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# Input Needed: Direction of Angular Velocity and Angular Acceleration Vectors

## Question

Hi All,

I had a friend and colleague ask me today about why the angular velocity and angular acceleration vectors point in directions given by the right-hand-rule, as highlighted here... When I read my response, I realized that my answer wasn't much better than that given in the link... I thought about it some more and started thinking that it probably related to:

${\vec \tau _{net}} = \vec r \times \vec F = I\vec \alpha$

and since the cross product of r and F is perpendicular to both r and F, with the positive direction given by RHR, the angular acceleration (and similarly angular velocity) vector must be consistent.

Still not a very pleasing or clear explanation. So, why not turn to the experts? If someone were to ask you, how would you explain the direction of the angular velocity and angular acceleration vectors in a manner that was as clear and straightforward as possible?

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[ATTACH=CONFIG]69[/ATTACH]My colleague has offered to "kick up" the challenge a notch. Details below:

The prize: a \$20 Starbucks gift card.

The challenge: Provide the best step-by-step explanation (ala a geometric proof) using only concrete, physical descriptors; in other words, NO MATH equations, detailing why the angular velocity and angular acceleration vectors point along the axis of rotation.

Your audience: students currently taking a full year AP C mechanics course. Half of them are currently in Calc BC; the rest are in AB. Only 10 of the 30 students took AP B last year; the rest took Physics 1 or no physics before this current AP C course.

Winner will be determined by a class vote from our "audience."

Please post challenge responses below... enter as often as you like, but remember to keep your audience in mind as you write your explanations!

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Similar to what Mr. Fullerton proposed, Torque is the rotational equivalent of Force. It then follows that because acceleration (and therefore velocity) are dependent on and follow the direction of forces that angular acceleration (and velocity) must too follow angular force or torque.

Still all this depends on whether we trust the right hand rule to be true or on whether we trust the cross product to give us what we want. Why is it that the right hand rule works? I can only show that the magnitude half of the cross product holds true. The magnitude of a Torque depends on three things: Distance from the axis of rotation R, magnitude of the force being applied F, and the degree to which those two vectors are perpendicular given by an angle @. The more perpendicular they are, the greater the Torque and only the perpendicular component of the force creates a torque so the sine function is used. Put all together is the magnitude that takes into account R, F, and the degree of perpendicularity. The cross product provides this magnitude and also provides what I can give no explanation for, the direction being perpendicular to both R and F. If R and F are on the x,y Cartesian plane, the only way to be perpendicular to both is on the z axis in or out of the page.

My last thought is that to cause a rotation or torque, there needs to be a perpendicular force and the only way to perpendicular at any point on a rotating disk is choosing the axis of rotation, the z axis.

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