THEORY

I assigned everyone a section, feel free to change!! – Sam
Report total: 100pts
THEORY (10pts):
Swings work by converting potential energy into kinetic energy. At rest, the swing is at equilibrium. When the swing is pulled back to a certain height, potential energy due to the weight force is increased. When the swing is released from this height, potential energy is converted into kinetic energy and the swing begins to move. The motion of a swing is often compared to that of a pendulum, which is really describing a particular type of circular motion as an object travels along a circular path at a fixed radius from a center point.

But what accounts for the acceleration of a swing, and how might we increase it? Is it the weight of a child on the swing that increases or decreases the swing’s acceleration, or is it the length of the swing itself that increases or decreases the swing’s acceleration?

The acceleration of the swing can be described by the following equations:

Ft=mgsin(ϴ) where Ft. is the force along the t direction of the circle (ϴ is the displacement angle of the swing measured against the vertical)

By Newton’s Second Law, Ft=mat where a is the acceleration

So, combining these two equations we get:

mat=mgsin(ϴ) . at=gsin(ϴ)

Thus, we can clearly see that mass will not affect acceleration.

To evaluate the effect of radius on acceleration, we use the following equation for angular acceleration:

Thus, we can clearly see that swing height (L) will affect acceleration.

PROCEDURE (30pts): Explain what you did, at a level such that someone in your class could reasonably reproduce your results. Include at least one diagram.
This section still needs: diagrams and explanation of LoggerPro data analysis process
Materials needed for this lab:
1 smartphone to record swing launches
1 smartphone with Measure app to level the amplitude (angle of the swing)
Tripod with attached level
Meter stick/Measure app on iPhone
Chain clamps to change length of swing chains
Small child (GUYS THIS SOUNDS WEIRD), Zephyr
Larger child (please help me make this sound okay??), Stella
LoggerPro for data analysis

We first set up our tripod (with an attached level) and recording smartphone in a way that we could clearly record the swing’s launch for each of our four trials:
Larger child at long swing length
Larger child at short swing length
Smaller child at long swing length
Smaller child at short swing length
We began with Stella, the larger child at a long swing length. We recorded the length of the long swing chain length. Then, one group member stood behind the tripod to record and level the amplitude of the launch. We used an amplitude of 35 degrees with respect to the vertical for each trial. Once we reached the angle, our second group member released Stella and we recorded one pendulum swing at the long swing length.
We repeated this same process with Stella a second time; however this time we used a shorter swing chain length. We used chain clamps to shorten the chain length, and then we recorded the short swing chain length for our data collection. After reaching the correct amplitude, the same group member released Stella and we recorded one pendulum swing at the short swing length.
Next, we ran through the same two trials (long swing length and short swing length) with Zephyr, the smaller child. We started with the same long swing chain length we used for Stella. After reaching the correct amplitude, the same group member released Zephyr and we recorded one pendulum swing at the long swing length.
Finally, we used the same chain clamps to shorten the swing chain length to the same shortened chain length we used for Stella, confirmed the correct amplitude and the same group member released Zephyr. We recorded one pendulum swing at the short swing length.
We uploaded the videos onto Logger Pro, and using the video analysis feature, plotted the X/Y position of each child’s center of mass. The software automatically calculates X and Y velocity. We calculated X and Y acceleration by taking the derivative of the X and Y velocity.
DATA (20pts):

Starting Data
Zephyr’s Mass 15.5 kg
Stella’s Mass 19 kg
Long Swing Height 77” 土 1/16” = 195.58 cm 土 0.15875 cm
Short Swing Height 67” 土 1/16” = 170.18 cm土 0.15875 cm
Level length 17.5” 土 1/16” = 44.45 cm 土 0.15875 cm
Launch Angle 35 degrees with respect to the horizontal

Experimental Data
For each of the four tables, we have included the time stamp, X position, Y position, X velocity, Y velocity, and calculated the X acceleration and Y acceleration as derivatives of the velocity (using Logger Pro).
Zephyr – short swing

Time (s) X Y X Velocity Y Velocity Acceleration X (m/s2) Acceleration Y (m/s2)
16.68166667 124.0198438 192.3998464 30.03503852 -33.94863526 78.70257763 -60.60423092
17.415 143.7464063 171.0600026 36.05710768 -40.27391021 177.1693434 -112.3240737
17.44833333 146.8753125 163.9115651 116.5154747 -115.7867788 1431.770356 -744.8586019
17.48166667 155.93 158.7396901 207.6179108 -133.3553964 1808.297025 -189.3993129
17.515 163.2971875 154.0521901 263.9160157 -121.608073 1296.542956 315.3218333
17.54833333 173.0432813 150.5639089 297.8971354 -94.70377603 438.8530252 358.1543832
17.58166667 184.5628125 148.2826589 290.2766927 -86.13932291 -260.6342825 33.13741784
17.615 193.3635938 145.5990651 258.1119792 -98.69466149 -267.1766494 -136.7919922
17.64833333 200.6604688 141.6146901 254.4108073 -109.2024739 280.2029074 68.94260011
17.68166667 209.289375 137.3685964 281.438802 -94.8893229 766.5066195 324.6826169
17.715 219.414375 135.2162526 315.4720053 -77.63346356 970.8116322 365.3672961
17.74833333 230.4495313 132.7943776 348.6132812 -71.73828124 1009.383137 460.8588326
17.78166667 242.2464063 130.4154714 389.0299479 -61.51041665 790.0472014 907.7690969
17.815 256.7307813 127.2045339 414.6582033 -11.68945313 192.9687501 1323.052301
17.84833333 271.1839063 129.2279714 403.9908853 49.12109374 -474.0912552 1114.822048
17.88166667 284.2854688 131.8295339 369.5345052 73.73046874 -776.8337662 574.769423
17.915 295.0589063 134.7709401 349.622396 77.93945315 -900.4530167 276.9178603
17.94833333 308.8089063 137.0482839 304.0397135 78.91927082 -821.3216151 382.9752602
17.98166667 314.5784375 139.4740651 278.001302 97.02148436 -331.0275599 699.5442709
18.015 325.383125 143.0248464 301.5364585 130.2115886 -343.2554864 865.9633671
18.04833333 337.0628125 148.0170339 266.9889323 167.5097656 -690.1080201 609.5067996
18.08166667 342.7073438 154.7279714 242.7311198 182.8417968 -770.5651383 0.774956621
18.115 352.8479688 161.0990651 220.7529217 170.933384 -922.1732026 -635.0125606
18.14833333 358.4690625 167.8568776 176.9032785 114.9198573 -854.9755155 -561.2920069
18.18166667 364.5315625 166.6498464 143.3467709 113.0140608 -294.3700311 59.06216977
18.21666667 366.3635938 174.3451589 157.1120811 143.0874994 162.0076194 76.29883049
18.25 374.4104688 178.0873464 190.7294677 132.7363938 -347.4389415 -307.4539442
18.28333333 381.8401563 183.2748464 151.4675878 119.0397404 -1383.76507 -663.0408052
18.31666667 385.664375 186.9076589 73.26497395 76.3313802 -1759.805488 -695.6852607
18.35 385.664375 186.9076589 17.1061198 70.7291667 -1523.645349 -605.5990804
18.38333333 387.2932813 192.6420339 -41.50065103 41.49414062 -811.6301618 -621.1417867
18.41666667 381.2151563 189.5834401 -48.71093748 16.78710938 185.4695636 -327.3331704
18.45 382.1526563 192.4271901 -4.218749983 26.3105469 485.6632489 -214.6997071
18.48333333 382.6175 192.4271901 2.31608073 9.210611979 302.2121851 -247.8838434
18.51666667 382.6175 192.4271901 4.661458332 4.739583333 168.5356985 -194.9994574

Zephyr – long swing

Time (s) X Y X Velocity Y Velocity Acceleration X (m/s2) Acceleration Y (m/s2)
19.35 101.9807813 180.7201589 61.17628597 -67.23190656 -923.7278211 1154.785481
19.38333333 105.2698438 176.6342214 19.13834256 -10.96852884 -447.4078604 470.4161227
19.41666667 104.2229688 179.7201589 3.775006668 -6.333011522 -18.12483494 4.758228877
20.885 107.7776563 172.0365651 4.513847903 -8.837374204 9.418455941 -25.55766123
20.91833333 111.9846875 165.8256276 17.94939178 -57.88090933 165.523195 -312.7204647
20.95166667 109.8010938 164.9935964 22.08303469 -47.42403518 993.9932417 -355.530274
20.985 111.46125 163.6810964 101.1328125 -73.56770833 1332.052984 -635.6204973
21.01833333 117.3870313 160.4506276 147.2753906 -99.55729167 634.1857153 -602.0182124
21.05166667 123.6995313 156.4193776 133.4016927 -115.0227865 177.5583001 -545.3947927
21.085 125.7190625 152.8607839 129.8893229 -130.9570313 461.3172743 -576.6059028
21.11833333 130.8167188 148.1107839 165.1334635 -156.6080729 767.1929253 -516.1349826
21.15166667 136.8284375 141.9857839 199.625651 -170.1790365 607.8884549 -308.7754991
21.185 144.9495313 136.7045339 212.0768229 -179.3619792 289.5182292 -27.19726563
21.21833333 151.8128125 129.5404714 204.2545573 -172.9036458 365.6846788 270.578342
21.25166667 157.77375 124.8217214 217.281901 -154.9283854 955.8865017 372.0703125
21.285 164.4807813 119.5014089 279.2057292 -144.1113281 1290.180122 338.8427734
21.31833333 177.5471875 115.0248464 318.3691406 -130.0195313 1175.417752 257.8097873
21.35166667 185.7620313 111.4623464 353.2519531 -130.8528646 1046.044922 327.8727214
21.385 200.1878125 106.1771901 403.515625 -123.0826823 499.9077691 777.9269748
21.41833333 215.0823438 101.8021901 389.6386719 -78.39518229 -85.35427517 1185.777452
21.45166667 226.1057813 100.5248464 374.9186198 -21.06119792 -57.77452257 998.9230686
21.485 239.1409375 102.2435964 381.9661458 -9.593098958 107.0014106 722.8651259
21.51833333 251.7151563 99.47797138 387.1744792 10.94075521 148.0170356 827.9459635
21.55166667 264.555 102.3139089 401.9986979 46.56575521 -119.3684896 897.4229601
21.585 279.6448438 102.5521901 384.5345052 81.45507813 -563.7071398 695.6027561
21.61833333 290.3479688 108.2162526 358.1184896 102.1614583 -830.8892144 282.0909288
21.65166667 303.5159375 110.8178151 326.9791667 85.95377604 -918.9588759 271.6145833
21.685 313.0432813 112.8490651 281.0253906 99.72981771 -502.2650825 791.9704861
21.71833333 321.1526563 116.0756276 271.0253906 149.469401 411.7865668 1014.49924
21.75166667 328.14875 123.1928151 333.6132813 190.3450521 577.6692708 587.9747179
21.785 344.7425 129.9779714 358.9160156 195.6282552 -549.9620226 -81.39919705
21.81833333 355.9495313 137.0053151 283.1608073 175 -1335.145399 -490.0960286
21.85166667 362.8245313 141.6420339 222.7604167 150.2832031 -875.4648088 -497.8702149
21.885 368.5198438 146.5912526 219.938151 137.7148438 -190.2722603 -317.5426801
21.91833333 377.0940625 150.9037526 235.2766927 127.3502604 -144.0105867 -81.24133883
21.95166667 385.27375 154.4818776 222.9773857 133.3011848 -512.0747031 86.21590632
21.985 392.2698438 159.4506276 198.4930837 146.0699691 -844.6701177 -95.02618841
22.01833333 399.1604688 165.3842214 155.7969903 129.873026 -866.6845385 -349.5025964
22.05333333 402.2190625 168.5248464 130.9882895 108.4445918 -645.6527394 -282.9341815
22.08666667 407.6096875 171.7514089 119.127096 110.3272994 -572.3814354 -206.8064504
22.12 410.789375 176.1146901 94.52156508 106.2123902 -545.925976 -417.4896752
22.15333333 413.633125 179.5404714 77.03125 83.08268229 -414.221347 -655.6183112
22.18666667 415.39875 181.7748464 72.171875 55.88476563 -425.084329 -722.1090888
22.22 419.0003125 183.1342214 55.16601563 30.94889323 -590.5282584 -674.3404661
22.25333333 419.0003125 183.1342214 26.953125 12.78645833 -732.750651 -605.0884332

Stella – short swing

Time (s) X (m) Y (m) X Velocity (m/s) Y Velocity (m/s) Acceleration X (m/s2) Acceleration Y (m/s2)
5.9716667 132.9456 193.8764 106.15234 -162.39583 170.03526 600.96191
6.005 136.8714 187.7983 103.73535 -140.65592 407.49566 610.93587
6.0383333 138.4456 184.7475 128.21875 -128.40625 546.66097 725.85368
6.0716667 144.805 178.5288 163.72721 -99.446615 123.98627 950.47499
6.105 151.3089 177.0522 146.21745 -45.449219 -466.24051 596.18245
6.1383333 155.7542 177.0522 102.71159 -37.978516 -201.09266 -341.8826
6.1716667 157.0003 176.0366 96.01888 -76.471354 959.29905 -998.87153
6.205 158.1253 172.6811 189.98698 -134.25456 1443.5601 -768.89106
6.2383333 170.5159 165.8569 259.91536 -155.04232 199.22689 225.02441
6.2716667 181.0667 159.6889 188.3138 -102.82878 -664.662 757.74468
6.305 181.7152 159.6538 146.61458 -66.429036 274.8291 226.74696
6.3383333 187.0355 157.0795 205.05534 -80.908203 1358.6968 -512.58138
6.3716667 195.4534 155.2749 272.92643 -124.73633 1480.344 -560.09928
6.405 205.5316 148.1967 330.07487 -150.36133 789.41243 311.67535
6.4383333 219.5433 142.6538 328.26172 -102.77669 -8.9057075 1168.7581
6.4716667 228.6292 140.9506 294.60938 -40.826823 58.816189 1154.8475
6.505 236.2152 141.1967 327.22982 -10.299479 288.8058 655.48213
6.5383333 250.7855 141.1967 350.64779 -7.5195313 -293.91502 378.93054
6.5716667 263.1761 140.3959 291.07422 0.2604167 -467.7696 503.92801
6.605 267.9027 140.3959 283.85863 27.34028 219.2099 663.81511
6.6383333 279.8519 142.5014 333.21632 52.399583 263.73848 643.42799
6.6716667 292.9534 144.5561 316.42875 62.051303 -98.690049 758.80087
6.7066667 300.7933 145.0092 313.54685 109.30209 -233.13531 708.49539
6.74 314.2542 153.0639 307.05709 130.24695 -532.59428 179.68982
6.7733333 322.1644 155.4272 276.49705 107.1645 -756.42194 -16.014964
6.8066667 332.2894 158.7084 254.19596 117.22005 -890.55837 85.404185
6.84 339.9261 163.2749 210.93424 124.39779 -827.12841 -57.955366
6.8733333 345.4691 167.7475 191.27604 115.22786 -528.08402 -242.21716
6.9066667 352.3284 170.9936 182.28516 101.74479 -356.85764 -259.94466
6.94 358.2659 174.2905 163.41797 94.889323 -112.23687 -159.65169
6.9733333 362.1448 177.4858 172.97201 87.373047 58.873155 20.524089
7.0066667 369.2034 179.2827 190.57617 99.407552 -426.1556 77.338325
7.04 376.4769 184.1069 162.16797 109.6224 -1326.8446 -308.96267
7.0733333 382.1527 188.3256 77.851563 75.559896 -1556.645 -623.5498
7.1066667 380.3714 188.5874 24.833984 52.294922 -856.655 -533.52919
7.14 381.6448 191.1459 30.139974 43.997396 -316.30561 -490.68359
7.1733333 383.1917 192.2436 22.3125 21.291016 -264.69645 -468.15592
7.2066667 383.43 192.2436 11.560872 8.2291667 -248.50559 -349.60368
7.24 383.43 192.2436 4.1666667 1.8294271 -244.9311 -263.58398

Stella – long swing

Time (s) X Y X Velocity Y Velocity Acceleration X (m/s2) Acceleration Y (m/s2)
3.00333333 100.281563 185.032659 17.8190104 -56.855469 99.3999566 125.054253
3.03666667 101.066719 182.505315 16.8684896 -45.224609 292.816026 -204.87196
3.07 101.066719 182.505315 25.8164063 -50.429688 562.366536 -696.82617
3.10333333 101.551094 181.247503 62.0214844 -105.77799 639.21658 -836.78928
3.13666667 105.68 174.110784 86.2141927 -128.84766 300.701497 -402.94868
3.17 108.969063 171.341253 70.5501302 -119.55078 289.138997 -266.04546
3.20333333 109.480781 167.142034 80.2539063 -132.26237 927.335612 -478.58615
3.23666667 112.519844 163.134221 137.639974 -159.3099 1433.96539 -521.55219
3.27 118.613594 156.102971 199.570313 -178.35612 1315.14214 -258.10818
3.30333333 127.097969 150.419378 230.947266 -173.72396 978.776042 -9.2393663
3.33666667 134.265938 145.056096 251.933594 -177.34701 904.966905 193.318685
3.37 142.894844 138.005315 293.916016 -164.30013 760.818142 429.717339
3.40333333 154.621406 133.481878 314.6875 -137.46094 323.185221 364.713542
3.43666667 164.644844 129.907659 311.529948 -134.82747 26.6140408 149.932183
3.47 175.472969 124.64594 306.01888 -138.35938 82.7528212 172.097439
3.50333333 185.351875 119.575628 297.988281 -118.28451 659.714084 113.178168
3.53666667 193.066719 118.075628 349.671224 -127.38932 1226.64931 20.719401
3.57 208.418281 111.181096 410.143229 -136.88477 982.804362 421.004232
3.60333333 222.543281 106.852971 417.633464 -94.205729 589.211697 702.376302
3.63666667 235.441719 105.970159 439.270833 -75.022786 408.365885 622.718641
3.67 252.08625 102.204534 449.876302 -61.591797 161.56684 740.454102
3.70333333 266.2425 101.040471 442.278646 -29.583333 156.206597 969.637044
3.73666667 280.832344 100.306096 453.11849 4.44335938 316.465929 1209.06576
3.77 296.195625 101.278753 470.498047 46.7578125 239.615885 1493.37294
3.80333333 312.301094 102.513128 482.220052 109.700521 -271.17242 1516.82943
3.83666667 328.96125 108.231878 468.024089 174.908854 -1216.6938 831.811523
3.87 345.308906 116.876409 398.264974 169.664714 -1970.0792 66.0508898
3.90333333 356.58625 120.392034 303.697917 141.588542 -1730.0727 266.175673
3.93666667 364.219063 123.587346 249.082031 178.447266 -587.94217 730.04286
3.97 369.734688 132.212346 284.752604 219.661458 18.375651 472.409397
4.00333333 384.226875 140.079534 298.59375 215.810547 -664.24154 -16.015625
4.03666667 392.769844 146.376409 232.014974 209.866536 -1226.3645 -346.24566
4.07 398.390938 154.087346 189.759115 195.452474 -1064.8166 -654.53559
4.10333333 405.066719 160.388128 159.215495 159.508464 -731.07096 -671.8533
4.13666667 409.164375 164.829534 133.567708 126.634115 -222.68066 -82.866753
4.17 411.96125 166.399846 157.246094 154.970703 -124.40592 426.972114
4.20333333 420.750313 174.73969 151.337891 193.639323 -713.70985 -77.655707
4.23666667 423.922188 182.274846 97.7864583 156.705729 -1026.9043 -816.02376
4.27 426.246406 185.200628 64.827474 113.180339 -856.11979 -918.86664
4.30333333 427.777656 188.767034 44.1210938 91.0677083 -680.05914 -831.26899
4.33666667 429.594063 191.626409 19.2513021 63.2454427 -474.71625 -804.69618
4.37 428.52375 193.126409 9.15039063 34.2545573 -214.05626 -690.97385
4.40333333 429.762031 193.634221 11.3606771 12.8841146 -125.1766 -464.74555
4.43666667 429.762031 193.634221 3.26367188 3.27734375 -115.72347 -240.75195
4.47 429.762031 193.634221 1.54785156 0.63476563 -73.784451 -110.34532
4.50333333 429.762031 193.634221 0 0 -58.470595 -47.381727

ANALYSIS (20pts):
Explain how you got from your data to your result. No need to show every step of your
derivations, but give enough explanation that a classmate could reproduce your results.
Our analysis began with redirecting our focus on the law which we were testing and proving through our experiment.

CONCLUSION (20pts):
What did you find? Numbers should include uncertainties. What went wrong? What might have affected your results (sources of error)? How could the experiment be improved?