Fun question on math behind angular momentum and torque

Question

ganeshie8 · Answer

attachment

ganeshie8 · Answer

(btw i had to put "math" keyword there because this is math group hmm )

parthkohli · Answer

clockwise rotation: negative angular momentum
anticlockwise rotation: positive angular momentum

negative (clockwise): 1, 7
positive (anticlockwise): 4, 6
the rest are zero

ganeshie8 · Answer

If we compare just the magnitudes of $\omega$, do we have a tie on $6$, $7$  ?

ganeshie8 · Answer

1, 4, 7 should be negative right ?

parthkohli · Answer

oh yes, kinda got screwed among all those numbers switching tabs

ganeshie8 · Answer

what about the magnitude of $\omega$ provided by 6, 7
do they tie ?

intuition tells me 6 should be more but the math seems to say otherwise..

parthkohli · Answer

6 > 7

ganeshie8 · Answer

they both are at same distance from the axis of rotation

ganeshie8 · Answer

how do you say 6 > 7

parthkohli · Answer

well because we have a sin(theta) term in angular momentum too

the easy way to calculate angular momentum is to extend the velocity vector till it is perpendicular to $\vec r$ (we can kinda do that since angular momentum remains conserved)|dw:1446026866721:dw|