A taught string $CP$ is unwrapped from a fixed drum, center $O$ and radius $R,$ with a uniform angular velocity $.$ The end of the string $P$ is initially in contact with the drum at $A,$ then traces out a planar curved path $AB$ as shown in Figure $2.$

a$)$ Find an expression for the position vector of $P$ relative to $O$ in terms of the intrinsic coordinate unit vectors $e_t$ and $e_n$ as shown $($note that this is a rare case for which the position vector is readily found in intrinsic coordinates$).$

b$)$ By differentiation find the velocity $r^{},$ and acceleration $r^{}$ of $P$ in this coordinate system.

c$)$ From the acceleration verify that the radius of curvature of the path of $P$ is equal to $CP.$ Would you expect this?

Note: This path is known as an 'involute' and is the most commonly used profile for gear teeth.