Acceleration Vector in Polar Coordinates

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Theorem

Consider a particle $p$ moving in the plane.

Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$.


Then the acceleration $\mathbf a$ of $p$ can be expressed as:

$\mathbf a = \left({r \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} + 2 \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \theta} {\mathrm d t} }\right) \mathbf u_\theta + \left({\dfrac {\mathrm d^2 r} {\mathrm d t^2} - r \left({\dfrac {\mathrm d \theta} {\mathrm d t} }\right)^2}\right) \mathbf u_r$

where:

$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$
$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$


Proof

Let the radius vector $\mathbf r$ from the origin to $p$ be expressed as:

$(1): \quad \mathbf r = r \mathbf u_r$


MotionInPolarPlane.png


From Derivatives of Unit Vectors in Polar Coordinates:

\((2):\quad\) \(\displaystyle \dfrac {\mathrm d \mathbf u_r} {\mathrm d \theta}\) \(=\) \(\displaystyle \mathbf u_\theta\)
\((3):\quad\) \(\displaystyle \dfrac {\mathrm d \mathbf u_\theta} {\mathrm d \theta}\) \(=\) \(\displaystyle \mathbf u_r\)


From Velocity Vector in Polar Coordinates:

$\mathbf v = r \dfrac {\mathrm d \theta} {\mathrm d t} \mathbf u_\theta + \dfrac {\mathrm d r} {\mathrm d t} \mathbf u_r$

where $\mathbf v$ is the velocity of $p$.


The acceleration of $p$ is by definition the rate of change in its velocity:

\(\displaystyle \mathbf a\) \(=\) \(\displaystyle \dfrac {\mathrm d \mathbf v} {\mathrm d t}\)
\(\displaystyle \) \(=\) \(\displaystyle r \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} \mathbf u_\theta + \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \theta} {\mathrm d t} \mathbf u_\theta + r \dfrac {\mathrm d \theta} {\mathrm d t} \dfrac {\mathrm d \mathbf u_\theta} {\mathrm d t} + \dfrac {\mathrm d^2 r} {\mathrm d t^2} \mathbf u_r + \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \mathbf u_r} {\mathrm d t}\) Product Rule for Derivatives
\(\displaystyle \) \(=\) \(\displaystyle r \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} \mathbf u_\theta + \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \theta} {\mathrm d t} \mathbf u_\theta + r \dfrac {\mathrm d \theta} {\mathrm d t} \dfrac {\mathrm d \mathbf u_\theta} {\mathrm d \theta} \dfrac {\mathrm d \theta} {\mathrm d t} + \dfrac {\mathrm d^2 r} {\mathrm d t^2} \mathbf u_r + \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \mathbf u_r} {\mathrm d \theta} \dfrac {\mathrm d \theta} {\mathrm d t}\) Chain Rule
\(\displaystyle \) \(=\) \(\displaystyle r \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} \mathbf u_\theta + \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \theta} {\mathrm d t} \mathbf u_\theta + r \dfrac {\mathrm d \theta} {\mathrm d t} \mathbf u_r \dfrac {\mathrm d \theta} {\mathrm d t} + \dfrac {\mathrm d^2 r} {\mathrm d t^2} \mathbf u_r + \dfrac {\mathrm d r} {\mathrm d t} \mathbf u_\theta \dfrac {\mathrm d \theta} {\mathrm d t}\) substituting from $(2)$ and $(3)$
\(\displaystyle \) \(=\) \(\displaystyle \left({r \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} + 2 \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \theta} {\mathrm d t} }\right) \mathbf u_\theta + \left({\dfrac {\mathrm d^2 r} {\mathrm d t^2} - r \left({\dfrac {\mathrm d \theta} {\mathrm d t} }\right)^2}\right) \mathbf u_r\) gathering terms

$\blacksquare$


Sources