201m Pendulum swings vs. Period

Physics 207 Length of pendulum(cm) Period(s) 131 2.3 100 2.048 70 1.712 40 1.372 10 0.83 Density of wooden block: Length(l): 3.793cm Width(w): 2.874cm Height(h): 18.75cm

Physics 207 ρ = m/ ?𝑤ℎ V= ?𝑤ℎ V= (3.793cm) (2.874cm) (18.75cm) · · = 204.40 𝑐? 3 mass= 143 grams ρ = m / V density= (143g) / (204.40 = .6996 g/ 𝑐? 3 ) 𝑐? 3 Measurement of fish, uncertainty measurement: Length of fish: 12.7 cm δL = 0.25/2= 0.05cm Fish length: 12.7±0.05cm Discussion Questions: 1.) Discuss the limitations of these methods for calculating π. Which method was the best? The worst? Are there improvements to be made? The limitations of these calculations lie in the tools used for calculating the diameter and circumference. For instance, when using toothpicks to estimate π we made a shape that was almost a circle, not exactly. The shape was more so an octagon. This could have greatly impacted the result as octagons have sharp edges and angles, while circles do not. Additionally, when using the tool on google maps to measure diameter and circumference of a circle, it was very difficult to plot and connect the points which resulted in more of a hexagonal or octagonal shape rather than a round shape with no sharp edges. The method of estimating π using google maps was better as it allowed us to actually measure around a circle instead of creating a rough, more inaccurate one with sticks. When it comes to improvements, learning to really use the google

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Physics 207 tool on maps and making the spaces between each point plotted smaller can create more of a circular shape, allowing for a more accurate estimate of π . 2.) Determine the uncertainty of your density measurement. This will involve some algebraic calculation. You can find some pointers here: Info about error analysis (http://physicslabs.ccnysites.cuny.edu/errors.php). Next, find an online database that you can use to look up the density of various woods. Try to match your piece of wood with a known tree. Discuss whether you can be certain of your identification. Using the database from the United States Department of Agriculture ( https://www.srs.fs.usda.gov/pubs/gtr/gtr_so088.pdf ), we found that the density of our woods falls within the density of Diospyros mindanaensis. The density of Diospyros mindanaensis is around 0.69g/ . In order to find out the uncertainty of the density we have to add (( 𝑐? 3 uncertainty of mass/mass) + (uncertainty of volume/volume)) multiplied by the density. The uncertainty of the mass is ± .0025 and the uncertainty of the volume is ±0.00025. Therefore, the uncertainty of our density is [(0.0025/143g)+(0.00025/204.40 )] 0.6996) = ± .000013 𝑐? 3 · ( g/ . So the density is around 0.6996±.000013 g/ . 𝑐? 3 𝑐? 3 3.) Based only on your experimental data, can you say how the time for one swing relates to the length of the pendulum? Is there a clear functional dependence? What could you do to make the experiment better?

Physics 207 Based solely on my experimental data I am able to graph it and see quite a clear direct relationship between the length of the pendulum and the amount of time it takes to swing one oscillation. In my graph there is a linear line with a positive slope, indicating that as the length of the rope increases, so does the time it takes to swing one oscillation. Although there seems to be a clear functional dependence, there were most definitely limitations and room for error. The person swinging the pendulum could have had his arm at a different angle during each trial, which could have greatly impacted the results. Additionally, the person who was recording the time could have stopped it too early or too late, which also impacts results. In order to make the experiment better there should be some hook or screw where the pendulum is being held at a constant angle, rather than be held by someone’s hand. 4.) Is there a correlation between the circumference of someone's head and the time between heartbeats? Would you expect there to be one? From my experimental data and the graph in which the points were plotted, there seems to be no correlation between the circumference of one’s head and the time between heartbeats. The data is not really clustered together in one direction; it is more so scattered, which indicates a lack of correlation. I truly would not expect there to be a correlation between the circumference of someone’s head and the time between heartbeats because someone can be physically very fit and have a large head, possibly resulting in more time between heartbeats. Additionally, someone can have a smaller head and could also be very physically fit, possibly with more between heartbeats. Both variables depend on a multitude of other factors for them to be dependent on each other in any way. Conclusion:

Physics 207 This lab allowed us to visualize and put into practice the uses for measurements and different kinds of measurements. In our first exercise we were asked to measure the circumference of our heads and count the time between heartbeats. We did this in order to see if there was some mathematical relationship between the two variables, with the circumference of our heads being the independent variable on the x-axis and the time between heartbeats being the dependent variable on the y-axis. When it is graphed we noticed a lack of correlation between the two variables as the data was scattered throughout rather than being close together. To reduce the error in this part of the experiment we decided to count more heartbeats. In the second exercise we were asked to estimate π . Pi is the constant that is present in our formula for circle area ( π ) and circumference ( C = 2 πr). We learned that all circles 𝐴 = 𝑟 2 have one thing in common which is when we divide the circumference by the diameter we end up with π . In this case we had to estimate the exact value of π using toothpicks to create circles and looking at circles in our world through google maps. Through this we estimated π was close to 3.2 . We were also asked to find the circumference and diameters of different circular objects in the room to estimate the value of π which was around 3.1286. To improve our results we could have learned to better use the tool on google maps attempting to trace a more round circle. For our third exercise we touched upon the possibility of uncertainty in measurements. We learned that our measurements can only be so accurate and that there is always room for error. We measured the length of a paper fish and estimated the value of uncertainty and came to a conclusion that it was around 12.7±0.05cm. To improve the result we could have used a measuring tool with smaller units.

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Physics 207 Furthermore, we were able to see how numerical data allows us to gain specific information about a specimen. In our fourth exercise we found the density of a wooden block and found the species of tree our block came from. This allowed us to understand measurements are an extremely important indicator in specimen identification, which is very important in scientific fields like biology and earth science. To improve the experiment and results we could have tared the balance and made sure it was completely at zero before placing the block on the scale. Lastly, our last exercise consisted of looking at a basic pendulum and finding a relationship between the length of the pendulum and the time it took for one oscillation. From gathering data and plotting it we were able to see a very clear direct relationship between the two variables. This allowed us to see how measurements can allow us to make correlations, which are seen in real life. In order to reduce error in our results we ensured we counted ten oscillations and then divided the time by ten to get a more accurate time per oscillation.

phys lab 1 000

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