diff --git a/plot1/plot.R b/plot1/plot.R
index a50998a..112cc29 100644
--- a/plot1/plot.R
+++ b/plot1/plot.R
@@ -14,6 +14,9 @@ data_t1 <- read_csv(
0.080,0.095,0.780"
)
+data_t1$cat <- rep("Theoretical", 8)
+colnames(data_t1) <- c("m1", "m2", "a_measured", "Theoretical")
+
a_theoretical <- function(m) {
9.81 * m
}
@@ -40,6 +43,14 @@ lm_eqn <- function(df) {
as.character(as.expression(eq))
}
+th_eqn <- function() {
+ eq <- substitute(
+ italic(y) == b %.% italic(x),
+ list(b = format(unname(9.81), digits = 3))
+ )
+ as.character(as.expression(eq))
+}
+
data_t1["M"] <- m_from_masses(data_t1["m1"], data_t1["m2"])
data_t1["a"] <- data_t1$M * 9.8
@@ -49,18 +60,31 @@ plt <- ggplot(
mapping = aes(x = M, y = a_measured)
) +
geom_point(
- col = "#D55E00",
- size = 2,
+ aes(color = "D55E00"),
+ size = 8,
) +
geom_smooth(
se = FALSE,
method = "lm",
- linewidth = 0.8,
+ linewidth = 2.6,
col = "#D55E00",
) +
+ geom_segment(aes(x = min(data_t1$M),
+ y = a_theoretical(min(data_t1$M)),
+ xend = max(data_t1$M),
+ yend = a_theoretical(max(data_t1$M)),
+ linetype = "Theoretical"),
+ size = 2.2,
+ color = "#0072B2"
+ ) +
theme(
- axis.text = element_text(size = 14),
- axis.title = element_text(size = 20),
+ axis.text = element_text(size = 32),
+ axis.title = element_text(size = 50),
+ panel.grid.minor = element_line(size = 1.5),
+ panel.grid.major = element_line(size = 2),
+ legend.title = element_blank(),
+ legend.text = element_text(size = 36),
+ legend.key.size = unit(8, "line")
) +
labs(
x = x_lab(),
@@ -70,29 +94,25 @@ plt <- ggplot(
label = lm_eqn(data_t1[, c("a_measured", "M")]),
x = 0.076,
y = 0.58,
- size = 5,
+ size = 18,
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",
+ label = th_eqn(),
x = 0.068,
y = 0.74,
col = "#0072B2",
- size = 5,
- )
+ size = 18,
+ parse = TRUE,
+ ) +
+ scale_color_manual(name = "", values = c("#D55E00"), labels = "Measured") +
+ scale_linewidth_manual("Theoretical", values = c("Theoretical" = 2))
plt
-ggsave("plot.pdf",
+ggsave("plot.svg",
plot = plt,
- device = cairo_pdf,
+ width = 30,
+ height = 25,
)
diff --git a/plot1/plot.pdf b/plot1/plot.pdf
index 4cf46e9..cb8de3d 100644
Binary files a/plot1/plot.pdf and b/plot1/plot.pdf differ
diff --git a/plot1/plot.png b/plot1/plot.png
index a0210e0..5a332c6 100644
Binary files a/plot1/plot.png and b/plot1/plot.png differ
diff --git a/plot1/plot.svg b/plot1/plot.svg
new file mode 100644
index 0000000..d4a5c7c
--- /dev/null
+++ b/plot1/plot.svg
@@ -0,0 +1,118 @@
+
+
diff --git a/plot1/plot10.svg b/plot1/plot10.svg
new file mode 100644
index 0000000..ff238f2
--- /dev/null
+++ b/plot1/plot10.svg
@@ -0,0 +1,535 @@
+
+
diff --git a/plot1/plot11.svg b/plot1/plot11.svg
new file mode 100644
index 0000000..87474cd
--- /dev/null
+++ b/plot1/plot11.svg
@@ -0,0 +1,118 @@
+
+
diff --git a/plot1/plot12.svg b/plot1/plot12.svg
new file mode 100644
index 0000000..8e2ad32
--- /dev/null
+++ b/plot1/plot12.svg
@@ -0,0 +1,118 @@
+
+
diff --git a/plot1/plot13.svg b/plot1/plot13.svg
new file mode 100644
index 0000000..c0fd09e
--- /dev/null
+++ b/plot1/plot13.svg
@@ -0,0 +1,118 @@
+
+
diff --git a/plot1/plot14.png b/plot1/plot14.png
new file mode 100644
index 0000000..8642b20
Binary files /dev/null and b/plot1/plot14.png differ
diff --git a/plot1/plot14.svg b/plot1/plot14.svg
new file mode 100644
index 0000000..c0fd09e
--- /dev/null
+++ b/plot1/plot14.svg
@@ -0,0 +1,118 @@
+
+
diff --git a/plot1/plot2.png b/plot1/plot2.png
new file mode 100644
index 0000000..5a332c6
Binary files /dev/null and b/plot1/plot2.png differ
diff --git a/plot2/plot.R b/plot2/plot.R
new file mode 100644
index 0000000..ec5440f
--- /dev/null
+++ b/plot2/plot.R
@@ -0,0 +1,33 @@
+library(ggplot2)
+library(readr)
+library(latex2exp)
+
+dat <- read_csv("M,a
+ 0,0")
+
+x_lab <- function() {
+ TeX(r"($M$)")
+}
+
+y_lab <- function() {
+ TeX(r"($a \ \small{m \cdot s^{-2}}$)")
+}
+
+plt <- ggplot(data = dat, aes(x = M, y = a)) +
+ geom_segment(x = 0,
+ y = 0,
+ xend = 1,
+ yend = 9.8,
+ col = "#0072B2",
+ size = 1) +
+ labs(x = x_lab(),
+ y = y_lab()) +
+ expand_limits(y = c(0, 10), x = c(0, 1))
+
+plt
+
+ggsave("plot.svg",
+ plot = plt,
+ width = 5,
+ height = 5,
+)
diff --git a/plot2/plot.png b/plot2/plot.png
new file mode 100644
index 0000000..14f8a38
Binary files /dev/null and b/plot2/plot.png differ
diff --git a/plot2/plot.svg b/plot2/plot.svg
new file mode 100644
index 0000000..381c7a8
--- /dev/null
+++ b/plot2/plot.svg
@@ -0,0 +1,84 @@
+
+
diff --git a/report.pdf b/report.pdf
index c14a0cd..e7aa94d 100644
Binary files a/report.pdf and b/report.pdf differ
diff --git a/report.typ b/report.typ
index 902838e..4a631d1 100644
--- a/report.typ
+++ b/report.typ
@@ -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) $
+$ sum F_(y,m_2) = F^T_(S,m_2) + F^G_(g,m_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 $
+$ F^G_(g,m_1) = m_1 g $
+$ F^G_(g,m_2) = m_2 g $
+
+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) $
+
+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) $
+
+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 $
+$ sum F_(y,m_2) = F^T_(S,m) + m_2 g $
+
+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 $
+$ m_2 a_(y,m_2) = F^T_S + m_2 g $
+
+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 $
+$ m_2 a_y = F^T_S + m_2 g $
+
+@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 $
+$ -m_1 a_y = m_2 g + (- m_1 a_y - m_1 g) $
+$ m_2 a_y = m_2 g - m_1 a_y - m_1 g $
+
+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 $
+$ a_y = (m_2 g - m_1 g)/(m_2 + m_1) $
+
+Pulling $g$ out of the right side of the equation gives
+
+$ a_y = 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_y = g M $
+
+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) $
-$ 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 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 $
-$ -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 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
+#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.
]
#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)) $
+$ 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% $
+$ "% 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.
+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.