Confused by radial vs. centripetal acceleration

[eventob]

Hi

I've been working through some examples from the course material we use in physics class, but one thing keeps confusing me: What is the difference between centripetal and radial acceleration?

For instance, when we have a particle traveling in a circular path, the acceleration towards the center of the circle may be written as Ar (a sub r)=-Ac= - v^2/r, while other times it is written simply as Ac=v^2/r. The textbook (JS Physics for Scientists and Engineers) seems to use both.

Where is the negative sign coming from? I made a quick sketch. Am I right if i think that the radial acceleration is negative in the first circle (to the left) and it is positive in the circle to the right? Is it just due to how I pick the axis and how I define positive direction?


Thanks in advance.

 

[Doc Al]

but one thing keeps confusing me: What is the difference between centripetal and radial acceleration?

Same thing.

For instance, when we have a particle traveling in a circular path, the acceleration towards the center of the circle may be written as Ar (a sub r)=-Ac= - v^2/r, while other times it is written simply as Ac=v^2/r. The textbook (JS Physics for Scientists and Engineers) seems to use both.

v^2/r is the magnitude of the radial acceleration; the direction is toward the center. Whether that's positive or negative just depends on how you define your sign convention.

Where is the negative sign coming from? I made a quick sketch. Am I right if i think that the radial acceleration is negative in the first circle (to the left) and it is positive in the circle to the right? Is it just due to how I pick the axis and how I define positive direction?

Yes.

 

[eventob]

Thank you very much. :)

 

[K^2]

Radial acceleration is equal to centripetal acceleration when the radius remains constant (with a +/- sign depending on definition). If radius changes as a function of time, you have to add the explicit second derivative of radius with respect to time.

$$a_r = a_c + \ddot{r} = -\omega^2 r + \frac{d^2r}{dt^2}$$

Similarly, tangential acceleration will pick up a term that depends on the second derivative of angle with respect to time and a Coriolis Effect term.

 

[PJC01]

Hello,

I can understand why you may be confused about the difference between centripetal and radial acceleration. Let me try to clarify this for you.

Centripetal acceleration is the acceleration that is directed towards the center of a circular path. This acceleration is necessary to keep an object moving in a circular path, as without it, the object would move in a straight line. This acceleration is always directed towards the center of the circle, and its magnitude is given by the equation Ac = v^2/r, where v is the velocity of the object and r is the radius of the circle.

On the other hand, radial acceleration is the component of acceleration that is directed along the radius of the circle. This means that it can be either towards the center of the circle or away from it, depending on the direction of motion of the object. In your example, the negative sign in the equation Ar = -Ac = -v^2/r indicates that the radial acceleration is directed towards the center of the circle, while the positive sign in the equation Ac = v^2/r indicates that the centripetal acceleration is directed away from the center of the circle.

In terms of your sketch, you are correct in thinking that the radial acceleration is negative in the first circle (to the left) and positive in the second circle (to the right). This is because the direction of motion changes in the two circles, and therefore the direction of the radial acceleration also changes.

I hope this helps to clarify the difference between centripetal and radial acceleration. Remember, the key difference is in the direction of the acceleration - centripetal is always towards the center of the circle, while radial can be either towards or away from the center, depending on the direction of motion. If you have any further questions, please don't hesitate to ask. Good luck with your physics studies!