Polar Coordinates

🌀 Motion in a Plane Using the Plane Polar Coordinate System

1. Introduction

In the plane polar coordinate system, the position of a particle is described by its radial distance r from a fixed pole and its angular coordinate θ. Unlike Cartesian coordinates, the unit vectors eᵣ and eθ rotate with the particle. This makes polar coordinates particularly useful for describing motion involving radial and angular changes.

2. Animation Details

Coordinate System: Plane Polar (r, θ)

Animation: Planar motion

Trajectory: r(θ) = 3 + 1.2 cos(2θ)

Vectors Displayed: Position, Velocity, Acceleration, eᵣ, eθ

Learning Objective: Visualize particle kinematics in a rotating polar coordinate system.

3. Kinematic Equations

4. Key Observations

• eᵣ and eθ rotate with the particle.

• Velocity is always tangent to the trajectory.

• Velocity and acceleration generally vary in both magnitude and direction.

5. Watch For

• eᵣ always points away from the pole.

• eθ remains perpendicular to eᵣ and points in the direction of increasing θ.

• Observe how the rotating unit vectors influence the velocity and acceleration vectors.

🛢️ Motion in Three Dimensions Using the Cylindrical Polar Coordinate System

1. Introduction

The cylindrical polar coordinate system extends the plane polar coordinate system by introducing an additional vertical coordinate z. The position of a particle is therefore described by its radial distance r, angular coordinate θ, and height z. It is particularly useful for analysing three-dimensional motion possessing rotational or cylindrical symmetry.

2. Animation Details

Coordinate System: Cylindrical Polar (r, θ, z)

Animation: Three-dimensional helical motion

Trajectory: r(u) = 2 + 0.25 sin(u), θ(u) = u, z(u) = 0.25u

Vectors Displayed: Position, Velocity, Acceleration, eᵣ, eθ, ez

Learning Objective: Visualize particle kinematics in a cylindrical polar coordinate system and understand the orientation of its unit vectors.

3. Kinematic Equations 

4. Key Observations

• Cylindrical polar coordinates extend plane polar coordinates by introducing the vertical coordinate .

• The unit vectors eᵣ and eθ rotate with the particle, while ez remains fixed.

• The velocity vector is always tangent to the trajectory.

• The unit vectors eᵣ, eθ, and ez form a right-handed orthogonal coordinate system.

5. Watch For

• eᵣ always points radially outward from the pole.

• eθ remains perpendicular to eᵣ and points in the direction of increasing θ.

• ez remains fixed and is perpendicular to both eᵣ and eθ.

• Observe the translucent cylinder that provides the geometric basis for the cylindrical polar coordinate system.

• Compare the governing equations with those of plane polar coordinates and identify the additional z-component.