_Acceleration and Circular Motion (Lab Report 2)

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Acceleration and Circular Motion Lab Sophia Tran October 25rd, 2023 Physics 1301W, Professor: Dr. Dan Cronin-Henessey, TA: Evan Skinner Abstract In this experiment, the purpose was to relate the acceleration and circular motion of a mass to a spring constant. This showed scientists that if they spun the mass attached to the spring in a circular motion causing circular acceleration the mass would cause the spring to stretch. In this experiment, the magnitude of the hanging mass vs. the stretched length was 6.53N/m with an uncertainty of ± 0.035m. The calculated spring constant (k) was 0.0245N/m with an uncertainty of 0.049m, this value was taken from the angular velocity^2 graph. After analyzing the data they concluded the relationships were linear. Introduction Hypothetically, scientists used a spring catapult that uses circular motions to a launch ball. What is the relation between the ball’s mass and the spring constant from the catapult? Scientists put the mass of the ball in a tube on a rotating platform that is hooked onto the spring from the catapult to calculate the relationship. They did that by measuring the mass and measuring the stretched length of the spring. Procedure The setup of this lab was a tube zip-tied to a rotating platform, within that tube there was a spring attached to a stationary object, and the other side of the spring was hooked onto a mass. This way as the platform rotates the mass slides away from the spring causing the spring to stretch. They then took videos of the platform rotating for multiple time intervals with four

different masses. Then used Vernier Video Analysis to further analyze and measure data points. Using the software's data, they made the weight vs. spring stretch length graph. The slope from said graph would be the predicted spring constant (k). With the predicted k they were able to get the X/R ratio by dividing k by the radius squared of the circular motion. Afterward, find ω² using (2p 𝜋 /Δt)². Using those values to make a graph showing the relationship between the X/R ratio and the ω². The slope of that graph will be the calculated spring constant (k). The position uncertainty is the length of the spring when it is unstretched. To determine X/R uncertainty, find the maximum of the following equation: (Spring length 土 position uncertainty)/(radius 土 position uncertainty). A possible error could be skewed data plots on Vernier Video Analysis, causing inaccurate data for graphs resulting in uncertainties in the calculated spring constant (k). Another possible cause of error could be the lack of calculation for friction. Analysis (Figure 1) Vernier data plot at weight vs. Distance stretched, the slope of the line of best fit was y=6.53x+0.0897, meaning the predicted spring constant was 6.53 N/m with an uncertainty of 土 0.035m.

(Figure 2) Vernier data plot at X/R ratio vs Angular Velocity Squared, the slope of the line of best fit was y=0.0245*x+0.489, meaning that the m/k was 0.0245 with an uncertainty of 土 0.489

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Figure 1 shows that the slope of the graph is 6.53, which means the spring constant (k) was 6.53 N/m. which corresponds with the predicted value of 6.53 N.m 土 0.035m. Our predicted value was within the uncertainty. The data suggest that mg/x=k. Figure 2 shows that the slope of the graph is 0.0245, which means that the m/k is 0.0245 土 0.0489. Using the equation 1/(slope of X/R Ratio vs. ω²)/(M), there is the ability to calculate k. After converting the m/k ratio found as the slope from the previous question to isolate the spring constant, the math yields a value around 12.78N/m with an uncertainty of 土 1.191 which is much larger than the predicted constant from Figure 1 . However, the uncertainty of the values within Figure 1 still allows a value of 6.53 N/m to remain within the range of the error bars. Possible errors that could have occurred are inaccurate data points on Vernier, poor camera quality, and low frames per second when recording. All those reasons could lead to skewed data. Another error could be the negligence of friction between the mass and the platform. The predicted

value does not take into account friction, which would make the prediction less than the lab results because friction makes the constant of the spring seem greater. Scientists are able to conclude that the relationship between the mass and the spring constant is linear due to the information from the lab. Mass does not change the spring constant. Since our prediction did not agree with the actual result, it is inconclusive. Also ignoring fiction is an error that skewed our results by a value of μ in all equations that include gravity. Conclusion A mass was placed onto a rotating platform with a spring hooked onto it. The platform was then rotated, making the spring stretch. Scientists measured the stretch length using Vernier and then used the data to plot graphs on Google Sheets. The predicted value was 6.54 N/m 土 0.035m, and the calculated value was 0.0245N/m 土 0.489m. Hypothetically, if scientists wanted to know the relationship between the mass of a projectile and the spring constant in a catapult, they could calculate the constant using 1/(slope of X/R Ratio vs. ω²)/(M).

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