Velocity and Acceleration in Polar Coordinates

When one is solving problems in classical mechanics on the two dimensional plane and you are using polar coordinates, it is always a challenge to remember what the velocity/acceleration in the radial and angular directions (vr,vθ,ar,aθ)(v_r,v_\theta,a_r,a_\theta) are. Here's one way using complex numbers that made things really easy:

z=reiθz˙=r˙eiθ+irθ˙eiθ=(r˙+irθ˙)eiθ\begin{aligned} z&=re^{i\theta} \\ \dot z&=\dot re^{i\theta}+ir\dot\theta e^{i\theta} \\&=(\dot r+ir\dot\theta)e^{i\theta} \end{aligned}

From the above expression, we can obtain vr=r˙v_r=\dot r and vθ=rθ˙v_\theta=r\dot\theta.

z¨=(r¨+ir˙θ˙+irθ¨)eiθ+(r˙+irθ˙)iθ˙eiθ=(r¨+ir˙θ˙+irθ¨+ir˙θ˙−rθ˙θ˙)eiθ=(r¨−r(θ˙)2+i(rθ¨+2r˙θ˙))eiθ\begin{aligned} \ddot z&=(\ddot r+i\dot r\dot\theta+ir\ddot\theta)e^{i\theta}+(\dot r+ir\dot\theta)i\dot\theta e^{i\theta} \\&=(\ddot r+i\dot r\dot\theta+ir\ddot\theta+i\dot r\dot\theta-r\dot\theta\dot\theta)e^{i\theta} \\&=(\ddot r-r(\dot\theta)^2+i(r\ddot\theta+2\dot r\dot\theta))e^{i\theta} \end{aligned}

From this we can obtain ar=r¨−r(θ˙)2a_r=\ddot r-r(\dot\theta)^2 and aθ=rθ¨+2r˙θ˙a_\theta=r\ddot\theta+2\dot r\dot\theta with absolute ease.

#Mechanics

Note by Brian Lie
2 years, 8 months ago

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