Doubt with centripetal acceleration

  • Thread starter pc2-brazil
  • Start date
  • #1
205
3
Good afternoon,

For a body in a curvilinear motion moving with velocity [tex]\vec{v}[/tex], 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 (vt or [tex]v_{\bot}[/tex]), which changes the direction of the radius vector.
The angular velocity is determined by the the tangential velocity:
[tex]\omega = \frac{v_t}{r}[/tex]
And the centripetal acceleration is given by:
[tex]a_c = \frac{v^2}{r}[/tex] (1)
My question is: I have seen, in many places, that the centripetal acceleration can be written as:
[tex]a_c = r\omega^2 = \frac{v_t^2}{r}[/tex] (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.
 
Last edited by a moderator:

Answers and Replies

  • #2
mathman
Science Advisor
8,022
526
At any given point on the circle, the velocity is the tangential velocity.
 
  • #3
205
3
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 [tex]\omega[/tex] = vt/r, while it seems that it should use v, not vt, in order to be compatible with (1) (I must be doing some confusion here).
 
Last edited:
  • #4
mathman
Science Advisor
8,022
526
I am not sure what the confusion is, but vt=v.
 

Related Threads on Doubt with centripetal acceleration

  • Last Post
Replies
15
Views
1K
Replies
4
Views
10K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
1
Views
727
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
7
Views
886
Replies
1
Views
3K
Top