Problem

An important and useful property of vectors is they may be easily transformed from one Car...

An important and useful property of vectors is they may be easily transformed from one Cartesian coordinate system to another. That is, if the x and y components of a vector are known, the t and n components can be found (or vice versa) by applying the formulas

In these equations, and are unit vectors in the t and n directions, respectively; ϕ is measured positive counterclockwise from the positive x direction to the positive t direction: and the y and n directions must be oriented 90° counterclockwise from the positive x and t directions, respectively.

(a) Derive the above transformation that gives rt, and rn, in terms of vx and ry. Hint: First consider a vector that acts in the x direction, and resolve this into components in t and n directions. Then consider a vector that acts in the y direction, and resolve this into components in t and n directions. Vectorially adding these results yields the transformation.

(b) For the eyebolt and post of Example 2.7, the x and y components of the resultant force are given by Eq. (4) of Example 2.6. Use these x and y components with the above transfomiation equations to obtain the t and n components of the resultant force, and verify these are the same as those in Eq. (4) of Example 2.7.