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Cylindrical Polars
Level 1 - Math II (Physics) topic page in Kinematics.
Principle
Cylindrical polar coordinates extend plane polar coordinates by adding a vertical coordinate \(z\). They are useful when a physical system has an axis of symmetry, such as rotation about a shaft, flow in a pipe, or fields around a long straight wire.
The radial and angular basis vectors behave as in plane polar coordinates, while \(\mathbf e_z\) is fixed along the axis.
Notation
\(r\)
distance from the z-axis
\(\mathrm{m}\)
\(\theta\)
angle around the z-axis measured in radians
\(z\)
height along the z-axis
\(\mathrm{m}\)
\(\mathbf e_r\)
radial unit vector in the horizontal plane
\(\mathbf e_\theta\)
azimuthal unit vector in the direction of increasing theta
\(\mathbf e_z\)
fixed unit vector along the z-axis
Method
Step 1: Write the position vector
The position has a horizontal radial part and a vertical part:
Cylindrical position
\[\mathbf r=r\mathbf e_r+z\mathbf e_z\]
Step 2: Differentiate to get velocity
The polar part changes exactly as in plane polar coordinates, and the vertical part contributes \(\dot z\mathbf e_z\):
Cylindrical velocity
\[\mathbf v=\dot r\mathbf e_r+r\dot\theta\mathbf e_\theta+\dot z\mathbf e_z\]
Step 3: Differentiate again to get acceleration
The vertical acceleration adds independently to the polar acceleration:
Cylindrical acceleration
\[\mathbf a=(\ddot r-r\dot\theta^2)\mathbf e_r+(r\ddot\theta+2\dot r\dot\theta)\mathbf e_\theta+\ddot z\mathbf e_z\]
Start from position
\[\mathbf r=r\mathbf e_r+z\mathbf e_z\]
Differentiate radial part
\[\frac{d}{dt}(r\mathbf e_r)=\dot r\mathbf e_r+r\dot\theta\mathbf e_\theta\]
Differentiate vertical part
\[\frac{d}{dt}(z\mathbf e_z)=\dot z\mathbf e_z\]
Differentiate velocity
\[\mathbf a=(\ddot r-r\dot\theta^2)\mathbf e_r+(r\ddot\theta+2\dot r\dot\theta)\mathbf e_\theta+\ddot z\mathbf e_z\]
Rules
Basis derivatives
\[\dot{\mathbf e}_r=\dot\theta\mathbf e_\theta,\quad \dot{\mathbf e}_\theta=-\dot\theta\mathbf e_r,\quad \dot{\mathbf e}_z=\mathbf 0\]
Cylindrical speed
\[|\mathbf v|=\sqrt{\dot r^2+r^2\dot\theta^2+\dot z^2}\]
Cartesian conversion
\[x=r\cos\theta,\quad y=r\sin\theta,\quad z=z\]
- The coordinate \(r\) is non-negative.
- The coordinate \(\theta\) is angular, so calculus formulas use radians.
- The vertical direction \(\mathbf e_z\) does not rotate with \(\theta\).
- Cylindrical formulas reduce to plane polar formulas when \(z\) is constant.
Examples
Question
A point has cylindrical coordinates
\[r=2\]
\[\theta=\pi/2\]
and \[z=5\]
What are its Cartesian coordinates?Answer
Use
\[x=r\cos\theta\]
and \[y=r\sin\theta\]
Then \[x=2\cos(\pi/2)=0\]
\[y=2\sin(\pi/2)=2\]
and \[z=5\]
The Cartesian coordinates are \[(0,2,5)\]
Checks
- Do not confuse radial distance \(r\) with the full position vector \(\mathbf r\).
- Include the \(\dot z\mathbf e_z\) and \(\ddot z\mathbf e_z\) terms when vertical motion is present.
- Do not treat \(\mathbf e_r\) and \(\mathbf e_\theta\) as fixed vectors when \(\theta\) changes.
- Check whether the system has axial symmetry before choosing cylindrical coordinates for simplification.