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Final Draft (Hopefully)

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Ethan Simmons
2024-03-13 21:59:22 -05:00
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@@ -7,6 +7,8 @@
author: "Ethan Simmons",
)
#set page(numbering: "1")
#set text(
font: "Times New Roman",
size: 12pt,
@@ -18,26 +20,29 @@
justify: true,
)
#show par: set block(spacing: 2em)
#set heading(numbering: "1.")
#set math.equation(numbering: it => {
locate(loc => {
let count = counter(heading).at(loc).last()
numbering("1.1", count, it)
numbering("(1.1)", count, it)
})
})
#show heading: it => {
let count = locate(loc => [
#counter(heading).at(loc).last()
#text(")")
#text()[#counter(heading).at(loc).last()]#text(".")#h(0.05em)
])
set text(size: 16pt)
v(0.5em)
block()[
#count
#underline(it.body)
]
v(0.5em)
counter(math.equation).update(0)
}
#show figure.caption: it => {
@@ -61,30 +66,150 @@
#text(size: 16pt)[
Ethan Simmons
] \
Submission Date: 02/27/23
Submission Date: 03/13/23
Lab Section 12 \
TA: \
TA: Seburne
]
#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.
An Atwood's machine shows the relationship between forces and acceleration.
It consists of two weights connected by a string. By changing the mass of these weights,
the acceleration can be measured. Analyzing the measured accelerations and weights used
allows for an experimental measurement of gravity.
= Theory
The forces in an Atwoods machine can be modeled by drawing a free body diagram for each weight ($m_1$ and $m_2$).
#figure(
grid(
columns: (50%, 50%),
rows: (auto),
box(height: 12.5em)[
#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: "north-east", padding: 0.13, [$a$])
circle(ForceStart, radius: 0.02, fill: black)
})
]
],
box(height: 12.5em)[
#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: "south-west", padding: 0.13, [$a$])
circle(ForceStart, radius: 0.02, fill: black)
})
]
]
),
caption: [The free body diagrams for $m_1$ and $m_2$ where $m_2 > m_1$]
)
#v(1em)
The sum of forces in the y direction for each weight can be found by adding
the two forces in each diagram.
$ sum F_(y,m_1) = F^T_(S,m_1) + F^G_(g,m_1) $ <eq-1>
$ sum F_(y,m_2) = F^T_(S,m_2) + F^G_(g,m_2) $ <eq-2>
Taking the downward direction to be positive, $F^G_(g,m_1)$ and
$F^G_(g,m_2)$ can be found with the equation:
$ F = m a $ <eq-3>
$ F^G_(g,m_1) = m_1 g $ <eq-4>
$ F^G_(g,m_2) = m_2 g $ <eq-5>
Since the string is not stretching, $m_1$ and $m_2$ are each
exerting equal forces on the string
$ F^T_(m_1,S) = F^T_(m_2,S) = F^T_(m,S) $ <eq-6>
Since the tension force acting on the weight and the force that the weight exerts on
the string is a force pair, the forces by the string acting on the weights can be found:
$ F^T_(S,m) = F^T_(m,S) $ <eq-7>
Using the values found in @eq-4, @eq-5, @eq-7,
the equations can be simplified to:
$ sum F_(y,m_1) = F^T_(S,m) + m_1 g $ <eq-8>
$ sum F_(y,m_2) = F^T_(S,m) + m_2 g $ <eq-9>
Using $sum F_y = m a_y$, the forces can now be related
to the weights' accelerations
$ m_1 a_(y,m_1) = F^T_S + m_1 g $ <eq-10>
$ m_2 a_(y,m_2) = F^T_S + m_2 g $ <eq-11>
Given that the string is still not stretching and that
the weights' masses are not the same, the acceleration of
the two weights should be equal in magnitude but opposite
in direction
$ - m_1 a_y = F^T_S + m_1 g $ <eq-12>
$ m_2 a_y = F^T_S + m_2 g $ <eq-13>
@eq-12 can now be solved for $F^T$ and can be plugged into @eq-13
$ F^T = - m_1 a_y - m_1 g $ <eq-14>
$ -m_1 a_y = m_2 g + (- m_1 a_y - m_1 g) $ <eq-15>
$ m_2 a_y = m_2 g - m_1 a_y - m_1 g $ <eq-16>
Isolating $a$ then gives an equation for acceleration in terms of $m_1$ and $m_2$
$ m_1 a_y + m_2 a_y = m_2 g - m_1 g $ <eq-17>
$ a_y = (m_2 g - m_1 g)/(m_2 + m_1) $ <eq-18>
Pulling $g$ out of the right side of the equation gives
$ a_y = g ((m_2 - m_1)/(m_1 + m_2)) $ <eq-19>
Using $M$ to represent $(m_2 - m_1)/(m_1 + m_2)$, the equation used for this procedure is found:
$ a_y = g M $ <eq-20>
This equation will be used in the procedure using $M$ as the independent
variable, and $a$ as the dependent variable to represent the theoretical line.
This equation can be used to find the accuracy of the results.
#align(center)[
#figure(
image("./plot2/plot.png", width: 60%),
caption: [A sketch of the theoretical line for $a(M)$]
)
]
= Procedure
#grid(
columns: (50%, 50%),
columns: (40%, 60%),
rows: (auto),
box(width: 100%)[
= T#h(0.02em)heory
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:
An Atwood Machine was created by suspending a string from a wheel attached to a lab support.
A photogate was set up so that it was blocked multiple times as the wheel spun.
On each end of the string, weights were attached of varying masses.
],
figure(
image("./001.png", width: 80%),
@@ -92,97 +217,8 @@ By varying the weights and measuring acceleration, the relationship between forc
)
)
#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) $ <eq-1>
$ sum F_(y,m_1) = F^T_(S,m_1) + F^G_(g,m_1) $ <eq-2>
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 $ <eq-3>
$ F^G_(g,m_1) = m_1 g $ <eq-4>
$ F^G_(g,m_2) = m_2 g $ <eq-5>
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 $ <eq-6>
Since the rope is not stretching, the objects are
accelerating with the same magnitude but in
opposite directions. Using this fact,
the values found in @eq-4, @eq-5, and @eq-6,
and that $sum F_y = m a_y$ the equations can be simplified to:
$ m_1 a = F^T + m_1 g $ <eq-7>
$ -m_2 a = F^T + m_2 g $ <eq-8>
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 $ <eq-9>
$ -m_2 a = m_2 g - (m_1 a - m_1 g) $ <eq-10>
$ -m_2 a = m_2 g - m_1 a + m_1 g $ <eq-11>
Isolating $a$ then gives an equation for acceleration in terms of $m_1$ and $m_2$
$ m_1 a - m_2 a = m_2 g + m_1 g $ <eq-12>
$ a = (m_2 g + m_1 g)/(m_1 - m_2) $ <eq-13>
Pulling $g$ out of the right side of the equation gives
$ a = g ((m_2 + m_1)/(m_1 - m_2)) $ <eq-14>
Using $M$ to represent $(m_2 + m_1)/(m_1 - m_2)$, the equation used for this procedure is found:
$ a = g M $ <eq-15>
= 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
#h(2em)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
@@ -190,15 +226,18 @@ 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.
The last 2 trials used a different change in weight between $m_1$ and $m_2$
this was done to try to decrease the error from the first 6 trials by calculating
with values of $M$ greater than in the first trials. Care was taken to
make sure that the no damage was done to any equipment due to the increased
acceleration. The acceleration data was collected from the
PASCO Capstone software and written down for later use. $M$ was then
calculated from the masses of the weights used for the trial.
#pagebreak()
#align(center)[
#box(width: 85%)[
#box(width: 100%)[
#figure(
caption: [A table containing all of the values collected during the experiment]
)[
@@ -223,50 +262,57 @@ any equipment due to the increased acceleration.
] <tab_1>
#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]
image("plot1/plot2.png"),
caption: [A graphical representation of the measured values, the line of best fit of
for the measured values from @tab_1, and the theoretical line]
)
]
]
= Data Analysis
Since the function should yield a linear function with slope $g$,
@eq-20 shows 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)) $ <eq-16>
$ g_("experimental") = 9.13 unit(meter/(second^2)) $ <eq-18>
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 $ <eq-17>
$ "% deviation" = abs((9.81 - 9.13)/9.81) dot 100 $ <eq-18>
$ "% deviation" = 6.9% $ <eq-19>
$ "% deviation" = abs((T - E)/T) dot 100 $ <eq-21>
$ "% deviation" = abs((9.81 - 9.13)/9.81) dot 100 $ <eq-22>
$ "% deviation" = 6.9% $ <eq-23>
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.
This error is relatively high but is still a reasonable result.
Two causes of error are air resistance and friction. Both are systematic errors
that cause the calculated $g$ to be lower than the theoretical $g$.
The systematic error can be seen in the graph by the decreased slope
compared to the theoretical line. However, the decreased slope could also be
caused by the random error. Both errors are very difficult to remove completely.
The best way to improve these errors would be to account for them in the calculations.
However, this would increase the complexity of the procedure exponentially as accounting
for it would require many more measurements to find the friction and air resistance.
Another possible method would be to use more specialized equipment.
The error due to air resistance could be almost completely removed by running the experiment
in a vacuum. The error due to friction could be reduced by using a more
efficient bearing to allow the wheel to turn.
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.
Another likely dominant cause of error is measurement inaccuracy.
The photogate was reading very inconsistent values and trials often had to be rerun
before they yielded a usable result. The measurement inaccuracy is
mostly random error. If there is systematic error caused by it,
there are not enough trials to know in which direction it skewed
the results. This error is also hard to remove completely, but
could easily be improved by running more trials. This would decrease
the random error as running more trials will bring the result closer to the average.
Another simple but expensive way to decrease this error would be to use better equipment.
= 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
Although experimental errors caused a rather large error of 6.9%, it is still a reasonable
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.
but would likely lead to a much higher complexity, the need for new measurement
equipment, or both.