# Tangential and centripetal acceleration in circular motion

• beasteye
In summary, the acceleration in circular motion can be broken down into a tangential and radial component, and this occurs at every point along the path. When the speed is uniform, the velocity vector only changes in direction, but the tangential component of acceleration is not equal to zero. However, with differentiation, the instantaneous rates of change for both components approach zero as the time interval approaches zero. This is an important concept to understand in circular motion.
beasteye
I'm trying to understand the geometrical meaning of acceleration in circular motion. When I consider a particle moving in a circular path, I can clearly see that the acceleration vector can be composed of a tangential and radial component. But since the change in velocity happens over a period of time, the particle has to move a small distance along the circle (arc), so my question is this: when we say that the acceleration has a radial and tangential component, at what point on the path are we talking about? Because we could say tangential and radial to the initial point, or the final point.

Now another thing I don't understand is that if the speed (magnitude of velocity vector) is uniform, the particle's velocity vector consequently only changes in direction. This has to mean that the initial and final velocity vectors have to be of equal length, but drawing them with tails in the same origin still shows that the acceleration has a tangential component not equal to zero.
I know I'm very off track somewhere in this thinking, because it clearly makes sense in an intuitive way, but not geometrically.

beasteye said:
But since the change in velocity happens over a period of time, the particle has to move a small distance along the circle
Yes, that's true for the average. That's where differentiation comes in: the instantaneous rate of change is the limit of the change divideed by the delta time.

That way you have a unique tangential and a radial acceleration at each moment and at each point alogng the trajectory.

Very important concept, differentials

beasteye said:
Now another thing
Not really another thing. The little difference vector has a vanishing tangential component for ##\Delta t\downarrow 0## and also a vanishing component for the radial component. But the first one goes like cosine -1 and the second one like the sine of ##\Delta \theta##. So divided by ##dt## the first one gives zero and the second one ##-\omega^2 r ## in the limit ##dt \downarrow 0##.

There are more accesible explanations -- is this adequate or is differentiation new for you ?

Oh yes, this actually makes sense now. You explained it very nicely, and yes I'm familiar with differentiation, just wasn't quite sure what happened in a geometrical sense. Anyways, thank you for your help :)

You're welcome

## 1. What is tangential acceleration in circular motion?

Tangential acceleration is the rate of change of the tangential velocity of an object moving in a circular path. It is a vector quantity that is directed along the tangent to the circular path at any given point.

## 2. What is centripetal acceleration in circular motion?

Centripetal acceleration is the acceleration towards the center of a circular path that is required to keep an object moving in a circular motion. It is always perpendicular to the tangential velocity and points towards the center of the circle.

## 3. How are tangential and centripetal acceleration related?

Tangential and centripetal acceleration are related by the equation at = rω2, where at is tangential acceleration, r is the radius of the circular path, and ω is the angular velocity. This means that as the tangential velocity increases, so does the centripetal acceleration.

## 4. How do tangential and centripetal acceleration affect circular motion?

Tangential and centripetal acceleration are essential for circular motion as they work together to maintain the object's motion along a circular path. Tangential acceleration keeps the object moving at a constant speed, while centripetal acceleration ensures that the object stays on the circular path.

## 5. What is the difference between tangential and centripetal acceleration?

The main difference between tangential and centripetal acceleration is their direction. Tangential acceleration is directed along the tangent to the circular path, while centripetal acceleration is directed towards the center of the circle. Additionally, tangential acceleration affects the object's speed, while centripetal acceleration affects its direction of motion.

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