Tangential and radial coordinate problem. Confused about the FBD.

[theBEAST]

Homework Statement

Here is a picture of the problem with the free body diagram:
http://dl.dropbox.com/u/64325990/HW%20Pictures/problem101.PNG

I am confused about why the free body diagram has the force vector in the direction of the a_θ. When I did it I thought the force that the rod would exert on the particle would be in the tangential direction? In other words in the direction of a_t.

 

[spacelike]

I don't see a_t in your pic.

But anyway, I am pretty sure that the tangential direction is the theta direction as well.
So a_theta = a_t

In polar coordinates the radial direction is out from the origin, and the theta direction is perpendicular to that, which would be the tangential direction.

 

[theBEAST]

I don't see a_t in your pic.

But anyway, I am pretty sure that the tangential direction is the theta direction as well.
So a_theta = a_t

In polar coordinates the radial direction is out from the origin, and the theta direction is perpendicular to that, which would be the tangential direction.

No the unit vector in the tangential direction (u_t) is perpendicular to the normal direction. a_theta != a_t..

I edited the picture slightly to show the direction of a_t which is in the direction of it's unit vector u_t.

 

[tiny-tim]

i'll just add this to what spacelike says …

in polar coordinates, the actual coordinates are r,θ,

and so the unit vectors for increasing r (with fixed θ), and for increasing θ (with fixed r) are called e_r and e_θ, respectively

(or $\boldsymbol{\hat{r}}$ and $\boldsymbol{\hat{θ}}$)

and their velocity or acceleration components therefore have the same subscripts

(and of course, t isn't a coordinate )

 

[DrewD]

I think I understand your confusion. There are two tangential vectors you could define. One would be the vector that is tangential to the motion of the particle and the other is tangential to any point on the rod which is rotating (but at a constant distance from the origin). The second one must be in the theta direction which is perp. to the rod. The force from the rod MUST be that force which is normal to the surface of the rod. Unless there are strange things going on, the only interaction between the rod and particle must be normal to the rod. The force that you are thinking about that would be in the $u_t$ direction is the net force on the particle. If that were asked, you would be correct.