Motion Equation, Rotating Frame

Consider two co-ordinate systems S (X, Y, Z) and S’ (X’, Y’, Z’); S is stationary while S’ is rotating in space about Z’-axis. For convenience, let us assume that origins O and O’ of the two systems coincide and axis of rotation of S’ (i.e. Z’-axis) also coincide with Z-axis of S.

Let us now take a particle P whose position vectors in S and S’ are r and r’ at some instant t. In terms of components, we have

r = x e_x + y e_y + z e_z                                    (i)

and, r’ = x’ e_x’ + y’ e_y’ + z’ e_z’

Note that (e_x, e_y, e_z) are constant unit vectors defining co-ordinate system change their directions with time. If we assume that X’, Y’ axes are inclined at an angle Ø with X, y axes respectively, at time t, then we have:

e_x’ = cos Ø e_x + sin Ø e_y

and, e_y’ = –sin Ø e_x + cos Ø e_y …                     (ii)

e_z’ = e_z

Since origins of both the frames coincide, we have, in fact

r = r’


We interpret the first bracket as the velocity v’ of particle observed from S’:



The second bracket in (iii) arises because of co-ordinate system S’. From (ii), we find



Hence, second bracket becomes equal to × r’, where = Ø e_z’. Let us check:

× r’ = (ω e_z’) × (x’ e_x’ + y’ e_y’ + z’ e_z’)

= ω x’ e_y’ – ω y’ e_x’                     (ω = Ø)

Thus, we get v = v’ + × r’                                                                       (v)

Differentiating once more with respect to t, we have



v’ = v_x’ e_x’ + v_y’ e_y’ + v_z’ e_z’

and then repeat the steps from eqs. (iii) to (v). We find



Note that v’ and a’ are the velocity and acceleration of the particle as actually measured by an observer standing on the rotating frame S’. With respect to this observer, the co-ordinate system in S’ is stationary or non-rotating. Therefore it is sometimes loosely said (and wrongly understood) that v’ and a’ are the variables relative to a non-rotating frame S’.

Further note that above analysis assumes that Z’ and Z-axes coincide and constitute the common axis of rotation of S’.

Now, if the frame S’ is rotating about Z (or Z’) axis with constant angular velocity, Ø = ω (constant), then angular acceleration and hence the last term in eq. (vi) vanishes.

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