Acceleration of the rim of a wheel

Radial and Tangential Acceleration

A wheel (radius r), starting from rest undergoes a uniform angular acceleration of alpha about its fixed axle. What are the tangential and radial accelerations as a function of alpha, r and t. Also, show that the angle between the total acceleration vector at P on the edge of the wheel and the radial vector is inversely proportional to the number of revolutions the wheel has made.

a)

a_tangential= alpha r


a_radial=a_centripetal=omega2 r=(alpha t)²r

b) Let theta be the angle between the vector a and P and N the number of rotations (a real number, so it includes partial rotations).

Tan(theta)=a_tangential/a_radial=alphar/(alpha t)²r

Tan(theta)=1/(alpha t)²

But:

alpha t²=2 omega =2*theta

and

omega=2 pi N

then

Tan(theta)=1/4 pi N