Q.Write down the expression for the angle made by the resultant acceleration and the radius vector in non-uniform circular motion.

If the speed of an object moving on a circular path also changes with time (not just its direction), the motion is non-uniform circular motion — e.g. a pendulum bob swinging through a vertical circle moves fastest at the bottom and slowest near the top. Such a particle experiences both a centripetal acceleration $a_c=v^2/r$ (from the changing direction) and a tangential acceleration $a_t=dv/dt=r\alpha$ (from the changing speed) simultaneously, and these two are always mutually perpendicular ($a_c$ radial, $a_t$ tangential). The net, resultant acceleration is their vector sum:

$a_R=\sqrt{a_c^2+a_t^2}, \qquad \tan\theta=\frac{a_t}{a_c}$ …