PHYS 200 Lab Report 4

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Athabasca University, Athabasca *

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Feb 20, 2024

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Sherlow | ID 3309093 | PHYS 200 | Lab #4 Athabasca University PHYS 200 Lab Report #4 Hooke’s Law Ashley Sherlow October 15, 2023 Student Number: 3309093 Introduction Hooke's Law is a fundamental principle in the field of elasticity, which describes the behaviour of materials when subjected to deformation. This law states that the force required to deform a material (stretch or compress it) is directly proportional to the displacement of the material from its equilibrium position. Mathematically, Hooke's Law is expressed as: F = kx These variables represent: F → the force applied to the material. k → the spring constant (a material-specific constant). x → the displacement from the material's equilibrium position (change in length).
The experiment provides valuable insights into the behavior of elastic materials and their response to applied forces, making it an essential concept in the study of mechanics and materials science. Procedure In this experiment, I used a medium-sized rubber band approximately 20 cm long (8 inches). I gathered ten toonies and confirmed that each toonie was of a mint 2012 or newer, thus the mass of any coin used is 6.92g per Table L4.2. Additionally, I gathered a measuring tape, a ziploc bag and a clip to attach it to the rubber band. To investigate the relationship between the applied force and the change in the rubber band's length, I set up the experiment as follows. 1) I attached one end of the rubber band to a fixed support and secured the small plastic bag to the other end of the band, adding some weight into the bag to ensure the rubber band had a slight initial stretch. At this point, I measured and recorded the initial length (L 0 ) of the rubber band. 2) I began adding coins one at a time into the container. After each coin was added, I measured the length (L) of the rubber band and documented these measurements in Table L4.1 . I used my judgment to estimate the uncertainty of the measured length. The measuring tape I used had millimeters as the smallest unit and I could clearly define the start and end points of the rubber band. Therefore, I considered an uncertainty of 2 mm and an uncertainty of Δm=0.2 g in the mass of each coin to be a reasonable estimate. See below a picture of the setup which illustrates the experimental setup and provides visual context for the measurements taken during the experiment. 1
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Data 3
Table L4.1 Record of measurements and calculations for elastic band stretch and weight of coins No. of coins n Rubber band length L ± (cm) Rubber band stretch x ± (m) Weight of coins w ± (N) n = # of coins 0 21cm x = L - L 0 x = 21 cm - 21 cm = 0 cm 0 cm x 100 = 0 m w = n mg w = 0(0.00692g)(9.80m/s 2 ) w = 0N 1 21.1cm x = L - L 0 x = 21.1 cm - 21 cm = 0.1 cm 0.1 cm x 100 = 0.001 m w = n mg w = 1(0.00692g)(9.80m/s 2 ) w = 0.00678N 2 21.2cm x = L - L 0 x = 21.2 cm - 21 cm = 0.2 cm 0.2 cm x 100 = 0.002 m w = n mg w = 2(0.00692g)(9.80m/s 2 ) w = 0.0136N 3 21.3cm x = L - L 0 x = 21.3 cm - 21 cm = 0.3 cm 0.3 cm x 100 = 0.003 m w = n mg w = 3(0.00692g)(9.80m/s 2 ) w = 0.0203N 4 21.4cm x = L - L 0 x = 21.4 cm - 21 cm = 0.4 cm 0.4 cm x 100 = 0.004 m w = n mg w = 4(0.00692g)(9.80m/s 2 ) w = 0.0271N 5 21.5cm x = L - L 0 x = 21.5 cm - 21 cm = 0.5 cm 0.5 cm x 100 = 0.005 m w = n mg w = 5(0.00692g)(9.80m/s 2 ) w = 0.0339N 6 21.6cm x = L - L 0 x = 21.6 cm - 21 cm = 0.6 cm 0.6 cm x 100 = 0.006 m w = n mg w = 6(0.00692g)(9.80m/s 2 ) w = 0.0407N 7 21.7cm x = L - L 0 x = 21.7 cm - 21 cm = 0.7 cm 0.7 cm x 100 = 0.007 m w = n mg w = 7(0.00692g)(9.80m/s 2 ) w = 0.0475N 8 21.8cm x = L - L 0 x = 21.8 cm - 21 cm = 0.8 cm 0.8 cm x 100 = 0.008 m w = n mg w = 8(0.00692g)(9.80m/s 2 ) w = 0.0543N 9 21.9cm x = L - L 0 x = 21.9 cm - 21 cm = 0.9 cm 0.9 cm x 100 = 0.009 m w = n mg w = 9(0.00692g)(9.80m/s 2 ) w = 0.0610N 10 22cm x = L - L 0 x = 22 cm - 21 cm = 1.0 w = n mg w = 10(0.00692g) 4
cm 1.0 cm x 100 = 0.01 m (9.80m/s 2 ) w = 0.0678N Analysis & Discussion Below is a graphical representation of the data gathered and presented above. The slope can be calculated using the formula slope = rise/run = w/ x = 0.0678N/0.01 m = 6.78. Therefore, we can determine the equation for the line of best fit as F = (6.78)x where the slope value of 6.78 replaces the k found in the Hooke’s Law equation of F = kx. Further, this demonstrates that the data collected supports Hooke’s Law. Limitations to this experiment lies primarily in the measurement of the rubber band stretch, both with my ability to accurately read the measurement tool and the granularity of the measurement units of the measuring tape. Graph L4.1 Rubber band stretch as a Function of Added Coin Weight 5
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Conclusion This experiment effectively demonstrated the principles of Hooke's Law, which is a fundamental concept in the field of elasticity. Hooke’s Law can be observed in the findings of this experiment, demonstrating elastic behavior of materials. The relationship between the force exerted by the coins and the changes in the rubber band's length was analyzed to observe its linearity. As seen in Table L4.1 , less coins resulted in less of a change in length, whereas more coins resulted in a greater change. The graph in Figure 2 further depicts this relationship by way of a linear slope, which resulted in a line of best fit equation of F = (6.78)x where the slope value of 6.78 replaces the k found in the Hooke’s Law equation of F = kx. Ultimately, this experiment supports Hooke’s Law. There are a number of sources of error in this experiment leading to uncertainty in 6
the results, such as equipment / measurement error or imperfect elastic behaviour. Questions 1. Based on your best-fit analysis, estimate the proportionality constant k for your rubber band. slope = rise/run = △w/△x F = kx → k = F/x F = y value = w (weight of coins N) X = x value = m (rubber band stretch m) k = 0.0678N/0.01 m = 6.78 N/m The proportionality constant for the rubber band I used this this experiment is 6.78 N/m. 2. How many coins are necessary to stretch the elastic band by 1.00m? Would this be possible with the rubber band you used in this lab? F = k/x = 6.78*1 = 6.78 F = w = mg m = F/ g = 6.78/9.80 = 0.6918 m/weight of 1 coin = 0.6918/0.00692 = 99.97 = 100 coins 100 coins would be necessary to stretch the rubber band by 1.00m, but this length would likely not be possible as it would likely hit the fracture point before stretching to 1.00m (Hooke’s Law is only obeyed for small deformations and a deformation of 1.00m for this rubber band would be large). 7