Unit 1 · Topic 07

Polar coordinates

A position in the plane can also be described with a radial distance and an angle, instead of Cartesian components. This basis changes direction with motion, which introduces additional terms in velocity and acceleration.

Unit 1Polar coordinatesOpen navigation

Concept 01

Coordinates r and θ and a moving basis

Essential The minimum you should retain

A planar position can be described by radial distance r and angle θ measured from a reference axis.

r^ \hat r points radially outward and θ^ \hat\theta is perpendicular to r^ \hat r in the direction of increasing θ.

UnderstandInterpret and connect

Unlike i and j, polar unit vectors change direction when θ changes. Although their magnitudes remain one, their derivatives are not zero.

DeepenFormulation and conditions

The relations d r^ \hat r /dt = θ˙ \dot\theta θ^ \hat\theta and d θ^ \hat\theta /dt = − θ˙ \dot\theta r^ \hat r explain the additional terms that appear when differentiating position and velocity.

ExploreConnections for further study

Polar coordinates are natural when a distinguished centre exists. In problems without radial symmetry, a Cartesian basis may give a simpler description.

A particle lies on a ray at forty-five degrees. The radial unit vector points outward and the transverse unit vector is perpendicular.rr̂θ̂(r, θ)θ

r^ \hat r points from the origin toward the particle and θ^ \hat\theta is perpendicular in the direction of increasing θ. Both unit vectors rotate when the angle changes.

Concept 02

Velocity in polar coordinates

Essential The minimum you should retain

Velocity combines radial and angular change: one part follows r^ \hat r and another follows θ^ \hat\theta .

UnderstandInterpret and connect

r˙ \dot r measures how quickly distance from the origin changes. r θ˙ \dot\theta is the transverse speed caused by rotation.

DeepenFormulation and conditions

The expression follows by differentiating r→=rr^ \vec r=r\hat r and using the fact that r^ \hat r changes with θ. Differentiating only coordinate r is therefore insufficient.

ExploreConnections for further study

A spiral trajectory combines radial change and rotation. Depending on their signs, the object may approach the origin while moving in the direction of increasing θ.

Mathematical relation

Velocity in polar coordinates

v→=r˙r^+rθ˙θ^ \vec v=\dot r\hat r+r\dot\theta\hat\theta
Represents

The sum of the radial and transverse velocity components.

Physical interpretation

r˙ \dot r changes the distance from the origin; r θ˙ \dot\theta produces transverse motion.

DeepenVariables, conditions, and checks

Variables

v
velocity vector; usual unit: ms \mathrm{m/s}
r, ṙ
radius and its rate of change; usual unit: m; ms \mathrm{m/s}
θ˙ \dot\theta
angular speed; usual unit: rad/s

Conditions of application

  • r and θ describe position in a positively oriented plane polar basis.

Dimensional check

r˙ \dot r and r θ˙ \dot\theta have dimension LT L/T .

Concept 03

Polar acceleration and its relation to circular motion

Essential The minimum you should retain

Polar acceleration has radial and transverse components. Each can receive contributions from more than one type of change.

UnderstandInterpret and connect

For a circle of constant radius and speed, r˙=r¨=θ¨=0 \dot r=\ddot r=\ddot\theta=0 . The radial component reduces to −rθ˙2r^ -r\dot\theta^2\hat r , directed toward the centre.

DeepenFormulation and conditions

The transverse component contains r θ¨ \ddot\theta and 2 r˙ \dot r θ˙ \dot\theta . The second term appears when radial distance and angle change simultaneously.

ExploreConnections for further study

The polar basis is an example of curvilinear coordinates. Its usefulness depends on whether the problem's geometry makes the description simpler than a fixed Cartesian basis.

Mathematical relation

Acceleration in polar coordinates

a→=(r¨−rθ˙2)r^+(rθ¨+2r˙θ˙)θ^ \vec a=(\ddot r-r\dot\theta^2)\hat r+(r\ddot\theta+2\dot r\dot\theta)\hat\theta
Represents

The radial and transverse acceleration components in a moving basis.

Physical interpretation

For constant r and constant ω, only −r ω2 \omega^2 r^ \hat r remains.

DeepenVariables, conditions, and checks

Variables

a
acceleration vector; usual unit: ms2 \mathrm{m/s^2}
r, ṙ, r¨ \ddot r
radius and its time derivatives; usual unit: m; ms \mathrm{m/s} ; ms2 \mathrm{m/s^2}
θ˙ \dot\theta , θ¨ \ddot\theta
angular velocity and acceleration; usual unit: rad/s; rad/s²

Conditions of application

  • Planar motion described by twice-differentiable functions r→(t) \vec r(t) and θ(t).

Dimensional check

Every term has dimension LT2 L/T^2 .

Errors it helps prevent

  • Differentiating r and θ while ignoring that the unit vectors also change.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Treating r^ \hat r and θ^ \hat\theta as fixed vectors when differentiating.

Polar unit vectors rotate with θ; their derivatives generate additional radial and transverse terms.