Parametrization of the path described by the end of a thread

[rashomon]

Homework Statement

2. Consider a stationary circular spool of thread of radius R. Assume the end of the thread is initially located at (0; R). While keeping the thread taut, the thread is unwound in a clockwise direction.

(a) Parameterize the path described by the end of the thread as r(t) = x(t)i + y(t)j. You may assume that the radius of the spool does not change as the thread is unwound.

(b) Determine the arc length, s(t), of the path traced out by the end of the thread.

Homework Equations

The Attempt at a Solution

It seems to me that it is describing the taut thread as it is more or less rotated around the spool. I am confused on why there isn't a z(t) element too.

 

[lanedance]

consider the axis of the spool as aligning with the z axis, and the thread constrained to the xy plane