Introduction
Transforming between inertial and rotating frames requires accounting for time derivatives of rotating basis vectors. Position vectors are same, but velocities and accelerations transform with additional terms from frame rotation. These transformations are fundamental for dynamics on rotating Earth and in rotating machinery.
Velocity Transformation
v_inertial = v_rot + ω × r. Inertial velocity equals rotating frame velocity plus velocity due to frame rotation (ω × r). This cross product gives tangential velocity at position r in rotating frame. All points in rotating frame have this additional velocity component in inertial frame.
Acceleration Transformation
a_inertial = a_rot + 2ω × v_rot + ω × (ω × r) + (dω/dt) × r. Four terms: rotating frame acceleration, Coriolis, centrifugal, and Euler (from changing ω). Last three are fictitious accelerations seen in rotating frame. Centrifugal points radially outward, Coriolis is perpendicular to velocity.
\nIntroduction
Transforming between inertial and rotating frames requires accounting for time derivatives of rotating basis vectors. Position vectors are same, but velocities and accelerations transform with additional terms from frame rotation. These transformations are fundamental for dynamics on rotating Earth and in rotating machinery.
Velocity Transformation
v_inertial = v_rot + ω × r. Inertial velocity equals rotating frame velocity plus velocity due to frame rotation (ω × r). This cross product gives tangential velocity at position r in rotating frame. All points in rotating frame have this additional velocity component in inertial frame.
Acceleration Transformation
a_inertial = a_rot + 2ω × v_rot + ω × (ω × r) + (dω/dt) × r. Four terms: rotating frame acceleration, Coriolis, centrifugal, and Euler (from changing ω). Last three are fictitious accelerations seen in rotating frame. Centrifugal points radially outward, Coriolis is perpendicular to velocity.
\nGeneral Equation of Motion
In rotating frame: ma_rot = F_real - 2m(ω × v) - mω × (ω × r) - m(dω/dt) × r. Real forces plus three fictitious forces equal mass times rotating frame acceleration. Used for dynamics on rotating Earth and in rotating machinery. Euler term important for changing rotation rate.
Time Derivatives in Rotating Frames
For any vector A: (dA/dt)_inertial = (dA/dt)_rot + ω × A. Rate of change in inertial frame equals rate in rotating frame plus contribution from frame rotation. Fundamental relation for transforming all vector quantities including velocity, angular momentum, and magnetic fields.
Applications
Satellite orbital mechanics in Earth-fixed coordinates. Aircraft navigation accounting for Earth rotation. Molecular dynamics in rotating molecules. Industrial centrifuges and separators. Inertial navigation systems using gyroscopes. Weather and ocean current modeling on rotating Earth.
\nSolved Example: Object Falling on Earth
Calculate the eastward deflection of an object dropped from height h = 100 m at the equator due to Coriolis effect. Solution: At equator, ω is perpendicular to surface. Object initially has inertial velocity v0 = ωR eastward (R = Earth's radius). At height h, inertial velocity is v = ω(R+h). As it falls, it keeps this eastward velocity (conservation of angular momentum, ignoring air resistance). Ground at radius R moves at ωR. Relative to ground, falling object has excess eastward velocity: Δv = ω(R+h) - ωR = ωh. Time to fall: from h = 1/2gt2 (approximately, ignoring small correction), t = √(2h/g) = √(200/9.8) = 4.52 s. Eastward deflection: d = Δv × t = ωh × t = 7.27×10^-5 × 100 × 4.52 = 0.0329 m = 3.3 cm. More accurate calculation using Coriolis: a_c = 2ωv_vertical (eastward). Integrating with v_vertical = gt: deflection = 1/3ωgt3 = 1/3 × 7.27×10^-5 × 9.8 × (4.52)3 = 2.38×10^-4 × 92.4 = 0.022 m = 2.2 cm. The exact result is about 2.2 cm eastward deflection. Objects falling from towers deflect eastward due to conservation of angular momentum (they started with larger eastward velocity at higher altitude).
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