Lecture on Normal and Tangential Coordinate Systems and Polar Coordinate Systems

Key Concepts

• Velocity and Acceleration in Normal Tangential Coordinates

  • Velocity: ( \rho \beta \dot{et} )
  • Acceleration: ( \rho \beta \double dot et + \rho \beta \dot^2 en )
  • Analysis of Dot Products in Coordinate Systems
    • Using the right-hand rule to understand directionality of vectors
  • Golden Formula for Time Derivative of Unit Vectors: Angular velocity vector cross product with the unit vector itself

• Cross Products and Derivatives

  • Reviewed the cross product in Cartesian coordinates
  • ( A \times B = I(ay \cdot bk - by \cdot ak) - J(ax \cdot bk - ak \cdot bx) + K(ax \cdot by - ay \cdot bx) )
  • Utilizing cross products in calculating derivatives in time for vectors

• Application: Vehicles and Curves

  • Calculating maximum speed when normal acceleration is limited
  • Example of unbanked, level roads
  • Use of formula ( a_n = \frac{v^2}{\rho} )

• Problem Solving Examples

  • Discussed maximum speeds for cars A and B by calculating normal acceleration given constraints
  • Analyzed a truck's acceleration passing a hump: constant speed, radius of curvature, and height of the center of mass

• Derivatives in Polar Coordinates

  • Polar coordinate system when radius is not constant
  • Velocity and acceleration when radius varies with time
  • Right-hand rule in polar systems: ( er, e\theta, K )

• Examples of Polar and Normal Tangential Coordinates

  • Football trajectory: Calculating radius of curvature
  • Pendulum example: Understanding angular velocity and acceleration using different systems

• Polar Coordinate System

  • Introduction to polar coordinates for rotation where radius changes
  • Velocity: ( r\dot \cdot er + r\theta\dot \cdot e\theta )
  • Acceleration: ( r\double dot - r\theta\dot^2 \cdot er + (r\theta\double dot + 2r\dot\theta\dot) \cdot e\theta )

Future Topics

• Further exploration of pendulum problems in polar coordinates
• Application of polar systems in vibration and control classes
• More class problems to solidify understanding

Conclusion

• Understanding different coordinate systems is crucial in physics and engineering to accurately describe motion
• Using coordinate systems effectively simplifies problem-solving and deriving equations
• Upcoming classes will focus more on practical problems and applications using these concepts