HW9_prob1_GPS21_G22

HW9_prob1_GPS21_G22 - Problem GPS 4.21 (2nd ed. 4.22) If we...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Problem GPS 4.21 (2 nd ed. 4.22) If we neglect the “centrifugal force” term in GPS Eq.(4.91), the equation of motion for a particle of mass m in the rotating frame of the earth is m d v dt = m g +2 m v × ω , (1) where v is the particle velocity, ω is the earth’s angular velocity vector, and g is the acceleration of gravity at the earth’s surface. All the vectors and derivatives aremeasuredinabody f xed frame in which the z axis is normal to the earth’s surface and the x - y plane is tangent to the earth’s surface and is located at z =0 . Assume that we are in the Northern hemisphere, and let the y axis point north and the x axis point east. As we saw in class, the solution to Eq.(1) to f rst order in ω is r = r 0 + v 0 t + 1 2 g t 2 +( v 0 × ω ) t 2 + 1 3 ( g × ω ) t 3 , (2) where the subscript 0 denotes an initial value. Since the initial velocity is upward (vertical), we can write v 0 = v 0 e z , (3) and gravity acts downward, so g = g e z . (4) With r 0 =0 ,the f rst three terms of Eq.(2) are seen to determine the height h and total time t tot of the particle’s trajectory as t tot =2 t h =2 v 0 /g , (5) h = 1 2 gt 2 h , (6) where t h is the time required for the particle to fall to the earth when it is released at the height h above the surface. The fourth and f fth terms in Eq.(2) are responsible for the de F ection of the particle from the vertical. Because ω lies, generally, in the body y - z plane, the cross products generate vectors along the body x axis, i.e., e z × ω = ω y e x = ω sin θ e x , (7) where θ is the polar angle, or colatitude, measured from the earth’s rotation axis. (The equator is at θ = π/ 2 .) Thus
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 5

HW9_prob1_GPS21_G22 - Problem GPS 4.21 (2nd ed. 4.22) If we...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online