Code: Select all
\documentclass[fleqn]{article}
\usepackage{fullpage}
\usepackage[margin=1in]{geometry}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{multicol}
\usepackage{floatflt}
\usepackage[makeroom]{cancel}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{xfrac}
\usepackage{siunitx}
\sisetup{%
output-decimal-marker={.},
load-configurations=abbreviations,
group-separator={,},
per-mode=fraction,
quotient-mode = fraction
}\pagestyle{empty}
\begin{document}
\begin{enumerate}
\item
\raisebox{1.85ex}{\parbox[t]{2in}{\null\includegraphics[width=\linewidth]{Diagram1}}}
\hspace{0.5cm}
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
r&=\text{3840 \cancel{mi}}\left(\dfrac{\text{1.609 \cancel{km}}}{\text{1 \cancel{mi}}}\right)\left(\dfrac{\text{1000 m}}{\text{1 \cancel{km}}}\right)=\SI{6.18e6}{\m} \\
s&=\text{5150 \cancel{mi}}\left(\dfrac{\text{1.609 \cancel{km}}}{\text{1 \cancel{mi}}}\right)\left(\dfrac{\text{1000 m}}{\text{1 \cancel{km}}}\right)=\SI{8.29e6}{\m} \\
\theta&=\text{?}
\end{aligned}$
$\begin{aligned}[t]
s&=r \theta \\
\cfrac{s}{r}&=\theta \\
\cfrac{8.29}{6.18}&=\theta \\
\text{1.34 rad}&=\theta
\end{aligned}$
\end{minipage}
\item
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
\Delta t&=\text{9.09 \cancel{hr}}\left(\dfrac{\text{60 \cancel{min}}}{\text{1 \cancel{hr}}}\right)\left(\dfrac{\text{60 s}}{\text{1 \cancel{min}}}\right)=\text{32,724 s} \\
\omega &=\text{?} \\
\omega &=\dfrac{\theta}{\Delta t}=\dfrac{\text{1.34 rad}}{\text{32,724 s}}=\text{\num{4.09e-5} rad/s}
\end{aligned}$
\end{minipage}
\item
\raisebox{1.85ex}{\parbox[t]{2in}{\null\includegraphics[width=\linewidth]{Diagram2}}}
\begin{minipage}[t]{4cm}
$\begin{aligned}[t]
d&=\text{328 m} \\
r&=6.18 \\
T&=\text{24 \cancel{h}}\left(\dfrac{\text{60 \cancel{min}}}{\text{1 \cancel{h}}}\right)
\end{aligned}$
\end{minipage}
$\begin{aligned}[t]
v_{1}&=\dfrac{2\pi r}{T} & v_{2}&=\dfrac{2\pi(r+d)}{T} \\
Tv_{1}&=2\pi r & Tv_{2}&=2\pi(r+d) \\
\dfrac{Tv_{1}}{2\pi}&=r & \dfrac{Tv_{2}}{2\pi}-d&=r \\
\end{aligned}$
\vspace{-4.5cm}
\begin{align*}
\hspace{5cm}
\dfrac{Tv_{1}}{2\pi}&=\dfrac{Tv_{2}}{2\pi}-d \\
d&=\dfrac{Tv_{2}}{2\pi}-\dfrac{Tv_{1}}{2\pi} \\
d&=\dfrac{T}{2\pi}\left(v_{2}-v_{1}\right) \\
\dfrac{2\pi d}{T}&=\Delta v \\
\dfrac{2\pi(328)}{86,400}&=\Delta v \\
0.02385&=\Delta v \\
\text{\num{2.39e-2 m/s}}&=\Delta v
\end{align*}
\item
\raisebox{1.85ex}{\parbox[t]{2in}{\null\includegraphics[width=\linewidth]{Diagram3}}}
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
\theta&=\SI{40.1}{\degree} \\
T&=\text{24 \cancel{h}}\left(\dfrac{\text{60 \cancel{min}}}{\text{1 \cancel{h}}}\right)\left(\dfrac{\text{60 s}}{\text{1 \cancel{min}}}\right)=\text{86,400 s} \\
r&=\SI{6.37e6}{\m} \\
\cos\theta&=\dfrac{d}{r} \\
r\cos\theta&=d
\end{aligned}$
\end{minipage}
\begin{align*}
v&=\dfrac{2\pi d}{T} \\
v&=\dfrac{2\pi r\cos\theta}{T} \\
v&=\dfrac{2\pi(\SI{6.37e6}{\m})\cos(\SI{40.1}{\degree})}{\text{86,400 s}} \\
v&=\SI{354.34}{\m\per\s}
\end{align*}
\item
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
m&=\SI{9.11e-31}{\kg} & f&=\text{?} \\
r&=\SI{5.33e-11}{\m} \\
F&=\SI{9.15e-8}{\N}
\end{aligned}$
\end{minipage}
\begin{align*}
f&=\dfrac{\sqrt{\dfrac{Fr}{m}}}{2\pi r} \\
f&=\dfrac{\sqrt{\dfrac{(\num{9.15e-8})(\num{5.33e-11})}{\num{9.11e-31}}}}{2\pi(\num{5.33e-11})} \\
f&=\num{6.91e15}
\end{align*}
\item
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
m&=\text{1420 kg} \\
r&=\text{125 m} \\
v&=\text{25.4 m/s} \\
F&=\text{?} \\
\end{aligned}$
\end{minipage}
$\begin{aligned}[t]
F&=ma_{r} \\
F&=\dfrac{mv^{2}}{r} \\
F&=\dfrac{\text{(1420 kg)(25.4 m/s)}^{2}}{\text{125 m}} \\
F&=\text{7329 N} \\
F&=\SI{7.33e3}{N}
\end{aligned}$
\item
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
m&=\text{0.0255 kg} \\
r&=\text{0.2 m} \\
\omega&=\text{56.2}
\end{aligned}$
\end{minipage}
\item
\raisebox{1.85ex}{\parbox[t]{1.6in}{\null\includegraphics[width=\linewidth]{Diagram4}}}
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
r&=\text{3.44 m} \\
T&=\text{6.92 s} \\
t&=\text{18.1 s} \\
v&=\text{?}
\end{aligned}$
\end{minipage}
\hspace{-1in}
$\begin{aligned}[t]
T&=\dfrac{2\pi r}{v} \\
Tv&=2\pi r \\
v&=\dfrac{2\pi r}{T} \\
v&=\dfrac{2\pi(3.44)}{6.92} \\
v&=\text{3.12 m/s} \\
\end{aligned}$
\item
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
\theta_{f}&=\theta_{i}+\omega_{i}t+\dfrac{1}{2}\alpha t^{2} \\
\theta_{f}-\theta_{i}&=\dfrac{vt}{r}+\dfrac{1}{2}\left(\dfrac{\omega_{f}-\omega_{i}}{\cancel{t}}\right)t^{\cancel{2}} \\
\Delta\theta&=\dfrac{vt}{r}+\dfrac{1}{2}\left(\omega_{f}-\omega_{i}\right)t \\
\Delta\theta&=\dfrac{vt}{r}-\dfrac{\omega_{i}t}{2} \\
\Delta\theta&=\dfrac{vt}{r}-\dfrac{vt}{2r}
\end{aligned}$
\end{minipage}
\hspace{1cm}
$\begin{aligned}[t]
\Delta\theta&=\dfrac{vt}{r}\left(1-\dfrac{1}{2}\right) \\
\Delta\theta&=\dfrac{vt}{2r} \\
\Delta\theta&=\dfrac{(3.12)(18.1)}{2(3.44)}\left(\dfrac{\text{1 rev}}{2\pi \text{ rad}}\right) \\
\Delta\theta&=\text{1.31 rev} \\
\end{aligned}$
\item
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
\omega&=3640\dfrac{\text{\cancel{rev}}}{\text{\cancel{min}}}\left(\dfrac{2\pi \text{ rad}}{\text{1 \cancel{rev}}}\right)\left(\dfrac{\text{1 \cancel{min}}}{\text{60 s}}\right)=\text{381.2 rad/s} \\
a_{r}&=\text{?} \\
r&=\text{0.103 m}
\end{aligned}$
\end{minipage}
\begin{align*}
a_{r}&=r\omega^{2} \\
&=\text{(0.103 m)(381.2 rad/s)}^{2} \\
&=\text{\num{1.50e4} m/s}^{2}
\end{align*}
\item
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
m&=\text{1350 kg} \\
r&=\text{51 m} \\
v&=\text{14 m/s} \\
f_{s}&=\text{?}
\end{aligned}$
\end{minipage}
\hspace{-1in}
$\begin{aligned}[t]
F&=ma_{r} \\
F&=\dfrac{mv^{2}}{r} \\
f_{s}&=\dfrac{mv^{2}}{r} \\
f_{s}&=\dfrac{(1350)(14)^{2}}{51} \\
f_{s}&=\SI{5.2e3}{N}
\end{aligned}$
\item
\begin{minipage}[t]{1in}
$\begin{aligned}[t]
m&=\text{0.591 kg} \\
L&=\text{1.46 m} \\
T&=\text{11 N} \\
v&=\text{?}
\end{aligned}$
\end{minipage}
\raisebox{1.85ex}{\parbox[t]{1.6in}{\null\includegraphics[width=\linewidth]{Diagram5}}}
\raisebox{1.85ex}{\parbox[t]{0.7cm}{\null\includegraphics[width=\linewidth]{Diagram6}}}
\hspace{0.3cm}
$\begin{aligned}[t]
F&=ma_{r} \\
T-mg&=\dfrac{mv^{2}}{L} \\
\left(T-mg\right)L&=mv^{2} \\
\sqrt{\dfrac{\left(T-mg\right)L}{m}}&=v \\
\sqrt{\dfrac{\left(11-(0.591)(9.81)\right)(1.46)}{0.591}}&=v \\
\text{3.58 m/s}&=v
\end{aligned}$
\item
\begin{minipage}[t]{2in}
$\begin{aligned}[t]
r&=\text{2.615 m} \\
L&=\text{7.67 m} \\
t&=\text{10.2 s} \\
\theta&=\ang{16.7} \\
v&=\text{?}
\end{aligned}$
\end{minipage}
\hspace{-5cm}
\raisebox{1.85ex}{\parbox[t]{8.5cm}{\null\includegraphics[width=\linewidth]{Diagram7}}}
\vspace{-2.9cm}
\begin{align*}
\hspace{5cm}
\sin\theta&=\dfrac{x}{L} \\
L\sin\theta&=x \\
\text{2.204 m}&=x
\end{align*}
\raisebox{1.85ex}{\parbox[t]{4cm}{\null\includegraphics[width=\linewidth]{Diagram8}}}
\vspace{-5cm}
\begin{align}
\hspace{4cm}
T\cos\ang{73.3}&=\dfrac{mv^{2}}{r} \\
T\sin\ang{73.3}-mg&=0 \\
T\sin\ang{73.3}&=mg \nonumber \\
T&=\dfrac{mg}{\sin\ang{73.3}} \nonumber
\end{align}
\vspace{0.5cm}
\begin{align*}
\hspace{-0.5cm}
\left(\dfrac{mg}{\sin\ang{73.3}}\right)\cos\ang{73.3}&=\dfrac{mv^{2}}{r+L\sin\theta} \\
\left(\dfrac{\cos\ang{73.3}}{\sin\ang{73.3}}\right)mg&=\dfrac{mv^{2}}{r+L\sin\theta} \\
\sqrt{\left(r+L\sin\theta\right)\left(\dfrac{\cos\ang{73.3}}{\sin\ang{73.3}}\right)g}&=v \\
\sqrt{(2.615+7.67\sin\ang{16.7})\left(\dfrac{\cos\ang{73.3}}{\sin\ang{73.3}}\right)(9.81)}&=v \\
\text{3.77 m/s}&=v
\end{align*}
\item
\raisebox{1.85ex}{\parbox[t]{2in}{\null\includegraphics[width=\linewidth]{Diagram9}}}
\begin{minipage}[t]{1in}
$\begin{aligned}[t]
r&=\text{47.7 m} \\
v&=\text{?} \\[0.5cm]
\end{aligned}$
$\begin{aligned}[t]
\Sigma{F_{y}}&=0 \\
N-mg&=0 \\
N&=mg
\end{aligned}$
\end{minipage}
\hspace{1cm}
$\begin{aligned}[t]
mg&=\dfrac{mv^{2}}{r} \\
mgr&=mv^{2} \\
\dfrac{\cancel{m}gr}{\cancel{m}}&=v^{2} \\
\sqrt{gr}&=v \\
\text{\num{2.16e1} m/s}&=v
\end{aligned}$
\item
\begin{minipage}[t]{1in}
$\begin{aligned}[t]
r&=\text{19.8 m} \\
T&=\text{29.2 s} \\
v&=\text{?} \\
a_{r}&=\text{?} \\[0.5cm]
\end{aligned}$
$\begin{aligned}[t]
a_{r}&=\dfrac{v^{2}}{r} \\
a_{r}&=\dfrac{(4.26)^{2}}{\text{19.8 m}} \\
a_{r}&=\text{0.917 m/s}^{2} \\
a_{r}&=\text{\num{9.17e-1} m/s}^{2}
\end{aligned}$
\end{minipage}
$\begin{aligned}[t]
T&=\dfrac{2\pi r}{v} \\
Tv&=2\pi r \\
v&=\dfrac{2\pi r}{T} \\
v&=\dfrac{2\pi(19.8)}{29.2} \\
v&=\text{4.26 m/s}
\end{aligned}$
\item
$\begin{aligned}[t]
\dfrac{W_{app}}{W}=\dfrac{-ma+mg}{mg}=\dfrac{\cancel{m}(-a+g)}{\cancel{m}g}=\dfrac{-a+g}{g}=\dfrac{-(0.917)+(9.81)}{9.81}=0.907
\end{aligned}$
\item
$\begin{aligned}[t]
\dfrac{W_{app}}{W}=\dfrac{ma+mg}{mg}=\dfrac{\cancel{m}(a+g)}{\cancel{m}g}=\dfrac{a+g}{g}=1.093
\end{aligned}$
\item
$\begin{aligned}[t]
m&=\text{0.136 kg} \\
L&=\text{0.637 m} \\
x_{1}&=\text{2.07 m} \\
x_{2}&=\text{5.11 m} \\
T&=\text{?} \\
\end{aligned}$
\raisebox{1.85ex}{\parbox[t]{2.5in}{\null\includegraphics[width=\linewidth]{Diagram10}}}
\begin{align*}
$\begin{aligned}[t]
v_{f}^{2}&=v_{i}^{2}+2a\Delta d \\
0&=v_{i}^{2}+2a\Delta d \\
-2a\Delta d&=v_{i}^{2} \\
-2a(x_{2}-x_{1})&=v_{i}^{2}
\end{aligned}$
$\begin{aligned}[t]
\hspace{1cm}
-T&=\dfrac{mv_{i}^{2}}{L} \\
T&=\dfrac{-mv_{i}^{2}}{L} \\
T&=\dfrac{-m(-2a(x_{2}-x_{1}))}{L} \\
T&=\dfrac{2ma(x_{2}-x_{1})}{L} \\
T&=\dfrac{2(0.136)(9.81)(5.11-2.07)}{0.637} \\
T&=\SI{12.73}{N} \\
T&\approx\SI{12.7}{N}
\end{aligned}$
\end{align*}
\end{enumerate}
\end{document}