LVang+-+L10+-+Atwood's+Machine+Lab

Lue Vang Mr. Kellogg AP Physics - Pd. 7 29 November 2011 Performed On: 23-29 November 2011 Lab Partners: Amy Weaver, Evan Weaver

**Atwood's Machine Lab ** __**Purpose: **__ The purpose of this lab is to study the acceleration present in a pulley in respect to the difference in load mass. In doing so, the lab will also study the relationship between the masses on the Atwood machine and the resulting acceleration.

__**Background: **__ In a pulley, the heavier mass will accelerate downward while the lighter one accelerates upward at the same rate. The acceleration depends on the difference in the two masses and the total mass.

__**Hypothesis: **__ According to previous life experiences, as the difference in mass increases, the acceleration of the loads will increase. As for the relationship between acceleration and total mass, it is most likely that the acceleration will decrease as the total mass increases because a larger mass will have more inertia, thus have more opposition to acceleration.

__**Apparatus: **__
 * computer (preferably high-end and does not take five hours to load)
 * Vernier computer interface (1)
 * //Logger Pro// (1)
 * Vernier Photogate with Ultra Pulley (1)
 * mass sets (2)
 * string (1)

__**Procedures: **__
 * Part 1 - **
 * 1) Set up the Atwood's machine with at least 40cm available for the mass to fall
 * 2) Connect the Photogate with Super Pulley to DIG/SONIC 1 of the interface
 * 3) Open the file "10 Atwoods Machine" in the provided Logger Pro program
 * 4) Put 200g as m2 on one side of the pulley and place 200g as m1 on the other side. Record the data and acceleration.
 * 5) Move 5g from m2 to m1 and record the data.
 * 6) Position m1 as high as possible on the pulley and make sure it's steady. Click "Collect" on Logger Pro and let m1 drop.
 * 7) Click "Examine" in Logger Pro for the region of increasing velocity on the newly generated graph. Then, click "Linear Fit" to find the slope which is also the acceleration. Record the acceleration.
 * 8) Continue to move 5g increments of mass from m2 to m1 and repeat steps 6-7 for each combination. Do this at least 5 times.


 * Part 2 - **
 * 1) Do steps 1-3 of Part 1.
 * 2) Put 120g on m1 and 100g on m2.
 * 3) Repeat steps 6-7 of Part 1.
 * 4) Add 20g increments to both sides of the pulley, keeping the difference 20g. Record the masses and repeat steps 6-7 from Part 1. Do this at least 5 times.

__**Data: **__

**(g)** || **m1** **(g)** || **Acceleration** **(m/s2)** || **Δm** **(g)** || **mT** **(g)** ||
 * **Table 1: Part 1 – Total Mass Constant** ||
 * **Trial** || **m1**
 * 1 || 202 || 200 || 0 || 2 || 402 ||
 * 2 || 207 || 195 || 0.1771 || 12 || 402 ||
 * 3 || 212 || 190 || 0.3811 || 22 || 402 ||
 * 4 || 217 || 184 || 0.6281 || 32 || 401 ||
 * 5 || 222 || 180 || 0.8297 || 42 || 402 ||

**(g)** || **m1** **(g)** || **Acceleration** **(m/s2)** || **Δm** **(g)** || **mT** **(g)** ||
 * **Table 2: Part 2 – The Mass Difference Constant** ||
 * **Trial** || **m1**
 * 1 || 120 || 100 || 0.6538 || 20 || 220 ||
 * 2 || 140 || 120 || 0.5922 || 20 || 260 ||
 * 3 || 160 || 140 || 0.4111 || 20 || 300 ||
 * 4 || 180 || 160 || 0.3829 || 20 || 340 ||
 * 5 || 200 || 180 || 0.3487 || 20 || 380 ||

__**Analysis: **__

3.) The //Acceleration vs. Time Difference// graph obviously shows that the relationship is linear. In fact, the LSRL equation is //y = 0.021x - 0.061// or as a STATS doctor would say, the value of //acceleration = 0.021(mass difference) - 0.061//. Hence, the acceleration increases as the mass difference increases.

4.) In the linear regression for this relationship for //Acceleration vs. Total Mass//, the acceleration decreases the total weight increases with a LRSL equation of //y = -0.002x + 1.092// where (y) is acceleration and (x) is total mass.

Nonetheless, in a statistical analysis, the residual plot for the linear regression of the Part 2 graph shows that a linear fit does not best represent the data because results in a parabolic nature. According to the graph, it seems a quartic regression best represents the trend of the relationship between Acceleration and total mass.

5.) If //a = 0.021(mass difference) - 0.061// and //a = -0.002(total mass) + 1.092//, then when combined//,// //2a = 0.021(mass difference) -// //0.002(total mass)// //+ 1.031//. When simplified, //a = 0.0105(mass difference) -// //0.001(total mass)// //+ 0.5155//.

If instead, the quartic equation where // y = -7E-09x 4 + 9E-06x 3 - 0.003x 2 + 0.759x - 54.27 // is used, the combined equations will be  //2a = // //0.021(mass difference)// //- 7E-09 // //(total mass) 4  + 9E-06(total mass) 3 - 0.003(total mass) 2 + 0.759(total mass) - 54.331. //

When this is simplified, //a = 0.0105(mass difference) -// //3.5E-09////(total mass) 4 + 5E-06(total mass) 3 - 0.0015(total mass) 2 + 0.3795(total mass) // //- 27.1655//.

__**Conclusion: **__

The hypothesis of this lab is supported by the data collected. As shown by graph 1 of Part 1, the acceleration increases as mass difference increases. Moreover, acceleration decreases as total mass increases.

In analysis of graph 1, this result is most likely due to the fact that F = ma, so as mass increases, the net force in one direction increases. In Part 1 of the experiment, m1 gets bigger as m2 gets small in the same increment of grams. The prior results in a greater net force in m1's direction and since F = ma, the net force gradually gets bigger as m1 gets bigger and m2 gets smaller as seen in Table 1.

In part two, a linear model did not fit the data very well. This incapability can be seen in a residual plot of the data. Instead, the residual plot suggests a higher powered equation. From examining the general shape of the graph, it seems to be following a quartic pattern. Whether the regression model is linear or quartic, it's evident that, in general, acceleration decreases as total mass increases. The pattern can also be seen in Table 2. And as seen in Table 2, the acceleration decreases at a smaller rate as the total weight increases: in between each interval, the acceleration changes in the respectively 0.0616, 0.1811, 0.0282, then 0.0342 m/s^2.

All in all, the data for Part 1 looks fine, but Part 2 could be more accurate. The trend in Part 2 still seems a bit unclear, so more trails will be beneficial to get more points for a better regression line.

 <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"> //<span style="color: black; font-family: "Calibri","sans-serif"; font-size: 14.5pt; line-height: 115%; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;">-7E-09 ////<span style="font-family: "Calibri","sans-serif"; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;">(total mass) ////<span style="color: black; font-family: "Calibri","sans-serif"; font-size: 14.5pt; line-height: 115%; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;">4 ////<span style="color: black; font-family: "Calibri","sans-serif"; font-size: 14.5pt; line-height: 115%; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;"> + 9E-06 ////<span style="font-family: "Calibri","sans-serif"; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;">(total mass) ////<span style="color: black; font-family: "Calibri","sans-serif"; font-size: 14.5pt; line-height: 115%; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;">3 ////<span style="color: black; font-family: "Calibri","sans-serif"; font-size: 14.5pt; line-height: 115%; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;"> - 0.003 ////<span style="font-family: "Calibri","sans-serif"; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;">(total mass) ////<span style="color: black; font-family: "Calibri","sans-serif"; font-size: 14.5pt; line-height: 115%; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;">2 ////<span style="color: black; font-family: "Calibri","sans-serif"; font-size: 14.5pt; line-height: 115%; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;"> + 0.759 ////<span style="font-family: "Calibri","sans-serif"; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;">(total mass) //