From 6982c351c4fb0ed302adb0a35bf0bc87409391c8 Mon Sep 17 00:00:00 2001 From: Ethan Simmons Date: Wed, 13 Mar 2024 21:59:22 -0500 Subject: [PATCH] Final Draft (Hopefully) --- plot1/plot.R | 58 +++-- plot1/plot.pdf | Bin 25531 -> 25917 bytes plot1/plot.png | Bin 135140 -> 196485 bytes plot1/plot.svg | 118 +++++++++++ plot1/plot10.svg | 535 +++++++++++++++++++++++++++++++++++++++++++++++ plot1/plot11.svg | 118 +++++++++++ plot1/plot12.svg | 118 +++++++++++ plot1/plot13.svg | 118 +++++++++++ plot1/plot14.png | Bin 0 -> 75136 bytes plot1/plot14.svg | 118 +++++++++++ plot1/plot2.png | Bin 0 -> 196485 bytes plot2/plot.R | 33 +++ plot2/plot.png | Bin 0 -> 19166 bytes plot2/plot.svg | 84 ++++++++ report.pdf | Bin 236737 -> 510506 bytes report.typ | 320 ++++++++++++++++------------ 16 files changed, 1464 insertions(+), 156 deletions(-) create mode 100644 plot1/plot.svg create mode 100644 plot1/plot10.svg create mode 100644 plot1/plot11.svg create mode 100644 plot1/plot12.svg create mode 100644 plot1/plot13.svg create mode 100644 plot1/plot14.png create mode 100644 plot1/plot14.svg create mode 100644 plot1/plot2.png create mode 100644 plot2/plot.R create mode 100644 plot2/plot.png create mode 100644 plot2/plot.svg 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 4cf46e9a468dce8f0c9f72b71ee3ac04743811ca..cb8de3dfe4a082c647040d204c82ad8e460c1df6 100644 GIT binary patch delta 12132 zcmV-qFPqT2#{s>{0gz08?O98Y9Jvj?*RQB!4}z9ON}?oSlY@aEzydxv9v}z92cx~- zjWLs8?F75|_xVVb>eW4-8EaoU=O4!x z@BR4WmqDixOKnp=|1f^ML~2}r{mV$si_~1kUqJgFf1if8??&%`{PzWALsrC9s}9EcPyfa=s|(<62*-ewoBw(Y%_z z2QGyWOU0u6#YHXBT&XnPk3^JQ<}j1)cI(Xbih3cu@xYF1u8;uCkbgf0IEXQT-$(giuWee*Jk3OCx zE2R5slH`!g`c;y2(oS#0SJ%@U8N0U0&55$pO2k*xN)%jwKvJ&McGBWn|HNQ=R4I8f z{#dT|W6zF#PnT|oUtIIb$i^YO%S>J;o@L7*dz7NPrbXL+`nLj<-i-Ed}ju8F97UBco}j5rEv<4j4wD~&p8 zRma(Xc;Yy9$^a6%io{)n?#G?w**ob0D_pVQ7+ti&WTqtFPH`{D=|mR@jpsB|5R$>p zqeaMm+*w`@42%U-%)pGi7sg?V@JXo^4aH^fR8CGj1BnSpr&(CBF^Ywt#L$il4~l~I zKvv9E1Sl#F(n^CQyO@x(1r#wkz#5@!iJ}0107@m7i{=(6S{AVnuEsi8rWZuSo#j63 zu4{mca&h!7MsTs{C;_2y_!knx2v9PxEC6_`C|V3kD#%rz#*E-6QB47mslcb`%u&q? z(}HsniwICchUozPC@yeRK|v6~@MmU#y6sK?IxyCkpO(00+Mai}Uy3PKG|6=8B#+l2M-~v#rjL1o=QAAXPk58j6GgB-9HbIx7%|jiURD zU1c=jlvzWe`f9Mm*i0ssQ?+V@WqKoEKCL0xl?0?Y&K+xW-J9Y6lO8&aUfvD>q6`cMLfggT zGR*2W5zWVlfUE&%5Xam5;l;m$D1RSgU6>F+zlmr?4QZV(Vpq%-R<`q+sgY4MAeSh)a-O-;(4 z+fMwd=KT5gVn{#c6R+@pQMGzwfDNiAjQ$rsS?x^O?)~qwGg^r0(SH8WHVD|$uE_-; 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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 0000000000000000000000000000000000000000..14f8a3838d0bed0bd0f55bb0dcd090ce6926e199 GIT binary patch literal 19166 zcmb`vc|4Wt+cv%=6e2^&m|0P#NXFDc%Fr@XrbJT4WL$VI{XF0IdEfV+-{<}8Vs}s1bzj3d9LI6q(U%N#4$yPcqfn>=y64eG zC=}Hg@^>FC90~eLNP%BR9L^i*qfh~2C{$=T3bg@;LZ?xvJ2EKLPdgM!ISqy4^mzIX zqY5WzuU^zaqjr)1p4H|g!;$?S=Phr;Z@3cvF5W!bg+l4p=%O`E{08Sne6j3zcG}2m zA6J*pO5eS7ZjTT>ZaVMCu}8)MrX`EUm24f)GXt)+dHK|yYw6?@^k(n+5bcK(+P6>a z`3t)}g8Re2>c8z;NEA(%+W$abadTuKNJAhokhl8q#P$2{2|iBOPbJ|b=%b>dB%WRC 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#text()[#counter(heading).at(loc).last()]#text(".")#h(0.05em) ]) set text(size: 16pt) + v(0.5em) block()[ #count #underline(it.body) ] v(0.5em) + counter(math.equation).update(0) } #show figure.caption: it => { @@ -61,30 +66,150 @@ #text(size: 16pt)[ Ethan Simmons ] \ - Submission Date: 02/27/23 + Submission Date: 03/13/23 Lab Section 12 \ - TA: \ + TA: Seburne ] #pagebreak() = Introduction -An Atwood machine shows the relationship between forces and acceleration. -The masses of both weights can be measured and the forces can be calculated from the measured masses and gravity. -By varying the weights and measuring acceleration, the relationship between forces and acceleration can be calculated. +An Atwood's machine shows the relationship between forces and acceleration. +It consists of two weights connected by a string. By changing the mass of these weights, +the acceleration can be measured. Analyzing the measured accelerations and weights used +allows for an experimental measurement of gravity. += Theory + +The forces in an Atwoods machine can be modeled by drawing a free body diagram for each weight ($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.

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