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Pendulum (conservation of energy) lab report with a 20/20 final score.
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Group#4: Hachim-Jeongho-Wasim-Cado-Cano PHYS 3A Professor Rosa Alvis 10/5/ Lab 4: Pendulum (energy conservation) Learning Objectives: To design and analyze an experiment for the purpose of describing how variables (length, angle, and gravity field) affect the motion of a pendulum. During the process, we will utilize the concept of conservation of energy to measure the acceleration of the pendulum due to gravity ( 𝑔) and calculate the uncertainty of 𝑔using the statistical standard deviation of a sample formula. Overview: A simple way to describe a pendulum is a mass dangling from a central point of rotation on a string of length (ℓ). The mass in our experiment will be a metal object. In the physics domain, this string should be massless. When the mass is stationary, it is said to be at equilibrium. If the mass is set into motion it will oscillate back and forth due to the influence of gravity on the mass. This repetitive, back and forth motion is called simple harmonic motion. A simple harmonic oscillator is a system that oscillates or has recurrent movement around a position of equilibrium and there is zero net force on the system. The mass continuously tries to move toward equilibrium (rest position). At all points in the oscillation of the pendulum, the angle between the force of tension and its direction of motion is 90 degrees. Consequently, the force of tension does not do work upon the metal object. Since there are no external forces doing work, the total mechanical energy of the pendulum metal object is conserved.
The period of a pendulum is the amount of time it takes for the metal object to move through one oscillation. The period can be altered by some significant factors such as the angle of the pendulum and the length of the string. When a pendulum is released from a greater angle, gravity accelerates the mass for a greater share of the swing. Therefore the mass covers a greater swing in less time. Other factors such as the mass of the object have no effect on the period. In order to calculate the period (T), we can either use the time 10 oscillation took in seconds to complete and divide by 10, or we could use the equation: T = 2π , where ℓ = Length of the string = acceleration due to gravity. Rearranging ℓ 𝑔 𝑔 this equation allows us to solve for 𝑔: T = 2π. ℓ 𝑔 →^ 𝑇 2 = 4π^2 ℓ 𝑔 →^ 𝑔 =^ 4π^2 ℓ 𝑇 2
𝑖 (^ -^ ) (m/s
𝑖
The uncertainty of 𝑔= 𝝳 𝑔 = = = = 0. 1 𝑁− 𝑖= 𝑁 ∑ (𝑔 𝑖
2 Σ(𝑔𝑖−𝑔)^ 2 26
26 ± 𝑔 = 9. 6 ± 0. 3 𝑚/𝑠 2 The uncertainty of single-trial #25 = 𝝳 𝑔 = 25
𝝳ℓ ℓ )^ 2
𝑒𝑥𝑝
𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
𝑒𝑥𝑝
𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
𝑒𝑥𝑝 % Error = |( 𝑔 - x100 = |(9.80-9.6)/9.80|x100 = 0.02x100 = 2% 𝑒𝑥𝑝
𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
Error Analysis : Although the error percentage was as low as 2%, there were indications of errors based on the obtained experimental data. Since there were variations in the measured data exhibited by unidentical measurements while using exactly the same method under the same condition, it suggests random errors took place during the experiment. One possibility of a random error that could have happened in this experiment was the inconsistency of the reader’s reaction time. This could have altered the values of the raw data, creating fluctuations in the results. This error can be minimized by repeating the experiment many times instead of three times for each combination of a particular angle and length. Another possible source of random error is the friction between the swinging metal object and the surrounding air. This also can be minimized by using a heavier object instead of a lighter one. Lastly, inaccurate rounding of the calculated values could also have been a contributor. This well could have been prevented by mastering the rules of rounding, being more cautious, and paying more attention to numerical details. Conclusion:
In this lab, a physical pendulum was displaced from its equilibrium position and it behaved as a simple harmonic oscillator. This experiment has proved that the acceleration due to the gravity (g) is significantly affected by the period of a pendulum (T) and the length of the string (ℓ). Our data showed that as the string is lengthened, the period of the pendulum is increased. There is a direct relationship between the period and the length. Similarly, as the arc angle surpassed the 𝛉 20° threshold, it diverged the experimental values of (g) further away from the 𝑔 of 9.80 m/s^2. 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 The equation of the period of a simple pendulum was used to find the experimental value of g, which was 9. 6 ± 0. 3 𝑚/𝑠. This was just under the theoretical value of 2
𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
𝑒𝑥𝑝
𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
Nonetheless, the value of 𝑔𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 agreed with the value 𝑔𝑒𝑥𝑝after performing a discrepancy test, so the experiment was successful.