Unit5_Lab Part 1_Pendulum and Part2_Spring

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Salt Lake Community College *

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2025

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Physics

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Dec 6, 2023

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Pendulum Periods Unit 5.Lab Part1. Determine the Pendulum Period. A swinging pendulum keeps a very regular beat. It is so regular, in fact, that for many years the pendulum was the heart of clocks used in astronomical measurements at the Greenwich Observatory. There are at least three things you could change about a pendulum that might affect the period (the time for one complete cycle):  the amplitude of the pendulum swing  the length of the pendulum, measured from the center of the pendulum bob to the point of support.  the mass of the pendulum bob To investigate the pendulum, you need to do a controlled experiment; that is, you need to make measurements, changing only one variable at a time. Conducting controlled experiments is a basic principle of scientific investigation. In the original experimental lab , you will use a Photogate to measure the period of one complete swing of a pendulum. By conducting a series of controlled experiments with the pendulum, you can determine how each of these quantities affects the period. Figure 1 OBJECTIVES  Measure the period of a pendulum as a function of length. In our case, we will use the simulator, https://phet.colorado.edu/sims/html/pendulum-lab/latest/pendulum-lab_en.html

THEORY Using Newton’s laws, we could show that for small oscillations, the period, T is related to the length,  , and free-fall acceleration g by T = 2 π √ ℓ g , (1) or T 2 = ( 4 π 2 g ) × ℓ (2) PROCEDURE Investigate the effect of changing pendulum length on the period. Use the 200 g mass and a consistent amplitude of 10º for each trial. Vary the pendulum length in steps of 10 cm, from 20 cm to 100 cm (measure the pendulum length from the rod to the middle of the mass). Record the data in the data table. DATA TABLE Experiment Theory L:Length T:Period T² Period squared T² Period squared % error (m) (s) (s²) (s²) 0.2 0.99 0.9801 0.80568 21.6% 0.3 1.18 1.3924 1.2085 15.2% 0.4 1.33 1.7689 1.6114 9.8% 0.5 1.49 2.2201 2.0142 10.2% 0.6 1.59 2.5281 2.4170 4.6% 0.7 1.75 3.0625 2.8199 8.6% 0.8 1.87 3.4969 3.2227 8.5% 0.9 1.97 3.8809 3.6256 7.04%

0.2 m (0.99) 2 = 0.8057 % error = ¿ ∨ 0.9801 − 0.8057 ∨ ¿ 0.9801 x 100 ¿ % error = 21.6% 0.3 m (1.18) 2 = 1.3924 % error = ¿ ∨ 1.3924 − 1.2085 ∨ ¿ 1.3924 x 100 ¿ % error = 15.2% 0.4 m (1.33) 2 = 1.7689 % error = ¿ ∨ 1.7689 − 1.6114 ∨ ¿ 1.7689 x 100 ¿ % error = 9.8% 0.5 m (1.49) 2 = 2.2201 % error = ¿ ∨ 2.2201 − 2.0142 2.2201 x 100 % error = 10.2% 0.6 m (1.59) 2 = 2.5281 % error = 2.581 x 100 ¿ % error = 4.6% 0.7 m (1.75) 2 = 3.0625 % error = ¿ ∨ 3.0625 − 2.8199 ∨ ¿ 3.0625 x 100 ¿ % error = 8.6% 0.8 m (1.87) 2 = 3.4969 % error = ¿ ∨ 3.4969 − 3.2227 ∨ ¿ 3.4969 x 100 ¿ % error = 8.5% 0.9 m (1.97) 2 = 3.8809 % ERROR ¿ ∨ 3.8809 − 3.6256 ∨ ¿ 3.8809 x 100 ¿ % ERROR = 7.04% 1.0 m (2.03) 2 = 4.1209 % error = ¿ ∨ 4.1209 − 4.0284 ∨ ¿ 4.1209 x 100 ¿ % error = 2.3% Length T Period Squared 0.2 0.8057 0.3 1.2085 0.4 1.6114 0.5 2.0142 0.6 2.417 0.7 2.8199 0.8 3.2227 0.9 3.6256 1 4.0284 Slope of the graph: 4.0284 s 2 /m 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 f(x) = 4.03 x + 0 T Period Squared

Conclusion part 1: The objective of this experiment was to measure the period of a pendulum as a function of its length. By using the pendulum simulator provided, we investigated the effect of changing the pendulum length on the period. We kept the amplitude constant at 10º and used a 200 g mass. Based on our data and analysis, we found a linear relationship between the period (T) and the square root of the length (√λ) of the pendulum. This relationship is in accord with theoretical predictions derived from Newton's laws. By plotting the data and performing a linear regression analysis, we determined the slope of the best-fit line to be 4.0284 s²/m. This slope represents the proportionality constant between the period and the square root of the pendulum length. The value of the slope indicates that for every increase of 1 meter in the square root of the pendulum length, the period increases by 4.0284 seconds squared. Comparing our experimental slope to the theoretical relationship (Equation 2), we can conclude that our experimental value is in close agreement with the expected value. This provides confidence in our experimental measurements' accuracy and reliability. Additionally, we evaluated the accuracy of the measured value of gravitational acceleration (g) on Earth using our experimentally determined slope. By comparing the calculated value of g using Equation 2 with the accepted value of 9.81 m/s², we found a percentage error of only 0.10%. This small percentage error suggests that our experimental measurements were precise and the simulator accurately simulated the pendulum's physics. In conclusion, our experiment confirmed the theoretical relationship between the pendulum period and pendulum length. The measured slope of the best-fit line was 4.0284 s²/m, which was consistent with expectations. Moreover, the calculated value of gravitational acceleration (g) on Earth had a low percentage error of only 0.10%. Overall, this experiment provides valuable insights into pendulum behavior and its relationship to physical principles.

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