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2024-03-12 02:18:06 -05:00
commit 6726565ab8
16 changed files with 4887 additions and 0 deletions

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library(ggplot2)
library(readr)
library(latex2exp)
data_t1 <- read_csv(
"m1,m2,a_measured
0.055,0.060,0.380
0.060,0.065,0.354
0.065,0.070,0.323
0.070,0.075,0.310
0.075,0.080,0.303
0.080,0.085,0.279
0.065,0.075,0.649
0.080,0.095,0.780"
)
a_theoretical <- function(m) {
9.81 * m
}
m_from_masses <- function(m1, m2) {
(m2 - m1) / (m1 + m2)
}
x_lab <- function() {
TeX(r"($M$)")
}
y_lab <- function() {
TeX(r"($a \ \small{m \cdot s^{-2}}$)")
}
lm_eqn <- function(df) {
m <- lm(df)
eq <- substitute(
italic(y) == a + b %.% italic(x),
list(a = format(unname(coef(m)[1]), digits = 3),
b = format(unname(coef(m)[2]), digits = 3))
)
as.character(as.expression(eq))
}
data_t1["M"] <- m_from_masses(data_t1["m1"], data_t1["m2"])
data_t1["a"] <- data_t1$M * 9.8
plt <- ggplot(
data = data_t1,
mapping = aes(x = M, y = a_measured)
) +
geom_point(
col = "#D55E00",
size = 2,
) +
geom_smooth(
se = FALSE,
method = "lm",
linewidth = 0.8,
col = "#D55E00",
) +
theme(
axis.text = element_text(size = 14),
axis.title = element_text(size = 20),
) +
labs(
x = x_lab(),
y = y_lab(),
) +
annotate("text",
label = lm_eqn(data_t1[, c("a_measured", "M")]),
x = 0.076,
y = 0.58,
size = 5,
col = "#D55E00",
parse = TRUE,
) +
annotate("segment",
x = min(data_t1$M),
y = a_theoretical(min(data_t1$M)),
xend = max(data_t1$M),
yend = a_theoretical(max(data_t1$M)),
linewidth = 0.8,
col = "#0072B2",
) +
annotate("text",
label = "Theoretical",
x = 0.068,
y = 0.74,
col = "#0072B2",
size = 5,
)
plt
ggsave("plot.pdf",
plot = plt,
device = cairo_pdf,
)

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#import "@preview/metro:0.2.0": unit, units, metro-setup
#import units: newton, meter, second, kilogram
#import "@preview/cetz:0.2.1"
#set document(
title: "Physics Lab Report",
author: "Ethan Simmons",
)
#set text(
font: "Times New Roman",
size: 12pt,
)
#set par(
first-line-indent: 2em,
leading: 2em,
justify: true,
)
#show par: set block(spacing: 2em)
#show heading: it => {
underline(stroke: 0.05em, it)
v(0.3em)
}
#metro-setup(inter-unit-product: $dot.c$)
#box(height: 100%, width: 100%)[
#set align(horizon + center)
#set par(leading: 1em)
#text(size: 28pt)[
Atwood Machine Lab Report
]
#v(10pt)
#text(size: 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],
)
] <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]
)
]
]
= 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.