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

be greater, where time total is proportional 𝐵(sin−1(

𝐶+𝐷 𝐸𝐹+𝐺 𝐸𝐻+𝐼 ) − sin−1( 𝐽 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
• 𝐸1 = 0.293 𝑛, 𝐸2 = 0.493 𝑛, 𝐸3 = 0.593 𝑛, 𝐸4 = 0.693 𝑛, 𝐸5 = 0.893 𝑛
• 𝑛 = 481.4 𝑕 = 0.4814 𝑙𝑕
• 𝑙 = 0.9263 𝑂/𝑛
• 𝜄 = 4.4°
• 𝑕 = 9.8 𝑛/𝑡2
• ∆𝑦 𝐸 = |

𝑛𝑕 sin 𝜄− 𝑛𝑕 sin 𝜄 2+2𝐸𝑙𝑛𝑕 sin 𝜄 𝑙

|

• 𝑢𝑈 𝐸, ∆𝑦 = 1

𝑏 ∗ sin−1 𝑏∗𝑣 𝑡

|

− 𝑐

2𝑏

∆𝑦− 𝑐

2𝑏 +

2𝐸 𝑕 sin 𝜄

• 𝑏 =

𝑙 𝑛 , 𝑐 = 2𝑕 sin 𝜄 , 𝑑 = 2𝐸𝑕 sin 𝜄, 𝑡 = 𝑑 + 𝑐2 4𝑏

DIAGRAM

D 𝛦x 𝛦t ℎ𝑗 ℎ𝑔 𝐵

𝐶 𝐷

Stage AB 𝐺

𝑕

𝐺

𝑂

Stage BC 𝐺

𝑕

𝐺

𝑂

𝐺

𝑇

Stage C 𝐺

𝑕

𝐺

𝑂

𝐺

𝑇

Stage AB

• Car has not reach point of

equilibrium Stage BC

• Car reaches point of equilibrium

where 𝐺

𝑡 = 𝐺 𝑕 sin 𝜄

Stage C

• Car passes the point of

equilibrium and is on the way to reaching max stretch, where 𝐺

𝑡 > 𝐺 𝑕 sin 𝜄

Diagram of the experiment with the cars at different stages

DATA TABLE FOR ∆𝑦 VS D

Setting (D) xavg SD %RSD xT |% Error| Ei Ef |% Change| (m) (m) (m)

• f mavg

(m)

• f m

(J) (J)

• f 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 (𝛦x) vs the Different Starting Settings (D)

GRAPH FOR ∆𝑦 VS D

Graph 1: Max Stretch Length (𝛦x) x) vs. Different Starting Distance (D)

𝛦xavg = 0.6764D0.731 R² = 0.9864 𝛦xT = 0.5846D0.7652 R² = 0.9995

0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Max Stretch Length, 𝛦x x (m) Starting Distance Above Equilibrium Point, D (m)

Max Stretch Length vs. Starting Distance

x average x theoretical Power (x average) Power (x theoretical)

DATA TABLE FOR T VS D

Table 2: Time (t) vs the different starting settings (D)

Setting (D) tavg SD %RSD tT |% Error| (m) (s) (s)

• f tavg

(s)

• f 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

GRAPH FOR T VS D

Graph 2: Time (t) vs. Different Starting Distance (D)

tavg = 2.6845D0.2929 R² = 0.9587 tT = 2.0736D0.4564 R² = 1

0.8 1.3 1.8 2.3 2.8 3.3 0.2 0.4 0.6 0.8 1.0

Time, t (s) Starting Distance Above Equilibrium Point, D (m)

Time vs. Starting Distance

t average t theoretical Power (t average) Power (t theoretical)

ANALYSIS

• 𝛦x vs D
• The average percent error of 𝛦x 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 𝑆2 value of 0.9864 for the data collected for 𝛦x 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 𝑆2 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 𝛦x average theoretical have values of 0.731 and 0.7652

respectively, which could indicate that the equations of 𝛦x 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 𝛦x

average and t average for D values of 0.593 and 0.693 meters.

• Higher D value theoretically means a higher 𝛦x 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, 𝛦x was lower, but time was higher
• When D equals 0.693, 𝛦x 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 𝛦x 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

• Derivation of Equation ∆𝑦[𝐸]
• 𝑄𝐹𝑕𝑗 = 𝑄𝐹𝑡 + 𝑄𝐹𝑕𝑔
• 𝑛𝑕 𝐸 + ∆𝑦 sin 𝜄 + 𝑛𝑕ℎ𝑔 =

1 2 𝑙∆𝑦2 + 𝑛𝑕ℎ𝑔

• 𝑛𝑕 𝐸 + ∆𝑦 sin 𝜄 =

1 2 𝑙∆𝑦2

• 𝑛𝑕𝐸 sin 𝜄 + 𝑛𝑕 ∆𝑦 sin 𝜄 =

1 2 𝑙∆𝑦2

• 1

2 𝑙∆𝑦2 − 𝑛𝑕 ∆𝑦 sin 𝜄 − 𝑛𝑕𝐸 sin 𝜄 = 0

• ∆𝑦 𝐸 =

𝑛𝑕 sin 𝜄± (𝑛𝑕 sin 𝜄)2+4∗1

2𝑙𝑛𝑕𝐸 sin 𝜄

2∗1

2𝑙

• ∆𝑦 𝐸 =

𝑛𝑕 sin 𝜄± (𝑛𝑕 sin 𝜄)2+2𝑙𝑛𝑕𝐸 sin 𝜄 𝑙

• For this equation, take the absolute value of the negative root of ∆𝑦, which is: ∆𝑦 𝐸 = |

𝑛𝑕 sin 𝜄− (𝑛𝑕 sin 𝜄)2+2𝑙𝑛𝑕𝐸 sin 𝜄 𝑙

|

APPENDIX 1

• The reason for this is because ∆x is negative to indicate direction (down the ramp is being viewed as a negative value). The

equation also makes more sense with the negative root because when the value D equals 0 is plugged in (meaning the experiment starts at the equilibrium point), the output of ∆x will also be 0 meters, thus matching the car’s theoretical stretch length at the equilibrium point, which is also 0 meters.

APPENDIX 2

• Derivation of Equation of t[∆𝒚, D]
• Time can be separated into two parts: the part traveling D (starting height and the point of equilibrium), which has no force of the

bungee pulling it back, and the part traveling ∆x, where the force of the bungee increases as the car travels further down the ramp.

• Time it takes to Travel to D
• 𝐸 = 𝐸𝑝 + 𝑤𝑗𝑢 +

1 2 𝑏𝑢2

• 𝐸 =

1 2 𝑕𝑢2

• 𝑢 =

2𝐸 𝑕 sin 𝜄

• Velocity at equilibrium point
• 𝑤𝐹2 = 𝑤𝑗2 + 2𝑏𝑡
• 𝑤𝐹2 = 2𝑕 sin 𝜄 𝐸
• 𝑤𝐹 =

2𝐸𝑕 sin 𝜄

• Traveling ∆x
• 𝐺 = 𝑛𝑕 sin 𝜄 − 𝑙𝑦
• 𝑏[∆𝑦] = 𝑕 sin 𝜄 −

𝑙𝑦 𝑛

• ׬

𝑤1 𝑤2 𝑤 𝑒𝑤 = ׬ 𝑦 𝑏[𝑦]𝑒𝑦

• 1

2 𝑤2|𝑤1 𝑤2 = 𝑕 sin 𝜄 𝑦 − 𝑙𝑦2 2𝑛

• 1

2 𝑤22 − 1 2 𝑤12 = 𝑕 sin 𝜄 𝑦 − 𝑙𝑦2 2𝑛

• 𝑤22 = 2𝑕 sin 𝜄 𝑦 −

𝑙𝑦2 𝑛 + 𝑤12 – 𝑤1 - velocity at equilibrium

• 𝑤2[𝑦, 𝐸] =

2𝑕 sin 𝜄 𝑦 −

𝑙𝑦2 𝑛 + 2𝐸𝑕 sin 𝜄

• ׬

𝑢 𝑒𝑢 = ׬ ∆𝑦 1 𝑤2[𝑦,𝐸] 𝑒𝑦

• 𝑢 = ׬

∆𝑦 1 2𝑕 sin 𝜄𝑦−𝑙𝑦2

𝑛 +2𝐸𝑕 sin 𝜄

𝑒𝑦

• 𝑏 =

𝑙 𝑛 , 𝑐 = 2𝑕 sin 𝜄 , 𝑑 = 2𝐸𝑕 sin 𝜄 , 𝑡 = 𝑑 + 𝑐2 4𝑏

• 𝑢 = ׬

∆𝑦 1 −𝑏𝑦2+𝑐𝑦+𝑑 𝑒𝑦

• 𝑢 = ׬

− 𝑐

2𝑏

∆𝑦− 𝑐

2𝑏

1 −𝑏𝑣2+𝑑+𝑐2

4𝑏

𝑒𝑣

• 𝑢 = ׬

− 𝑐

2𝑏

∆𝑦− 𝑐

2𝑏

1 𝑇−𝑏𝑣2 𝑒𝑣

• 𝑢 = ׬

− 𝑐

2𝑏

∆𝑦− 𝑐

2𝑏

1 𝑇−𝑏𝑣2 𝑒𝑣

• 𝑢 𝐸, ∆𝑦 =

1 𝑏 ∗ sin−1 𝑏∗𝑣 𝑣

|

− 𝑐

2𝑏

∆𝑦− 𝑐

2𝑏

• 𝑢𝑈 𝐸, ∆𝑦 =

1 𝑏 ∗ sin−1 𝑏∗𝑣 𝑣

|

− 𝑐

2𝑏

∆𝑦− 𝑐

2𝑏 +

2𝐸 𝑕 sin 𝜄

• Derivation of k constant
• 𝑙 = 0.9263, which is the slope of the graph

APPENDIX 3

x Force (m) (N) 0.3 0.416 0.4 0.4955 0.2 0.3138 0.1 0.2346 0.6 0.7165 0.5 0.5648

F = 0.9263x + 0.1327 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Force, F (N) Change in x, (m)

Force vs X