Angular Acceleration and tangential acceleration

[Seph]

Hmmm. I have a question...

We know that the tangential acceleration (At) is equal to the radius (R) multiplied by angular acceleration (Ar), of which At and Ar are vector components.
At = R Ar

Then I was told that Ar = At/R is not a vector equation. Why is that true?

~Seph

 

[Pyrrhus]

Well mainly, because those relations refer to the magnitude of those vectors, the problem is angle displacement does not work with vectorial addition communitative property, but as it gets smaller it works with vectorial addition, that's why angular velocity and aceleration are vectors.

Also, you could define angular displacement as a vector, but you will need an unit vector for the radius and the angle, so you will give them direction.

 

[reilly]

Your equation is one component of the basic vector equation for angular motion:

V = omega X R where v is the velocity vector, omega is the angular velocity vector, and R is the position vector , and X indicates the vector cross product. (If an object is rotating around the z axis, then the omega vector points along the positive z axis for counterclockwise rotations.)

This vector approach is discussed in Resnik and Halliday, and in most more advanced texts on mechanics.


Regards,
Reilly Atkinson