# Why is there no tangential acceleration in uniform circular motion? [duplicate]

I need to know what all is remains constant in uniform circular motion. The tangential SPEED, angular velocity and centripetal acceleration right? Why is there no tangential acceleration in uniform circular motion? Isn’t the tangential velocity always changing direction? The magnitude of the tangential velocity,I.e,tangential speed is the same at every point but the VELOCITY is not, right? But then why does the formula : acceleration(tangential) = radius*acceleration(angular) tell me that the tangential acceleration is zero?

The velocity is changing only the direction, which is why there exists only a centripetal acceleration. If the velocity would have been changing its magnitude (in other words, if the speed was changing) then you'd see that tangential acceleration would exist. This can also be seen by the relevent formula of a body's acceleration in polar coordinates:

$$\mathbf a=(\ddot r -r\dot{\theta}^2)\mathbf{\hat r}+(r\ddot{\theta}+2\dot r \dot{\theta})\boldsymbol{\hat{\theta}}$$

Since it's uniform circular motion, $$\dot r$$, $$\ddot r$$ and $$\ddot{\theta}$$ are zero. Thus the value of acceleration simplifies to

$$\mathbf a=-r\dot{\theta}^2\mathbf{\hat r}$$

As you can see, this is precisely the centripetal acceleration, and the value of tangential acceleration is zero.

Why is there no tangential acceleration in uniform circular motion?

Because, then we wouldn't call it uniform.

Circular motion can have both tangential $$a_\parallel$$ and centripetal $$a_\perp$$ acceleration. They are then the components of an acceleration vector $$\vec a=(a_\parallel,a_\perp)$$ pointing at an angle. Those special cases where there is no tangential acceleration, $$\vec a=(0,a_\perp)\Leftrightarrow a=a_\perp$$, we call them uniform.

Isn’t the tangential velocity always changing direction?

Indeed it is. But that does not require a tangential acceleration.

• Changing the velocity magnitude requires tangential acceleration $$a_\parallel$$ (parallel, so elongating/shrinking the velocity vector).
• Turning it requires centripetal acceleration $$a_\perp$$ (perpendicular, so turning the velocity vector).

So, in situations where we only turn, we only have centripetal acceleration. Likewise, in situations where we only speed up or slow down, we only have tangential acceleration.