Saturday, 14 February 2015

Relative angular velocity and acceleration


Background: (Irodov 1.55) Two bodies rotate around intersecting perpendicular axes with angular velocities ˆω1,ˆω2. Relative to one body, what is the angular-velocity and -acceleration of the other?


Irodov's answer implies that


ˆω=ˆω1ˆω2

ˆα=ˆω1׈ω2.


I have a hard time grokking why the above are true above (for the first) vague analogies with linear velocity. Does anyone, willing to share, have an intuitive grasp on the above equations?



Answer



ja72's answer is probably right, but I was confused by his notation, so I will give my own answer. Suppose we have two object rotating with angular velocity ω1 and ω2. Then the velocity of a point r of object 1 in the lab frame is v1,lab=ω1×r. Similarly, the velocity of a point r of object 2 in the lab frame is v2,lab=ω2×r.


Now to someone in the second object, a point r that is stationary in the lab frame will have an apparent velocity v2,lab=ω2×r. From this, you can see that a point r in the first object will appear to have a velocity v1,labv2,lab=ω1×rω2×r=(ω1ω2)×r. Thus the first object appears to have angular velocity ω1ω2 to an observer in the second object.



Now since ω1 is stationary in the lab frame, it's time derivative is ω2×ω1=ω1×ω2 in the frame of the second object. Thus object one appears to have an angular acceleration of ω1×ω2 in the frame of the second object.


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