Doubt with centripetal acceleration

[pc2-brazil]

Good afternoon,

For a body in a curvilinear motion moving with velocity $$\vec{v}$$, velocity can be divided into two components (see http://en.wikipedia.org/wiki/Angular_velocity#Two_dimensions" Wikipedia article): a component parallel to the position vector (the radius), which changes the magnitude of the radius and is given by dr/dt, and a component perpendicular to the radius vector, called tangential velocity (v_t or $$v_{\bot}$$), which changes the direction of the radius vector.
The angular velocity is determined by the the tangential velocity:
$$\omega = \frac{v_t}{r}$$
And the centripetal acceleration is given by:
$$a_c = \frac{v^2}{r}$$ (1)
My question is: I have seen, in many places, that the centripetal acceleration can be written as:
$$a_c = r\omega^2 = \frac{v_t^2}{r}$$ (2)
But it doesn't seem right. Why is it defined like this so frequently?
EDIT: I realized that, in definition (2), "r" is the polar radius, and not necessarily the radius of curvature of the trajectory. Definition (1) is true when "r" is the radius of curvature. Am I right?

Thank you in advance.

 

[mathman]

At any given point on the circle, the velocity is the tangential velocity.

 

[pc2-brazil]

At any given point on the circle, the velocity is the tangential velocity.

Yes, but I defined tangential velocity as the component of velocity perpendicular to the position vector, like the Wikipedia article to which I linked.
What I'm trying to understand is why definition (2) uses $$\omega$$ = v_t/r, while it seems that it should use v, not v_t, in order to be compatible with (1) (I must be doing some confusion here).

 

[mathman]

I am not sure what the confusion is, but v_t=v.

 

[PJC01]

Hello,

Thank you for your question. It is not uncommon to see different definitions or equations for centripetal acceleration in different sources. This can be confusing, but it is important to understand the context in which these equations are being used.

The first equation you mentioned, a_c = \frac{v^2}{r}, is the most general form of the centripetal acceleration equation. It applies to any object moving in a curved path, regardless of the shape or size of the curve. This equation is derived from the definition of centripetal acceleration, which is the acceleration that is always directed towards the center of the curve.

However, in some cases, it may be more useful to express the centripetal acceleration in terms of the angular velocity, \omega, instead of the tangential velocity, v_t. This is where the second equation, a_c = r\omega^2, comes into play. This equation is derived from the first equation, by substituting v_t = r\omega. This equation is commonly used when dealing with circular motion, where the tangential velocity is constant, and the angular velocity is the only variable.

So, to answer your question, both definitions (1) and (2) are correct, but they are used in different contexts. In definition (1), "r" represents the radius of curvature, while in definition (2), "r" represents the polar radius. It is important to understand the context in which each equation is being used, in order to use the correct equation for your specific problem.

I hope this helps to clarify the confusion. Keep questioning and seeking answers, as that is an important part of being a scientist. Best of luck in your studies.

Best regards,