254 lines
8.4 KiB
Typst
254 lines
8.4 KiB
Typst
#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.
|