How Does An Object Move With a Constant Speed When Travelling on a Circle Segments

Question

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.

Answer 1

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!]

Answer 2

You forget that there are other forces acting than just gravity.

Sure, if only gravity acted then on the way up you would slow down and on the way down you would speed up. Like a roller-coaster cart in a circular loop.

But what if you with your car engine apply a force forwards when going up and brake when going down? What if you adjust this perfectly to counteract gravity? Then you do have a uniform (constant-speed) circular motion through the circle section.

In any horisontal circular path that takes place we don't necessarily have any default always-acting force like gravity. Your described confusion should thus only appear with vertical circular paths. Just think of driving around a roundabout. Here we must supply the force that causes the turning, which might be friction or so - here the speed is only constant if the car engine constantly applies a driving force that perfectly balances out any counteracting rolling friction, just like it does when driving straight ahead on non-curving roads.

We are taught that centripetal acceleration exists when an object is in uniform circular motion

Just a note to the terminology: centripetal acceleration exists even when the speed changes. Whenever turning takes place, the perpendicular acceleration causing the turning is called the centripetal (meaning centre-seeking) acceleration regardless of any parallel, tangential acceleration taking place simultaneously. In the specific case of constant speed, we use the word uniform to indicate that the speed doesn't change.