SPRING ENERGY LAB WILLIAM HUNG K SECTION INTRODUCTION Purpose - PowerPoint PPT Presentation

SPRING ENERGY LAB WILLIAM HUNG K SECTION

INTRODUCTION • Purpose • The purpose of this lab is to design and perform an experiment which analyzes the conservation of energy in a spring-based system. • Researchable Question • How does the initial height of a weight that starts at rest on a ramp, placed above the equilibrium point, and attached to a relaxed bungee affect the time it takes to reach the bungee’s max stretch after the weight’s release? • Hypothesis • If the initial start height of the weight in increased, the time it takes for the bungee to reach its max stretch will 𝐶+𝐷 𝐸𝐹+𝐺 𝐽 be greater, where time total is proportional 𝐵(sin −1 ( 𝐸𝐻+𝐼 ) − sin −1 ( 𝐸𝐻+𝐼 )) + 𝐾𝐸 , where D is the difference between the starting point and the equilibrium point based on the markings on the ramp.

PROCEDURE • Tape a bungee spring with a length of 105.65 cm onto the top of the ramp • On the side with the taped bungee, lift the ramp and set it at an incline of 4.4 degrees so that the bungee string’s stretch at the maximum setting would not allow the car to go over the length of the ramp • Find the equilibrium point, the cart was let go and left untouched until it stops. The recorded equilibrium point was the 100.7 cm mark on the ramp. • Place the front of the car at the 130.0 cm mark on the ramp • Release from the height to obtain a rough estimate of where on the ramp the bungee would reach its max stretch

• Use one phone to start the timer immediately when the car was released • Use a second phone to take a video with the view of the whole ramp and the timer on the first phone in slow motion in order to check how much time it takes for the car to reach its max stretch • Use the third phone to take a slow-motion video of the area where the maximum stretch occurred, based on the estimate from 3 steps before • Record the data and repeat experiment for 9 more times • Repeat the whole process by holding the front of car at 150 cm, 160 cm, 170 cm, and 190 cm

MATERIALS • Bungee String • Metal Ramp • Car • Tape • Heights (to hold the height of the ramp constant)

CONSTANTS AND EQUATIONS • D = difference between the starting point and equilibrium point, based on markings on the ramp • 𝐸 1 = 0.293 𝑛, 𝐸 2 = 0.493 𝑛, 𝐸 3 = 0.593 𝑛, 𝐸 4 = 0.693 𝑛, 𝐸 5 = 0.893 𝑛 • 𝑛 = 481.4 𝑕 = 0.4814 𝑙𝑕 • 𝑙 = 0.9263 𝑂/𝑛 • 𝜄 = 4.4° • 𝑕 = 9.8 𝑛/𝑡 2 𝑛𝑕 sin 𝜄 2 +2𝐸𝑙𝑛𝑕 sin 𝜄 𝑛𝑕 sin 𝜄− • ∆𝑦 𝐸 = | | 𝑙 ∆𝑦− 𝑐 • 𝑢 𝑈 𝐸, ∆𝑦 = 1 𝑏∗𝑣 2𝑏 + 2𝐸 𝑏 ∗ sin −1 | − 𝑐 𝑡 𝑕 sin 𝜄 2𝑏 𝑐 2 • 𝑏 = 𝑙 𝑛 , 𝑐 = 2𝑕 sin 𝜄 , 𝑑 = 2𝐸𝑕 sin 𝜄, 𝑡 = 𝑑 + 4𝑏

DIAGRAM D Stage AB 𝛦 t - Car has not reach point of equilibrium 𝐵 Stage BC ℎ 𝑔 𝛦 x 𝐶 - Car reaches point of equilibrium where 𝐺 𝑡 = 𝐺 𝑕 sin 𝜄 Stage C 𝐷 - Car passes the point of ℎ 𝑗 equilibrium and is on the way to reaching max stretch, where 𝐺 𝑡 > 𝐺 𝑕 sin 𝜄 𝐺 𝑂 𝐺 Stage BC 𝐺 Stage AB Stage C 𝑂 𝑂 𝐺 𝐺 𝑇 𝑇 𝐺 𝑕 𝐺 𝐺 𝑕 𝑕 Diagram of the experiment with the cars at different stages

DATA TABLE FOR ∆𝑦 VS D Setting (D) x avg SD %RSD x T |% Error| E i E f |% Change| (m) (m) (m) of m avg (m) of m (J) (J) of J IV1 0.293 0.269 0.010 3.66 0.227 18.44 0.203 0.033 83.5 IV2 0.493 0.420 0.006 1.36 0.343 22.41 0.330 0.082 75.3 IV3 0.593 0.456 0.012 2.66 0.394 15.79 0.380 0.096 74.6 IV4 0.693 0.537 0.009 1.61 0.442 21.25 0.445 0.133 70.0 IV5 0.893 0.599 0.010 1.59 0.532 12.64 0.540 0.166 69.3 Avg 2.18 Avg 18.10 Avg 74.5 Table 1: Max Stretch ( 𝛦 x) vs the Different Starting Settings (D)

GRAPH FOR ∆𝑦 VS D Max Stretch Length vs. Starting Distance 0.7 𝛦 x avg = 0.6764D 0.731 R² = 0.9864 0.6 x (m) Max Stretch Length, 𝛦 x 0.5 𝛦 x T = 0.5846D 0.7652 0.4 R² = 0.9995 0.3 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Starting Distance Above Equilibrium Point, D (m) x average x theoretical Power (x average) Power (x theoretical) Graph 1: Max Stretch Length ( 𝛦 x) x) vs. Different Starting Distance (D)

DATA TABLE FOR T VS D Setting (D) t avg SD %RSD t T |% Error| (m) (s) (s) of t avg (s) of s IV1 0.293 1.87 0.051 2.73 1.18 58.39 IV2 0.493 2.14 0.076 3.56 1.50 42.38 IV3 0.593 2.40 0.102 4.24 1.63 46.87 IV4 0.693 2.36 0.075 3.16 1.75 34.40 IV5 0.893 2.60 0.111 4.29 1.97 32.02 Avg 3.60 Avg 42.81 Table 2: Time (t) vs the different starting settings (D)

GRAPH FOR T VS D Time vs. Starting Distance 3.3 t avg = 2.6845D 0.2929 2.8 R² = 0.9587 2.3 Time, t (s) 1.8 t T = 2.0736D 0.4564 R² = 1 1.3 0.8 0.2 0.4 0.6 0.8 1.0 Starting Distance Above Equilibrium Point, D (m) t average t theoretical Power (t average) Power (t theoretical) Graph 2: Time (t) vs. Different Starting Distance (D)

ANALYSIS • 𝛦 x vs D • The average percent error of 𝛦 x is approximately 18.1%, which indicates a low accuracy • The average %RSD for the experiment is approximately 2.18, indicating a high precision • The high 𝑆 2 value of 0.9864 for the data collected for 𝛦 x indicates a strong mathematical model • T vs D • The average percent error of t is approximately 42.81%, indicating a low accuracy • The average %RSD still determined high precision, with a value of approximately 3.6 • The 𝑆 2 value of 0.9587 is also high, indicating that the mathematical model is also relatively strong. • Energy Loss • The experiment lost approximately 74.5% energy, indicating a low accuracy

• The exponentials of the functions for 𝛦 x average theoretical have values of 0.731 and 0.7652 respectively, which could indicate that the equations of 𝛦 x average and theoretical matches up similarly and are modeled almost the same way, with the thing that set them apart is the difference in coefficient values. • However, the exponential of the functions for t average theoretical have values of 0.2929 and 0.4564 respectively, which sets the two equations highly different between one another. • Two pieces of data that did not make sense from the experiment was the correlation between the 𝛦 x average and t average for D values of 0.593 and 0.693 meters. • Higher D value theoretically means a higher 𝛦 x value since more energy is inputted into the system, allowing more transfer of energy towards increasing the max stretch of the bungee string. • During that process, it will also take a longer time to reach that max stretch. • Conflict with results: • When D = 0.593, 𝛦 x was lower, but time was higher • When D equals 0.693, 𝛦 x was higher, but time was lower • Could possibly expect why the curve fit didn’t having matching exponential values

CONCLUSIONS • The experiment did not find the initial hypothesis to be correct, as it can be seen from the analysis that the difference between the equations of time theoretical and time average is significant. • Sources of Error for Time • The group started the timer as the person released it. There might be a slight delay in pressing the start button, and in an experiment where the values for time are considerably small, one small error in time could greatly vary the data. • Change in k constant due to the bungee string becoming more worn out as the experiment progresses which therefore increased the 𝛦 x average and varying the time as a result. • Sources of Error for Energy Loss • Friction and Air Resistance - When the car is sliding down the ramp, friction and air resistance does negative work against the car, thus decreasing the amount of energy initially inputted. • Future Extension • Varying the car’s initial mass • Changing the angle at which the car slides down the ramp.

PHOTO The Group Conducting the Experiment

APPENDIX 1 • Derivation of Equation ∆𝑦[𝐸] • 𝑄𝐹 𝑕𝑗 = 𝑄𝐹 𝑡 + 𝑄𝐹 𝑕𝑔 • 𝑛𝑕 𝐸 + ∆𝑦 sin 𝜄 + 𝑛𝑕ℎ 𝑔 = 2 𝑙∆𝑦 2 + 𝑛𝑕ℎ 𝑔 1 1 • 𝑛𝑕 𝐸 + ∆𝑦 sin 𝜄 = 2 𝑙∆𝑦 2 • 𝑛𝑕𝐸 sin 𝜄 + 𝑛𝑕 ∆𝑦 sin 𝜄 = 1 2 𝑙∆𝑦 2 1 • 2 𝑙∆𝑦 2 − 𝑛𝑕 ∆𝑦 sin 𝜄 − 𝑛𝑕𝐸 sin 𝜄 = 0 (𝑛𝑕 sin 𝜄) 2 +4∗ 1 𝑛𝑕 sin 𝜄± 2 𝑙𝑛𝑕𝐸 sin 𝜄 • ∆𝑦 𝐸 = 2∗ 1 2 𝑙 (𝑛𝑕 sin 𝜄) 2 +2𝑙𝑛𝑕𝐸 sin 𝜄 • ∆𝑦 𝐸 = 𝑛𝑕 sin 𝜄± 𝑙 (𝑛𝑕 sin 𝜄) 2 +2𝑙𝑛𝑕𝐸 sin 𝜄 • For this equation, take the absolute value of the negative root of ∆𝑦 , which is: ∆𝑦 𝐸 = | 𝑛𝑕 sin 𝜄− | 𝑙