Question

In the 8.01SC course notes, Two Dimensional Rotational Kinematics, the rotational velocity vector, omega, is directed towards the z-axis. The z-axis is also the axis of rotation. I can’t understand why the velocity vector is directed in the same direction as the axis of rotation. Can someone please explain why this is so and shouldn’t the velocity vector be directed along the circular path of its tangent?

Answer 1

I'm not positive about the situation you're describing but it sounds to me like what you're describing is analogous to the rotation of a screw. This analogy comes into play when doing cross-products (the right hand rule.) If the rotation is in a counter-clockwise direction, then there will be a velocity of the object upwards with the positive z-axis. Therefore, think of these rotational directions like a screw. (Clockwise and the screw goes down; counter-clockwise and the screw goes up). The velocity in the direction of the z-axis is the linear velocity of the object, not the angular velocity of its turning. Hope I understood your question right!

Answer 2

Yes, this stems from the right hand rule for cross products. Essentially the direction of the velocity vector will always be perpendicular to the plane that the disk is rotating in. So if the disk is rotating in the XY plane,the velocity will be in the z direction.

Answer 3

Thank you cbott17 and jsmerson. I think my confusion originated from my expectations that the direction of velocity is necessarily the direction of movement. I understand what you are saying but I am still slightly confused as to why, for example, a CD on a CD player doesn’t move along the z-axis and thus fly off the player!

Answer 4

I think your confusion stems from the difference between the convention e.g. velocity in the z direction and the actual motion.

Imagine the cd you refer to. Which direction is it spinning clock-wise or counter clock-wise? The answer is both of course! If your cd spins clockwise while looking at it from the top. It will seem to be spinning counter-clockwise from the bottom. Therefore clockwise and counter-clockwise are not efficient at giving the direction of a velocity with out giving your orientation.

To counter this problem we use the right hand rule. This creates a direction (positive or negative z in the case of an object spinning in the XY plane) which is independent of orientation of the viewer!