I'm experiencing extreme confusion. We are taught that centripetal acceleration exists when an object is in uniform circular motion, and that implies that the object has a constant speed. How then, could centripetal acceleration actually exist if the object is going through an arc?
Say, an arched ramp. Won't the object's speed change as it travels through the ramp? What about a car that goes through a wide dip on the road. Won't the car speed up as it travels down, then slow down as it travels back up?
How can an object maintain constant speed traveling vertically on a circle segment? And how can centripetal acceleration exist in that situation?
I apologize if the answer is obvious, but I'm just confused.
Best Answer
There is indeed centripetal acceleration for a body travelling in a circle at constant speed, but there is also centripetal acceleration for a body travelling in a circle at a smoothly varying speed, such as for a pendulum swinging through an arc of a circle.
The magnitude of the acceleration is constant in the former case, but varies in the latter. Nevertheless in both cases we can use the formula $$a_{centripetal}=\frac {v^2}{r}$$ In the latter case (varying speed) we simply use the speed, $v$, at a particular point in the motion, in order to find $a_{centripetal}$ at that point.
[Don't read this next bit if you're still not entirely happy, but we can also relax the need for the path to be circular. If we want to find the acceleration normal to the path at some point along a non-circular path, we substitute for $r$ the radius of curvature of the path at that point!]