Tangential acceleration

AP Physics 1 EK 2.9.A.3, word for word as C: Mechanics EK 2.10.A.3: tangential acceleration is the rate at which an object's speed changes and is directed tangent to the object's circular path.

Both equation sheets print it directly under $v = r\omega$:

AP Physics 1 lists it among the derived relationships in EK 5.2.A.2, alongside $s = r\theta$ and $v = r\omega$. The subscript is a capital T for tangential, not a radial component.

The two components do different jobs.

• $a_T = r\alpha$ points along the path, changes the speed, and is zero whenever the speed is steady.
• $a_c = v^2/r$ points at the center, changes only the direction, and is zero only when the object is momentarily at rest.

So uniform circular motion has $a_T = 0$ with $a_c$ possibly large, while a wheel released from rest has $a_T \ne 0$ and $a_c = 0$ at that first instant because $v = 0$.

They add as vectors, not as numbers. EK 2.9.A.4 states that the net acceleration of an object moving in a circle is the vector sum of the centripetal and tangential accelerations. They are perpendicular, so the magnitude is $\sqrt{a_c^2 + a_T^2}$, and a car speeding up through a bend accelerates neither straight inward nor straight forward.

Do not confuse $a_T$ with $\alpha$. Angular acceleration is in rad/s$^2$ and is the same for every point on a rigid body. Tangential acceleration is in m/s$^2$ and grows with distance from the axis, since $r$ connects them: a point on the rim of a disc has a larger $a_T$ than a point halfway in at the same $\alpha$.