Lecture: Coordinate Systems

Introduction

• Subject: Coordinate Systems in Mechanics
• Objective: To understand the position, velocity, and acceleration of an object
• Utility: BSc Non-medical, Computer Science, Physics Honors

Definition of Coordinate Systems

• Reference Frame: Used to find the position, velocity, and other motion parameters of an object.
• Origin: A predetermined point
• Axes: X, Y, Z

Types of Coordinate Systems

1. Cartesian Coordinate System
   • X, Y, Z axes
   • Unit Vectors: î, ĵ, k̂
   • Use: Determining positions within a box
2. Spherical Polar Coordinate System
   • Use: Spherically symmetric objects
   • Coordinates: r, θ, φ
3. Cylindrical Coordinate System
   • Coordinates: ρ, θ, z
   • Use: For cylinders

Vector Calculation in Cartesian System

• Position Vector: r = xî + yĵ + zk̂
• Velocity Vector: v = dx/dt î + dy/dt ĵ + dz/dt k̂
• Acceleration Vector: a = d²x/dt² î + d²y/dt² ĵ + d²z/dt² k̂
• Magnitude of Vector:
  • Velocity: |v| = √(vx² + vy² + vz²)
  • Acceleration: |a| = √(ax² + ay² + az²)

Plane Polar Coordinate System

• Use: To find the position of an object in a plane
• Coordinates: r (radius), θ (angle)
• Conversion from Cartesian to Plane Polar:
  • x = r cos θ
  • y = r sin θ
  • r = √(x² + y²)
  • θ = tan⁻¹(y/x)

Unit Vectors

• r̂ (Radially Outward): Unit vector in the radial direction
• θ̂ (Angular Direction): Unit vector in the angular direction

Conclusion

• Importance of Coordinate Systems: To measure and analyze the position and motion of objects
• In the next lecture, we will understand unit vectors in detail.

Note: These notes are prepared based on the examples and explanations given in the lecture.