Q.Derive the expression for the total (resultant) acceleration in non-uniform circular motion.

Step 1. Two accelerations present. In non-uniform circular motion, both the direction and the speed of the particle change. The changing direction produces the centripetal acceleration $a_c=v^2/r$ (derived in §2.11.6/Q6 above), directed radially toward the centre. The changing speed produces the tangential acceleration $a_t=\dfrac{dv}{dt}=r\alpha$ (from §2.11.5, differentiating $v=r\omega$ with $r$ constant), directed along the tangent to the circle (along $\vec v$, or opposite to it if slowing down).

Step 2. Perpendicularity. The radial direction (toward the centre) and the tangential direction (along the instantaneous velocity) are, by construction, always mutually perpendicular at every point on the circle.

Step 3. Vector sum. Since $\vec a_c$ and $\vec a_t$ are perpendicular components of the total (resultant) acceleration $\vec a_R=\vec a_c+\vec a_t$, ordinary Pythagoras applies to find the magnitude: …

Identify the two perpendicular acceleration components (centripetal from changing direction, tangential …

• Adding a_c and a_t as plain numbers instead of combining them as perpendicular vector components …