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Section 12.10

Up to now, it has been sufficient to do rotation with the convention of + for counterclockwise and - for clockwise. As we turn to the very last piece, angular momentum, we need to develop a vector description of rotation.

And rotation needs to be fully three dimensional, we will use the standard x,y, z coordinate.

As we have discussed already in the labs, angular velocity can be described as a vector. The direction is giving by the right hand rule.

The x-y-z coordinate system is right-handed. The positive direction of z is found by using the right hand wrapping from x to y and the thumb is in the +x direction. It then goes cyclically, y to z, thumb in in positive x. z to x, thumb is in positive y

For example for a rotation clockwise in the x-y plane.

Applying the right-hand rule to this rotation, we find that the angular velocity

\vec{\omega} = -\omega\hat{k}

Angular Acceleration and Torque

The angular acceleration is also vector that points in the direction of the rate of change of the angular velocity.

This allows us to also definite a direction for the torque using the cross product.

\vec{\tau}=\vec{r}\times\vec{F}

The magnitude of this product is what we have been using up to now |\tau| =rF\sin\theta where the angle is the angle between the two vectors. The direction is given by a new right-hand rule. The order is important for the cross product and the direction is given by the right-hand rule. There are two ways to do the right-hand rule. I prefer the second (to the right) in this image.

Make sure you differentiate the dot product (whose result is a number) from the cross product which result in a vector.

Try it out!

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