A vehicle particle moving relative to the earth is located i

A vehicle (particle) moving relative to the earth is located in terms of its position vector N_p B from the center of the earth. Place the b_x, b_y, b_z unit vectors with the origin at the center of mass of the moving vehicle, with b_z along N_p B. If the angular velocity of the b_x, b_y, b_z co-ordinate axes is: N_omega B = omega_x b+x + omega_y b_y + omega_z b_z show that particle acceleration is: N_a B = (omega_x omega_x r + omega_y r + 2 omega_y r) b_x +(omega_y omega_z r - omega_x r - 2 omega_x r) b_y + (r - omega_x^2 r- omega_y^2 r)b_x

Solution

earth -centred inertial coordinate frames have their origins at the center of mass of the earth. ECI frames are called intertical in cotrast to the earth- centred, earth-fixed frames which rotates in intertial space in order to remain fixed with respect to the surfface of the earth. it is conveninent to respect the position and velocities of trrestrial objects in ECEF coordinates or with latitude,longitude, and altitude. however for objects in space the equation of motion that describe orbital motion are simpler in a non rotating frame such as ECI. the ECI frame is also useful for specifing the direction toward celestial objects. ECI coordinate frames are not exactly intertial since the earth center of mass itself is accelerating as it travels in its orbit about the sun. in many cases it may be assumed that the ECI frames is inertial without adverse effect. however when computing the gravitional influnce of a thired body such as the moon on the dynamic of a spacecraft the acceleration of the ECI frame must be considered.