Chapter 5.4, Problem 76E

(Modeling) Roll of a Spacecraft The figure on the left below shows the three quantities that determine the motion of a spacecraft. A conventional three-dimensional spacecraft coordinate system is shown on the right.

Angle YOQ = θ and OQ = r. The coordinates of Q are (x, y, z). where

y = r cos θ and z = r sin θ.

When the spacecraft performs a rotation, it is necessary to find the coordinates in the spacecraft system after the rotation takes place. For example, suppose the spacecraft undergoes roll through angle R. The coordinates (x, y, z) of point Q become (x', y', z'), the coordinates of the corresponding point Q'. In the new reference system, OQ' = r and. because the roll is around the x-axis and angle Y' OQ' = YOQ = θ.

x' = x, y' = r cos(θ + R), and z' = r sin(θ + R).

(Source: Kastner, B., Space Mathematics. NASA.)

Write z' in terms of y, R, and z.