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 $\polar {r, \theta}$.
Then the acceleration $\mathbf a$ of $p$ can be expressed as:
- $\mathbf a = \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} } \mathbf u_\theta + \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2} \mathbf u_r$
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:
- $(1): \quad \mathbf r = r \mathbf u_r$
From Derivatives of Unit Vectors in Polar Coordinates:
\(\text {(2)}: \quad\) | \(\ds \dfrac {\d \mathbf u_r} {\d \theta}\) | \(=\) | \(\ds \mathbf u_\theta\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \dfrac {\d \mathbf u_\theta} {\d \theta}\) | \(=\) | \(\ds -\mathbf u_r\) |
From Velocity Vector in Polar Coordinates:
- $\mathbf v = r \dfrac {\d \theta} {\d t} \mathbf u_\theta + \dfrac {\d r} {\d t} \mathbf u_r$
where $\mathbf v$ is the velocity of $p$.
The acceleration of $p$ is by definition the rate of change in its velocity:
\(\ds \mathbf a\) | \(=\) | \(\ds \dfrac {\d \mathbf v} {\d t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds r \dfrac {\d^2 \theta} {\d t^2} \mathbf u_\theta + \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} \mathbf u_\theta + r \dfrac {\d \theta} {\d t} \dfrac {\d \mathbf u_\theta} {\d t} + \dfrac {\d^2 r} {\d t^2} \mathbf u_r + \dfrac {\d r} {\d t} \dfrac {\d \mathbf u_r} {\d t}\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds r \dfrac {\d^2 \theta} {\d t^2} \mathbf u_\theta + \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} \mathbf u_\theta + r \dfrac {\d \theta} {\d t} \dfrac {\d \mathbf u_\theta} {\d \theta} \dfrac {\d \theta} {\d t} + \dfrac {\d^2 r} {\d t^2} \mathbf u_r + \dfrac {\d r} {\d t} \dfrac {\d \mathbf u_r} {\d \theta} \dfrac {\d \theta} {\d t}\) | Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds r \dfrac {\d^2 \theta} {\d t^2} \mathbf u_\theta + \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} \mathbf u_\theta - r \dfrac {\d \theta} {\d t} \mathbf u_r \dfrac {\d \theta} {\d t} + \dfrac {\d^2 r} {\d t^2} \mathbf u_r + \dfrac {\d r} {\d t} \mathbf u_\theta \dfrac {\d \theta} {\d t}\) | substituting from $(2)$ and $(3)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} } \mathbf u_\theta + \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2} \mathbf u_r\) | gathering terms |
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.21$: Newton's Law of Gravitation: $(5)$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): acceleration
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): acceleration