Spherical Coordinates

🌐 Motion in Three Dimensions Using the Spherical Coordinate System

1. Introduction

The spherical coordinate system describes the position of a particle using its radial distance R, azimuthal angle θ, and polar angle ϕ. It is particularly useful for describing three-dimensional motion possessing spherical symmetry, such as planetary motion, radar systems, and electromagnetic fields.

Unlike Cartesian coordinates, the unit vectors rotate with the particle as it moves. The radial unit vector always points away from the centre of the sphere, while the angular unit vectors define directions tangent to the spherical surface. Together, they form a mutually perpendicular, right-handed coordinate system.

2. Animation Details

Coordinate System: Spherical (R, θ, ϕ)

Animation: Three-dimensional motion on a spherical surface

Trajectory: R(u) = 2.3, θ(u) = u, ϕ(u) = π/6 + 0.30 sin(2u)

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

Learning Objective: Visualize the spherical coordinate system and understand the orientation of its rotating unit vectors.

Convention: In this animation, the polar angle ϕ is measured from the positive x-axis.

3. Kinematic Equations

4. Key Observations

• The particle remains on the surface of a sphere since the radial distance R is constant.

• eᵣ always points radially outward from the centre of the sphere.

• eθ is tangent to the sphere and points in the direction of increasing θ.

• eϕ lies along the meridian and points in the direction of increasing ϕ.

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

5. Watch For

• Observe that the particle always remains on the spherical surface.

• Notice how eᵣ always aligns with the radius drawn from the centre of the sphere.

• Observe how eθ and eϕ rotate continuously as the particle moves.

• Compare the directions of eθ (around the sphere) and eϕ (along the meridian).

• Observe how the projection of the particle onto the x–y plane defines the azimuthal angle θ.

• Notice the geometric interpretation of the polar angle ϕ, measured from the positive z-axis.

Note:

This topic is not part of the official ME21205 syllabus and is included as supplementary material for students interested in extending the concepts of curvilinear coordinate systems to three-dimensional motion.