Centripetal vs Radial acceleration

[Amio]

I am reading introductory physics from Serway. Where they say if $$a_r$$ is radial acceleration and $$a_c$$ is centripetal acceleration then $$a_c = v^2/r$$ and $$a_r = -a_c = - v^2/r$$
But aren't the radial and centripetal acceleration same (correct me if I am wrong)? Why is there a minus sign?
The book explains by saying that the negative sign indicates that the direction of centripetal acceleration is towards the center of the circle representing the radius of curvature..
I don't understand this explanation because as the direction of the radial acceleration is also towards the center, shouldn't it be $$a_{radial} = a_{centripetal} ?$$ Why the minus sign?
Would someone please clarify?

 

[D H]

Do you know what spherical coordinates are? The radial component of a vector is the projection of that vector onto the radial unit vector. That vector points away from the origin. Thus the radial component of acceleration is positive if the acceleration vector is pointing away from the origin.

Centripetal acceleration is acceleration toward the center.

It's just a sign convention, nothing else.

 

[Amio]

Thank you.

 

[D H]

The term centripetal acceleration applies only when an object is following a curved path, and by definition the centripetal acceleration is always positive. On the other hand acceleration can have a positive or negative radial component. Consider two electrons moving toward another, for example, with the origin at the center of mass. The radial component of acceleration will be positive because like charges repel.

 

[PJC01]

Hello,

I can clarify the concept of centripetal and radial acceleration for you. First of all, let's define these two terms. Centripetal acceleration is the acceleration towards the center of a circular path, while radial acceleration is the acceleration in the direction of the radius of the circle.

Now, in the formula provided by the book, a_c = v^2/r, the minus sign is used to indicate the direction of the acceleration. Since the velocity, v, is always tangent to the circular path, the acceleration must be perpendicular to the velocity and towards the center of the circle. This is why the minus sign is used to indicate the direction of the acceleration, which is towards the center of the circle.

On the other hand, the radial acceleration, a_r, is also directed towards the center of the circle. However, it is defined as the change in direction of the velocity, which can be either towards or away from the center. This is why a_r = -a_c = -v^2/r, as the direction of the acceleration is opposite to the direction of the velocity.

In simpler terms, the minus sign is used to differentiate between the direction of the centripetal and radial acceleration, even though they both point towards the center of the circle. I hope this explanation helps to clarify any confusion you may have had. Keep studying and asking questions!