Difference between radial and centripetal acceleration?

[AMan24]

Homework Statement

I'm not understanding the difference between them, this is for Uniform Circular Motion.

Homework Equations

a_r = -a_c = -v²/r

The Attempt at a Solution

So what i know is radial acceleration goes in a direction towards the radius (perpendicular to velocity), and tangential goes with the direction of velocity.

For some reason my book, uses radial acceleration and centripetal acceleration interchangeably, but it doesn't make sense because of that equation i posted which was also in my book.

 

[kuruman]

The difference is that centripetal is always towards the center, but radial could be towards or away from the center in the radial direction.

 

[Doc Al]

radial acceleration goes in a direction towards the radius (opposite direction of velocity),

I think you mean perpendicular to the velocity.

 

[gneill]

I think you mean perpendicular to the velocity.

...if the motion is circular. Consider an elliptical orbit...

 

[haruspex]

Seems to be a variety of views.
I would define radial acceleration as that component of the acceleration vector which is orthogonal to the velocity vector. If the current position is at $\vec r$ relative to the instantaneous centre of curvature then $\vec r.\vec {a_r}=\vec r.\vec a=-\vec v.\vec v$.
We can deduce this by noting that by definition not only is $\vec r.\vec v=0$ but also its first derivative wrt t is zero: $0=\dot{\vec r}.\vec v+\vec r.\dot{\vec v}=\vec v.\vec v+\vec r.\vec a$.
If we think of centripetal acceleration as a vector then there is no difference between the two (other than that many would only use the term centripetal for circular motion).
I believe that all agrees with the description at https://en.m.wikipedia.org/wiki/Acceleration#Tangential_and_centripetal_acceleration.

Or, perhaps you could use centripetal acceleration to mean just the magnitude.

The difference is that centripetal is always towards the center, but radial could be towards or away from the center in the radial direction.

No, it must always be towards the instantaneous centre of curvature. Maybe you are thinking of radial in respect of an arbitrary origin?

...if the motion is circular. Consider an elliptical orbit...

It is always perpendicular to the velocity. Again, maybe you are using it in respect of an arbitrary origin.

 

[kuruman]

No, it must always be towards the instantaneous centre of curvature. Maybe you are thinking of radial in respect of an arbitrary origin?

I was thinking instantaneous center of curvature, but neglected to say so.

 

[gneill]

It is always perpendicular to the velocity. Again, maybe you are using it in respect of an arbitrary origin.

I agree that if you consider the instantaneous center of curvature along an arbitrary trajectory that the radial velocity will always be perpendicular to the velocity vector. But I suspect that this approach might be outside the OP's current level of study (I invite @AMan24 to correct me here!).

I think it more likely that original intention was to address circular motion about a fixed center. I added the allusion to elliptical orbits, where the center is conventionally taken to be Sun, so that the concept wouldn't be taken as universally true regardless of circumstance.

 

[haruspex]

Or, perhaps you could use centripetal acceleration to mean just the magnitude

Rereading post #1, I think that mght be the answer to the original query. In the relevant equation posted, it seems a_r is being taken as positive in the positive r direction (fairly standard) but a_c is either being taken as a magnitude or measured in the direction towards the instantaneous centre. The book, on the other hand, may be treating both as vectors, and from the same origin, making them interchangeable.

 

[haruspex]

I agree that if you consider the instantaneous center of curvature along an arbitrary trajectory that the radial velocity will always be perpendicular to the velocity vector. But I suspect that this approach might be outside the OP's current level of study (I invite @AMan24 to correct me here!).

I think it more likely that original intention was to address circular motion about a fixed center. I added the allusion to elliptical orbits, where the center is conventionally taken to be Sun, so that the concept wouldn't be taken as universally true regardless of circumstance.

Ok, but it comes down to the definition of "radial acceleration" (you meant that, not radial velocity, right?). According to the link I posted, it means the component of acceleration normal to the velocity, not the acceleration in the direction away from an arbitrarily defined origin.
But since the OP is only concerned with circular motion as yet, this will be a future matter.

 

[gneill]

Ok, but it comes down to the definition of "radial acceleration" (you meant that, not radial velocity, right?)

Yes, of course. My error in typing faster than I was thinking