AcademyKinematics
Academy
Polar Coordinates
Level 1 - Math II (Physics) topic page in Kinematics.
Principle
Plane polar coordinates describe position in a plane using a radial distance \(r\) from the origin and an angle \(\theta\) from a reference axis. The basis vectors change direction as \(\theta\) changes, so velocity and acceleration contain extra angular terms.
Angles are measured in radians. This convention makes derivatives such as \(d\theta/dt\) compatible with arc length and angular velocity formulas.
Notation
\(r\)
radial distance from the origin
\(\mathrm{m}\)
\(\theta\)
polar angle measured in radians
\(\mathbf e_r\)
unit vector pointing radially outward
\(\mathbf e_\theta\)
unit vector perpendicular to \mathbf e_r in the direction of increasing theta
\(\dot r\)
radial velocity component
\(\mathrm{m\,s^{-1}}\)
\(\dot\theta\)
angular velocity
\(\mathrm{rad\,s^{-1}}\)
Method
Step 1: Write position in the moving polar basis
The position vector has only a radial component:
Polar position
\[\mathbf r=r\mathbf e_r\]
Step 2: Differentiate the moving basis
The polar unit vectors rotate as \(\theta\) changes:
Polar basis derivatives
\[\dot{\mathbf e}_r=\dot\theta\mathbf e_\theta,\quad \dot{\mathbf e}_\theta=-\dot\theta\mathbf e_r\]
Step 3: Differentiate position for velocity
Use the product rule because both \(r\) and \(\mathbf e_r\) can change:
Polar velocity
\[\mathbf v=\dot r\mathbf e_r+r\dot\theta\mathbf e_\theta\]
Step 4: Differentiate velocity for acceleration
Differentiate again and collect radial and angular components:
Polar acceleration
\[\mathbf a=(\ddot r-r\dot\theta^2)\mathbf e_r+(r\ddot\theta+2\dot r\dot\theta)\mathbf e_\theta\]
Start from position
\[\mathbf r=r\mathbf e_r\]
Differentiate position
\[\mathbf v=\dot r\mathbf e_r+r\dot{\mathbf e}_r\]
Use radial basis derivative
\[\mathbf v=\dot r\mathbf e_r+r\dot\theta\mathbf e_\theta\]
Differentiate velocity
\[\mathbf a=(\ddot r-r\dot\theta^2)\mathbf e_r+(r\ddot\theta+2\dot r\dot\theta)\mathbf e_\theta\]
Rules
Radial acceleration component
\[a_r=\ddot r-r\dot\theta^2\]
Angular acceleration component
\[a_\theta=r\ddot\theta+2\dot r\dot\theta\]
Circular motion special case
\[r=R,\ \dot r=0:\quad \mathbf a=-R\dot\theta^2\mathbf e_r+R\ddot\theta\mathbf e_\theta\]
- The polar basis vectors depend on \(\theta\), so they are usually time-dependent.
- The term \(-r\dot\theta^2\mathbf e_r\) points inward for circular motion.
- The term \(2\dot r\dot\theta\mathbf e_\theta\) appears when radial distance and angle both change.
- Polar coordinates are useful for central-force and circular-motion problems.
Examples
Question
A particle has
\[r=2\]
m and \[\theta=\pi/3\]
What does \(r\) represent?Answer
The coordinate \(r\) is the distance from the origin to the particle. Here the particle is
\[2\]
m from the origin along the direction making angle \[\pi/3\]
with the reference axis.Checks
- Use radians for \(\theta\) in calculus formulas.
- Do not treat \(\mathbf e_r\) and \(\mathbf e_\theta\) as constant when \(\theta\) changes.
- The symbol \(r\) is a distance coordinate, while \(\mathbf r\) is the position vector.
- Check whether a problem is planar before using only polar coordinates.