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Transforming Velocities Between Rotating Coordinate Systems

Letandbe coordinate systems with the same originwith systemrotating with angular velocitywith respect toLetbe the position vector of a particle in

then

In the coordinate system s,are changing with time so

Sincesoandare perpendicular. Similarly forandandand

Hence

(1)

(2)

,(3)

Differentiatinggives(4)

Differentiatinggives(5)

Differentiatinggives(6)

Taking the dot product of (1) withgives

Taking the dot product of (2) withgives

Hence

Similarlyand

Hence

Letthen we can write


Then