Newton's Laws and Reference Frames in Aircraft Motion

Newton's First Law

• Definition: A body at rest stays at rest, and a body in motion continues at constant velocity unless acted upon by an external force.
• Inertial Frame of Reference: Necessary for Newton's first law to apply, meaning no acceleration.
• Example: On a train, a ball moves without an external force when brakes are applied due to non-inertial frame.
• Application: Important in deriving equations of motion for aircraft.

Earth Axis System

• Definition: Reference frame fixed to the Earth's surface.
• Non-Inertial Frame: Requires an acceleration to maintain circular motion at constant altitude.
• Centrifugal Force: Apparent force due to acceleration, similar to the train example.
• Acceleration Calculation:
  • Formula: ( \frac{V^2}{r} )
  • Typical Aircraft Velocity: 250 m/s or Mach 0.8.
  • Resulting Acceleration: 0.0097 m/s², negligible.
• Flat Earth Assumption: Earth’s rotation effects are negligible except at high altitudes and speeds.

Aircraft Reference Frames

• Moving Earth Axis System: Frame attached to the aircraft but oriented like the Earth-fixed frame.
• Body Axis System: Fixed to the aircraft body; pilot views along the x-axis.
• Air Path Axis System: Aligned with the airspeed vector, accounting for lift and drag.
• Flight Path Angle (( \gamma )): Angle of airspeed vector with the horizon.

Forces and Motion

• Forces on Aircraft:
  • Gravitational Force (W): Acts towards Earth's center; small altitude variation effect.
  • Aerodynamic Forces: Lift (perpendicular) and drag (parallel) to airspeed.
  • Thrust: Directional based on propulsion system.
• Acceleration Definitions:
  • Forward Acceleration: Change in velocity (dv/dt).
  • Perpendicular Acceleration: Centripetal acceleration (( \frac{V^2}{r} )).

Equations of Motion

• Newton's Second Law: ( F = ma ), forces as vectors.
• Free Body Diagram: Represents forces and motion for deriving equations.
• Parallel Forces Equation:
  • ( m \frac{dv}{dt} = T \cos(\alpha_t) - D - W \sin(\gamma) )
  • Simplified using approximations for thrust and weight.
• Perpendicular Forces Equation:
  • ( mV \frac{d\gamma}{dt} = L - W \cos(\gamma) )
  • Consider perpendicular thrust components in special cases.

Summary

• Derived two primary equations of motion for aircraft performance.
• Will use these equations in further lectures for performance calculations.