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report.typ
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report.typ
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#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 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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#show heading: it => {
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underline(stroke: 0.05em, it)
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v(0.3em)
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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: 02/27/23
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Lab Section 12 \
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TA: \
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]
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#pagebreak()
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= Introduction
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An Atwood machine shows the relationship between forces and acceleration.
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The masses of both weights can be measured and the forces can be calculated from the measured masses and gravity.
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By varying the weights and measuring acceleration, the relationship between forces and acceleration can be calculated.
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= Theory
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#grid(
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columns: (50%, 50%),
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rows: (auto),
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box(width: 100%)[An Atwood machine consists of two weights ($m_1$ and $m_2$) connected by a string ($S$).
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The string is placed on a wheel that allows the weights to move up and down.
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The system can be modeled with the two free body diagrams:],
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figure(
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image("./001.png", width: 80%),
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)
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)
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#grid(
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columns: (50%, 50%),
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rows: (auto),
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box(width: 100%)[
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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: "south-east", padding: 0.1, [$a$])
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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(width: 100%)[
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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: "north-west", padding: 0.1, [$a$])
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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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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. Since these are the only forces acting on the weights
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$ sum F_(y,m_1) = F^T_(S,m_1) + F^G_(g,m_1) $
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$ sum F_(y,m_1) = F^T_(S,m_1) + F^G_(g,m_1) $
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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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t
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$ F = m a $
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$ F^G_(g,m_1) = m_1 g $
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$ F^G_(g,m_2) = m_2 g $
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Assuming that the string is not stretching, $m_1$ and $m_2$ are each
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exerting equal forces on each of the weights
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$ F^T_(S,m_1) = F^T_(S,m_2) = F^T $
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Since the rope is not stretching, the objects are
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accelerating with the same magnitude but in
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opposite directions. Using this fact,
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the values for $F^G$ and $F^T$ found previously,
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and that $sum F_y = m a_y$ the equations can be simplified to:
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$ m_1 a = F^T + m_1 g $
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$ -m_2 a = F^T + m_2 g $
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The first equation can now be solved for $F^T$ and can be plugged into
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the second equation
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$ F^T = m_1 a - m_1 g $
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$ -m_2 a = m_2 g - (m_1 a - m_1 g) $
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$ -m_2 a = m_2 g - m_1 a + m_1 g $
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Isolating $a$ then gives us an equation for acceleration in terms of $m_1$ and $m_2$
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$ m_1 a - m_2 a = m_2 g + m_1 g $
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$ a = (m_2 g + m_1 g)/(m_1 - m_2) $
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Pulling $g$ out of the right side of the equation gives
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$ a = g ((m_2 + m_1)/(m_1 - m_2)) $
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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 = g M $
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= Procedure
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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 so that it was blocked multiple times while the wheel spun.
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On each end of the string, weights were attached of varying masses.
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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 difference 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 changing
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more than just $m_1$. Care was taken to make sure that the no damage was done to
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any equipment due to the increased acceleration.
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#pagebreak()
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#align(center)[
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#box(width: 85%)[
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#set par(leading: 0.5em)
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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/plot9.svg"),
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caption: [A graphical representation of the line of best fit of $a(M)$ 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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Since 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)) $
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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 $
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$ "% deviation" = abs((9.81 - 9.13)/9.81) dot 100 $
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$ "% deviation" = 6.9% $
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This error is somewhat high but is still a decent result.
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The two dominant errors causing this error are systematic and random error.
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The systematic error can be seen in the graph as the measured acceleration
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is consistently lower than the theoretical. The primary causes of systematic error
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were likely friction and air resistance. Although the systematic error has a greater
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influence on the difference in the theoretical and experimental values of $a$, it
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does not have as much on the calculated $g$. This is because $g$ is calculated from
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the slope.
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The random error is the primary cause of the error in g and
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can be seen especially in the lower values of $M$.
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The primary cause of this random error was inaccuracies in
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measurement. The readings from the photogate were very inconsistent.
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Another possible source of random error could be differences in the process
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of releasing the weight. Even with the care taken, it was likely at least a
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minor source of error in the experiment. Both errors included here are somewhat
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difficult to reduce given that they would require upgraded or new equipment.
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However, the influence of the random error could be reduced by performing more trials.
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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 relatively
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good 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 and the need for new measurement
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equipment.
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