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Motion /
Circular Motion.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=1cm]{geometry} \begin{document} {\large Circular Motion} \begin{align*} \text{Basic Equations:}\quad&\text{Tangental Force }F_T=m\:a_T\:,\quad\text{Norminal (Radial) Force }F_N=m\:a_N\:.\\ &\text{Let $l$ be the arc length travelled, and $\theta$ be the angle such arc spans, then }l=r\theta\:.\\ &v=\frac{dl}{dt}=r\frac{d\theta}{dt}=r\dot{\theta}\:.\quad \text{Angular velocity is defined as }\boxed{\omega=\dot{\theta}}.\quad\therefore\boxed{v=r\omega}\:.\\ &a_T=\dot{v}=r\dot{\omega}\quad\ldots\:\text{(1)}\\ \\ \text{Projections }&\text{on $x$-axis and $y$-axis:}\\ &\boxed{x=r\cos\theta}\\ &\dot{x}=-r\sin\theta\cdot\dot{\theta}\quad \therefore\boxed{\dot{x}=-r\omega\sin\theta=-v\sin\theta}\\ &\ddot{x}=-r\sin\theta\cdot\dot{\omega}-r\omega\cos\theta\cdot\dot{\theta}\quad \therefore\boxed{\ddot{x}=-r\dot{\omega}\sin\theta-r\omega^2\cos\theta}\\ &\boxed{y=r\sin\theta}\\ &\dot{y}=r\cos\theta\cdot\dot{\theta}\quad \therefore\boxed{\dot{y}=r\omega\cos\theta=v\cos\theta}\\ &\ddot{y}=r\cos\theta\cdot\dot{\omega}-r\omega\sin\theta\cdot\dot{\theta}\quad \therefore\boxed{\ddot{y}=r\dot{\omega}\cos\theta-r\omega^2\sin\theta}\\ \\ \text{Acceleration:}\quad&\text{Let $a$ be the magnitude of acceleration.}\\ a^2&=\ddot{x}^2+\ddot{y}^2\\ &=(-r\dot{\omega}\sin\theta-r\omega^2\cos\theta)^2+(r\dot{\omega}\cos\theta-r\omega^2\sin\theta)^2\\ &=r^2\left[(\dot{\omega}^2\sin^2\theta+2\dot{\omega}\sin\theta\cdot{\omega}^2\cos\theta+\omega^4\cos^2\theta)+(\dot{\omega}^2\cos^2\theta-2\dot{\omega}\cos\theta\cdot{\omega}^2\sin\theta+\omega^4\sin^2\theta)\right]\\ &=r^2\left[\dot{\omega}^2(\sin^2\theta+\cos^2\theta)+\omega^4(\sin^2\theta+\cos^2\theta)\right]\\ &=r^2(\dot{\omega}^2+\omega^4)\\ \therefore\quad&\boxed{a=r\sqrt{\dot{\omega}^2+\omega^4}}\\ &\boxed{a_T=r\dot{\omega}}\quad\text{From (1)}\\ &a^2={a_N}^2+{a_T}^2\:,\quad{a_N}^2=a^2-{a_T}^2=r^2(\dot{\omega}^2+\omega^4)-r^2\dot{\omega}^2=r^2\omega^4\:,\\ \therefore\quad&\boxed{a_N=r\omega^2}\\ &\text{As a result,}\\ &\boxed{\ddot{x}=-a_T\sin\theta-a_N\cos\theta}\\ &\boxed{\ddot{y}=a_T\cos\theta-a_N\sin\theta}\\ &\text{To obtain the direction of $a$:}\quad \tan\phi=\frac{\:\ddot{y}\:}{\ddot{x}}=\frac{a_T\cos\theta-a_N\sin\theta}{-a_T\sin\theta-a_N\cos\theta}\div\frac{-a_N\cos\theta}{-a_N\cos\theta} =\frac{-\tfrac{a_T}{a_N}+\tan\theta}{\tfrac{a_T}{a_N}\tan\theta+1}\\ &\therefore\tan\phi=\tan(\theta-\psi)\:,\quad\text{where }\psi=\frac{a_T}{a_N}\\ \\ \text{Uniform Circular Motion}:\quad&\text{$v$ is constant, so is $\omega$ .}\quad\therefore\dot{\omega}=0\:.\quad\text{Substitute into the general formulae }\ldots\\ &\boxed{a_T=0}\:,\quad\boxed{a=a_N=r\omega^2}\\ &\boxed{\ddot{x}=-r\omega^2\cos\theta=-a_N\cos\theta}\:,\quad\boxed{\ddot{y}=-r\omega^2\sin\theta-a_N\sin\theta}\\ &\text{Frequency $f$ \--- the number of revolutions ($\theta$ increased by $2\pi$) per second: }\boxed{f=\frac{\omega}{2\pi}\quad \sec^{-1}}\\ &\text{Period $T$ \--- the time for a revolution: }\boxed{T=\frac{1\:\sec}{f}=\frac{2\pi}{\omega}\quad\sec} \end{align*} \end{document}