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16pt)[ + Ethan Simmons + ] \ + Submission Date: 02/27/23 + + Lab Section 12 \ + TA: \ +] + +#pagebreak() + + += Introduction +An Atwood machine shows the relationship between forces and acceleration. +The masses of both weights can be measured and the forces can be calculated from the measured masses and gravity. +By varying the weights and measuring acceleration, the relationship between forces and acceleration can be calculated. + + += Theory + +#grid( + columns: (50%, 50%), + rows: (auto), + box(width: 100%)[An Atwood machine consists of two weights ($m_1$ and $m_2$) connected by a string ($S$). + The string is placed on a wheel that allows the weights to move up and down. + The system can be modeled with the two free body diagrams:], + figure( + image("./001.png", width: 80%), + ) + ) + +#grid( + columns: (50%, 50%), + rows: (auto), + box(width: 100%)[ + #align(center)[ + #cetz.canvas(length: 35%, { + import cetz.draw: * + let (ForceStart, ForceGravity, ForceTension, AcelStart, Acel) = ((0,0), (0,-0.6), (0,0.8), (-0.3, 0), (-0.3, 0.3)) + set-style(mark: (end: ">", fill: black)) + line(ForceStart, ForceGravity, name: "Fg") + content("Fg.end", anchor: "south-west", padding: 0.1, [$F^G_(g,m_1)$]) + line(ForceStart, ForceTension, name: "Ft") + content("Ft.end", anchor: "north-west", padding: 0.1, [$F^T_(S,m_1)$]) + line(AcelStart, Acel, name: "a") + content("a.end", anchor: "south-east", padding: 0.1, [$a$]) + circle(ForceStart, radius: 0.02, fill: black) + }) + ] + ], + box(width: 100%)[ + #align(center)[ + #cetz.canvas(length: 35%, { + import cetz.draw: * + let (ForceStart, ForceGravity, ForceTension, AcelStart, Acel) = ((0,0), (0,-1), (0,0.8), (0.3, 0), (0.3, -0.3)) + set-style(mark: (end: ">", fill: black)) + line(ForceStart, ForceGravity, name: "Fg") + content("Fg.end", anchor: "south-west", padding: 0.1, [$F^G_(g,m_2)$]) + line(ForceStart, ForceTension, name: "Ft") + content("Ft.end", anchor: "north-west", padding: 0.1, [$F^T_(S,m_2)$]) + line(AcelStart, Acel, name: "a") + content("a.end", anchor: "north-west", padding: 0.1, [$a$]) + circle(ForceStart, radius: 0.02, fill: black) + }) + ] + ] +) + +The sum of forces in the y direction for each weight can be found by adding +the two forces in each diagram. Since these are the only forces acting on the weights + +$ sum F_(y,m_1) = F^T_(S,m_1) + F^G_(g,m_1) $ +$ sum F_(y,m_1) = F^T_(S,m_1) + F^G_(g,m_1) $ + +Taking the downward direction to be positive, $F^G_(g,m_1)$ and +$F^G_(g,m_2)$ can be found with the equation +t +$ F = m a $ +$ F^G_(g,m_1) = m_1 g $ +$ F^G_(g,m_2) = m_2 g $ + +Assuming that the string is not stretching, $m_1$ and $m_2$ are each +exerting equal forces on each of the weights + +$ F^T_(S,m_1) = F^T_(S,m_2) = F^T $ + +Since the rope is not stretching, the objects are +accelerating with the same magnitude but in +opposite directions. Using this fact, +the values for $F^G$ and $F^T$ found previously, +and that $sum F_y = m a_y$ the equations can be simplified to: + +$ m_1 a = F^T + m_1 g $ +$ -m_2 a = F^T + m_2 g $ + +The first equation can now be solved for $F^T$ and can be plugged into +the second equation + +$ F^T = m_1 a - m_1 g $ +$ -m_2 a = m_2 g - (m_1 a - m_1 g) $ +$ -m_2 a = m_2 g - m_1 a + m_1 g $ + +Isolating $a$ then gives us an equation for acceleration in terms of $m_1$ and $m_2$ + +$ m_1 a - m_2 a = m_2 g + m_1 g $ +$ a = (m_2 g + m_1 g)/(m_1 - m_2) $ + +Pulling $g$ out of the right side of the equation gives + +$ a = g ((m_2 + m_1)/(m_1 - m_2)) $ + +Using $M$ to represent $(m_2 + m_1)/(m_1 - m_2)$, the equation used for this procedure is found: + +$ a = g M $ + + += Procedure + +An Atwood Machine was created by suspending a string from a wheel attached to a lab support. +A photogate so that it was blocked multiple times while the wheel spun. +On each end of the string, weights were attached of varying masses. + +The experiment consisted of 8 trials. The first 6 trials were calculated with +varying weights for $m_1$ and $m_2 = m_1 + 0.005 unit(kilogram)$ +The value of $M$ was calculated for each trial +The weight was held up until the PASCO Capstone software was recording and then +released. The $a$ was measured using the photogate until $m_1$ neared the top +of the machine. Care was taken to make sure that the weight was dropping the +same way for each trial. + +The last 2 trials used a different difference in weight between $m_1$ and $m_2$ +this was done to try to decrease the error from the first 6 trials by changing +more than just $m_1$. Care was taken to make sure that the no damage was done to +any equipment due to the increased acceleration. + +#pagebreak() + +#align(center)[ + #box(width: 85%)[ + #set par(leading: 0.5em) + + #figure( + caption: [A table containing all of the values collected during the experiment] + )[ + #table( + inset: 9pt, + columns: (1fr, 1fr, 1fr, 2fr, 2fr), + align: horizon, + [Trial], + [$m_1$ #text(size: 0.8em, $unit(kilogram)$)], + [$m_2$ #text(size: 0.8em, $unit(kilogram)$)], + [$M$ #text(size: 0.8em, $(m_2-m_1)/(m_1+m_2)$)], + [$a$ #text(size: 0.8em, $unit(meter / second^2)$)], + [1], [0.055], [0.060], [0.043], [0.380], + [2], [0.060], [0.065], [0.040], [0.354], + [3], [0.065], [0.070], [0.037], [0.323], + [4], [0.070], [0.075], [0.034], [0.310], + [5], [0.075], [0.080], [0.032], [0.303], + [6], [0.080], [0.085], [0.030], [0.279], + [7], [0.065], [0.075], [0.071], [0.649], + [8], [0.080], [0.095], [0.086], [0.780], + ) + ] + + #figure( + image("plot1/plot9.svg"), + caption: [A graphical representation of the line of best fit of $a(M)$ from @tab_1 and the theoretical line] + ) + ] +] + += Data Analysis + +Since the function should yield a linear function with slope $g$, +an experimental value for $g$ can be found by finding the line of +best fit of the function. +$ g_"experimental" = 9.13 unit(meter/(second^2)) $ + +Comparing the calculated $g$ to the accepted $g = 9.81 unit(meter/(second^2))$ the percent +deviation can be calculated + +$ "% deviation" = abs((T - E)/T) dot 100 $ +$ "% deviation" = abs((9.81 - 9.13)/9.81) dot 100 $ +$ "% deviation" = 6.9% $ + +This error is somewhat high but is still a decent result. +The two dominant errors causing this error are systematic and random error. +The systematic error can be seen in the graph as the measured acceleration +is consistently lower than the theoretical. The primary causes of systematic error +were likely friction and air resistance. Although the systematic error has a greater +influence on the difference in the theoretical and experimental values of $a$, it +does not have as much on the calculated $g$. This is because $g$ is calculated from +the slope. + +The random error is the primary cause of the error in g and +can be seen especially in the lower values of $M$. +The primary cause of this random error was inaccuracies in +measurement. The readings from the photogate were very inconsistent. +Another possible source of random error could be differences in the process +of releasing the weight. Even with the care taken, it was likely at least a +minor source of error in the experiment. Both errors included here are somewhat +difficult to reduce given that they would require upgraded or new equipment. +However, the influence of the random error could be reduced by performing more trials. + += Conclusion + +An Atwood Machine can be a good method for determining acceleration due to gravity. +Although experimental errors caused a rather large error of 6.9%, it is still a relatively +good approximation. The results could likely be improved by running more trials to +decrease the influence of random error. Other methods could be used to decrease error +but would likely lead to a much higher complexity and the need for new measurement +equipment.