Lecture Notes on Tangent and Normal Unit Vectors, Linear Momentum, and Angular Momentum

Overview of Topics

• Tangent and Normal Unit Vectors
• Linear Momentum and Impulse
• Angular Momentum

1. Tangent and Normal Unit Vectors

• Definition:
  • Tangent unit vector (U_T): Aligns with the tangent to the path.
  • Normal unit vector (U_N): Points inward on the curve.
• Acceleration Vector:
  • Acceleration can be expressed as:
    • A = V dU_T + V^2 / ρ U_N
  • Where ρ is the radius of curvature.
• Derivation of Radius of Curvature:
  • Formula:
    • ρ = dy/dx (1 + (dy/dx)^2)^(3/2) / |d²y/dx²|
  • Example with y = a sin(Kx).
  • Numerical calculation of radius of curvature at a specific point.
• Acceleration Calculation:
  • Example of a car driving on a curve with constant speed.
  • Centripetal acceleration calculated and its significance.

2. Linear Momentum and Impulse

• Newton's Second Law:
  • Sum of external forces (ΣF) = mass (m) × acceleration (a).
• Impulse:
  • Defined as the integral of forces over time leading to a change in momentum.
• Impulse-Momentum Relationship:
  • Final momentum (mv2) = Initial momentum (mv1) + Impulse.
• Example Problem:
  • Block sliding down a hill under gravity and friction.
  • Break down forces into components to find acceleration.

3. Angular Momentum

• Definition:
  • Angular momentum of a particle (h) with respect to a point is given by:
    • h = r × p, where p = mv (momentum).
• Torque:
  • Time derivative of angular momentum is equal to the torque applied to the object.
  • In planar motion often expressed as Iθ² (moment of inertia) or τ = Iα (torque = moment of inertia × angular acceleration).
• New Concept:
  • Angular momentum concerning a point not fixed at the center of mass.
  • Derivation shows how to account for torque and angular momentum with respect to arbitrary points.
• Practical Application:
  • Example of a rotating arm with a mass, calculating torque and forces experienced by the mass.
• Key Formulas:
  • Torque with respect to a fixed axis = time rate of change of angular momentum.
  • Special cases:
    • When the point about which momentum is calculated is moving and when it's at center of mass.

Conclusion

• Review of key properties of linear and angular momentum.
• Importance of understanding these concepts for solving physics problems.
• Reminder: Chapters on linear momentum and angular momentum should be reviewed for practice problems.