Aweaver+Atwoods+Machine


 * Atwood's **** Machine **** Lab: **

The purpose of the lab is to study acceleration in a pulley system with respect to varying masses. This can be used to analyze the relationship masses m1 and m2 on the Atwood's machine and the resulting accelerations.
 * __Purpose__: **

__** Background: **__ In a pulley system, the larger mass will accelerate downward while the lighter one accelerates upward. The accelerations occur at the same rate. Also, the overall acceleration depends on the difference between m1 and m2 and the total mass of the system.

=__** Hypothesis: ** __= As the difference between m1 and m2 increases, the acceleration of the system will increase. Also, as total mass increases acceleration should decrease because it is more difficult to initiate motion of heavier objects. =__** Apparatus: ** __= =__** Procedure: ** __= =** Part 1 - ** = ** Part 2 - **
 * Vernier computer interface (1)
 * Computer with //Logger Pro// (1)
 * Vernier Photogate with Ultra Pulley (1)
 * mass sets (2)
 * string (1)
 * Ringstand (1)
 * Clamp (1)
 * Small cups (2)
 * 1) Set up the Atwood's machine, at least 40cm above floor
 * 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) Place m1 of 200g on one side of the pulley and m2 of 200g on the other side
 * 5) Position m1 as high as possible and click "collect" on Logger Pro
 * 6) Let m1 drop
 * 7) Record the data and acceleration
 * 8) Move 5g from m2 to m1 and record the data.
 * 9) 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.
 * 10) Continue to move 5g increments of mass from m2 to m1 and repeat steps 6-7 for each combination.
 * 11) Repeat this process five times.
 * 1) Repeat steps 1-3 of procedure part 1
 * 2) Place 120 on m1 and 100 on m2.
 * 3) Repeat steps 6-7 of procedure part 1
 * 4) Add 20g increments to both sides of the pulley, keeping the difference 20g.
 * 5) Record the masses and repeat steps 6-7 from Part 1.
 * 6) Repeat this process five 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. mass difference graph shows a clear linear relationship. This means that the change in acceleration is directly proportional to the difference in mass between m1 and m2. 4.) In the linear regression for the graph of acceleration vs. total mass, the acceleration decreases as the total weight within the system increases with the regression equation of // y = -0.002x + 1.092 // where (y) is acceleration and (x) is total mass.

However, the residual plot for the linear regression of the Part 2 graph shows that a linear fit does not best represent the data. The results are more parabolic in nature. According to the graph, it seems a quartic regression best represents the trend of the relationship between acceleration and total mass in the Atwood's machine.

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. It was determined that acceleration increases as the difference between m1 and m2 increases. This is probably due to the fact that F=ma. Therefore, as mass increase, net force increases in the direction of m1, causing a greater acceleration. Also, it was found that acceleration decreases as the total mass of the system increases. As the total mass became greater, the acceleration decreased in a non-linear fashion.