319 lines
11 KiB
Typst
319 lines
11 KiB
Typst
#import "@preview/metro:0.2.0": unit, units, metro-setup
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#import units: newton, meter, second, kilogram
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#import "@preview/cetz:0.2.1"
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#set document(
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title: "Physics Lab Report",
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author: "Ethan Simmons",
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)
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#set page(numbering: "1")
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#set text(
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font: "Times New Roman",
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size: 12pt,
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)
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#set par(
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first-line-indent: 2em,
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leading: 2em,
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justify: true,
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)
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#show par: set block(spacing: 2em)
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#set heading(numbering: "1.")
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#set math.equation(numbering: it => {
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locate(loc => {
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let count = counter(heading).at(loc).last()
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numbering("(1.1)", count, it)
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})
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})
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#show heading: it => {
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let count = locate(loc => [
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#text()[#counter(heading).at(loc).last()]#text(".")#h(0.05em)
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])
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set text(size: 16pt)
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v(0.5em)
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block()[
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#count
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#underline(it.body)
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]
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v(0.5em)
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counter(math.equation).update(0)
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}
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#show figure.caption: it => {
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set par(leading: 0.8em)
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it
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}
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#metro-setup(inter-unit-product: $dot.c$)
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#box(height: 100%, width: 100%)[
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#set align(horizon + center)
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#set par(leading: 1em)
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#text(size: 28pt)[
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Atwood Machine Lab Report
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]
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#v(10pt)
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#text(size: 16pt)[
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Ethan Simmons
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] \
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Submission Date: 03/13/23
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Lab Section 12 \
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TA: Seburne
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]
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#pagebreak()
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= Introduction
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An Atwood's machine shows the relationship between forces and acceleration.
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It consists of two weights connected by a string. By changing the mass of these weights,
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the acceleration can be measured. Analyzing the measured accelerations and weights used
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allows for an experimental measurement of gravity.
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= Theory
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The forces in an Atwoods machine can be modeled by drawing a free body diagram for each weight ($m_1$ and $m_2$).
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#figure(
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grid(
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columns: (50%, 50%),
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rows: (auto),
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box(height: 12.5em)[
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#align(center)[
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#cetz.canvas(length: 35%, {
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import cetz.draw: *
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let (ForceStart, ForceGravity, ForceTension, AcelStart, Acel) = ((0,0), (0,-0.6), (0,0.8), (-0.3, 0), (-0.3, 0.3))
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set-style(mark: (end: ">", fill: black))
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line(ForceStart, ForceGravity, name: "Fg")
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content("Fg.end", anchor: "south-west", padding: 0.1, [$F^G_(g,m_1)$])
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line(ForceStart, ForceTension, name: "Ft")
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content("Ft.end", anchor: "north-west", padding: 0.1, [$F^T_(S,m_1)$])
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line(AcelStart, Acel, name: "a")
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content("a.end", anchor: "north-east", padding: 0.13, [$a$])
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circle(ForceStart, radius: 0.02, fill: black)
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})
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]
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],
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box(height: 12.5em)[
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#align(center)[
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#cetz.canvas(length: 35%, {
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import cetz.draw: *
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let (ForceStart, ForceGravity, ForceTension, AcelStart, Acel) = ((0,0), (0,-1), (0,0.8), (0.3, 0), (0.3, -0.3))
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set-style(mark: (end: ">", fill: black))
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line(ForceStart, ForceGravity, name: "Fg")
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content("Fg.end", anchor: "south-west", padding: 0.1, [$F^G_(g,m_2)$])
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line(ForceStart, ForceTension, name: "Ft")
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content("Ft.end", anchor: "north-west", padding: 0.1, [$F^T_(S,m_2)$])
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line(AcelStart, Acel, name: "a")
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content("a.end", anchor: "south-west", padding: 0.13, [$a$])
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circle(ForceStart, radius: 0.02, fill: black)
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})
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]
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]
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),
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caption: [The free body diagrams for $m_1$ and $m_2$ where $m_2 > m_1$]
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)
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#v(1em)
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The sum of forces in the y direction for each weight can be found by adding
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the two forces in each diagram.
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$ sum F_(y,m_1) = F^T_(S,m_1) + F^G_(g,m_1) $ <eq-1>
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$ sum F_(y,m_2) = F^T_(S,m_2) + F^G_(g,m_2) $ <eq-2>
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Taking the downward direction to be positive, $F^G_(g,m_1)$ and
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$F^G_(g,m_2)$ can be found with the equation:
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$ F = m a $ <eq-3>
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$ F^G_(g,m_1) = m_1 g $ <eq-4>
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$ F^G_(g,m_2) = m_2 g $ <eq-5>
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Since the string is not stretching, $m_1$ and $m_2$ are each
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exerting equal forces on the string
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$ F^T_(m_1,S) = F^T_(m_2,S) = F^T_(m,S) $ <eq-6>
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Since the tension force acting on the weight and the force that the weight exerts on
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the string is a force pair, the forces by the string acting on the weights can be found:
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$ F^T_(S,m) = F^T_(m,S) $ <eq-7>
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Using the values found in @eq-4, @eq-5, @eq-7,
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the equations can be simplified to:
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$ sum F_(y,m_1) = F^T_(S,m) + m_1 g $ <eq-8>
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$ sum F_(y,m_2) = F^T_(S,m) + m_2 g $ <eq-9>
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Using $sum F_y = m a_y$, the forces can now be related
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to the weights' accelerations
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$ m_1 a_(y,m_1) = F^T_S + m_1 g $ <eq-10>
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$ m_2 a_(y,m_2) = F^T_S + m_2 g $ <eq-11>
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Given that the string is still not stretching and that
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the weights' masses are not the same, the acceleration of
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the two weights should be equal in magnitude but opposite
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in direction
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$ - m_1 a_y = F^T_S + m_1 g $ <eq-12>
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$ m_2 a_y = F^T_S + m_2 g $ <eq-13>
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@eq-12 can now be solved for $F^T$ and can be plugged into @eq-13
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$ F^T = - m_1 a_y - m_1 g $ <eq-14>
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$ -m_1 a_y = m_2 g + (- m_1 a_y - m_1 g) $ <eq-15>
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$ m_2 a_y = m_2 g - m_1 a_y - m_1 g $ <eq-16>
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Isolating $a$ then gives an equation for acceleration in terms of $m_1$ and $m_2$
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$ m_1 a_y + m_2 a_y = m_2 g - m_1 g $ <eq-17>
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$ a_y = (m_2 g - m_1 g)/(m_2 + m_1) $ <eq-18>
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Pulling $g$ out of the right side of the equation gives
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$ a_y = g ((m_2 - m_1)/(m_1 + m_2)) $ <eq-19>
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Using $M$ to represent $(m_2 - m_1)/(m_1 + m_2)$, the equation used for this procedure is found:
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$ a_y = g M $ <eq-20>
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This equation will be used in the procedure using $M$ as the independent
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variable, and $a$ as the dependent variable to represent the theoretical line.
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This equation can be used to find the accuracy of the results.
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#align(center)[
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#figure(
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image("./plot2/plot.png", width: 60%),
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caption: [A sketch of the theoretical line for $a(M)$]
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)
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]
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= Procedure
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#grid(
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columns: (40%, 60%),
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rows: (auto),
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box(width: 100%)[
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An Atwood Machine was created by suspending a string from a wheel attached to a lab support.
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A photogate was set up so that it was blocked multiple times as the wheel spun.
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On each end of the string, weights were attached of varying masses.
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],
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figure(
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image("./001.png", width: 80%),
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caption: [A model of the Atwood machine used in the procedure]
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)
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)
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#h(2em)The experiment consisted of 8 trials. The first 6 trials were calculated with
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varying weights for $m_1$ and $m_2 = m_1 + 0.005 unit(kilogram)$
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The value of $M$ was calculated for each trial
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The weight was held up until the PASCO Capstone software was recording and then
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released. The $a$ was measured using the photogate until $m_1$ neared the top
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of the machine. Care was taken to make sure that the weight was dropping the
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same way for each trial.
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The last 2 trials used a different change in weight between $m_1$ and $m_2$
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this was done to try to decrease the error from the first 6 trials by calculating
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with values of $M$ greater than in the first trials. Care was taken to
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make sure that the no damage was done to any equipment due to the increased
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acceleration. The acceleration data was collected from the
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PASCO Capstone software and written down for later use. $M$ was then
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calculated from the masses of the weights used for the trial.
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#pagebreak()
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#align(center)[
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#box(width: 100%)[
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#figure(
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caption: [A table containing all of the values collected during the experiment]
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)[
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#table(
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inset: 9pt,
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columns: (1fr, 1fr, 1fr, 2fr, 2fr),
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align: horizon,
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[Trial],
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[$m_1$ #text(size: 0.8em, $unit(kilogram)$)],
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[$m_2$ #text(size: 0.8em, $unit(kilogram)$)],
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[$M$ #text(size: 0.8em, $(m_2-m_1)/(m_1+m_2)$)],
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[$a$ #text(size: 0.8em, $unit(meter / second^2)$)],
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[1], [0.055], [0.060], [0.043], [0.380],
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[2], [0.060], [0.065], [0.040], [0.354],
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[3], [0.065], [0.070], [0.037], [0.323],
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[4], [0.070], [0.075], [0.034], [0.310],
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[5], [0.075], [0.080], [0.032], [0.303],
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[6], [0.080], [0.085], [0.030], [0.279],
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[7], [0.065], [0.075], [0.071], [0.649],
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[8], [0.080], [0.095], [0.086], [0.780],
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)
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] <tab_1>
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#figure(
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image("plot1/plot2.png"),
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caption: [A graphical representation of the measured values, the line of best fit of
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for the measured values from @tab_1, and the theoretical line]
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)
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]
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]
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= Data Analysis
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@eq-20 shows the function should yield a linear function with slope $g$,
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an experimental value for $g$ can be found by finding the line of
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best fit of the function.
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$ g_("experimental") = 9.13 unit(meter/(second^2)) $ <eq-18>
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Comparing the calculated $g$ to the accepted $g = 9.81 unit(meter/(second^2))$ the percent
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deviation can be calculated
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$ "% deviation" = abs((T - E)/T) dot 100 $ <eq-21>
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$ "% deviation" = abs((9.81 - 9.13)/9.81) dot 100 $ <eq-22>
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$ "% deviation" = 6.9% $ <eq-23>
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This error is relatively high but is still a reasonable result.
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Two causes of error are air resistance and friction. Both are systematic errors
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that cause the calculated $g$ to be lower than the theoretical $g$.
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The systematic error can be seen in the graph by the decreased slope
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compared to the theoretical line. However, the decreased slope could also be
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caused by the random error. Both errors are very difficult to remove completely.
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The best way to improve these errors would be to account for them in the calculations.
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However, this would increase the complexity of the procedure exponentially as accounting
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for it would require many more measurements to find the friction and air resistance.
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Another possible method would be to use more specialized equipment.
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The error due to air resistance could be almost completely removed by running the experiment
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in a vacuum. The error due to friction could be reduced by using a more
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efficient bearing to allow the wheel to turn.
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Another likely dominant cause of error is measurement inaccuracy.
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The photogate was reading very inconsistent values and trials often had to be rerun
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before they yielded a usable result. The measurement inaccuracy is
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mostly random error. If there is systematic error caused by it,
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there are not enough trials to know in which direction it skewed
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the results. This error is also hard to remove completely, but
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could easily be improved by running more trials. This would decrease
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the random error as running more trials will bring the result closer to the average.
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Another simple but expensive way to decrease this error would be to use better equipment.
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= Conclusion
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An Atwood Machine can be a good method for determining acceleration due to gravity.
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Although experimental errors caused a rather large error of 6.9%, it is still a reasonable
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approximation. The results could likely be improved by running more trials to
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decrease the influence of random error. Other methods could be used to decrease error
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but would likely lead to a much higher complexity, the need for new measurement
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equipment, or both.
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