Consider the system described by the image below where K = k and is positive com

Question

Consider the system described by the image below where K = k and is positive coming out of the page. Let I-J-K be an inertial frame. The i-j-k frame rotates and is fixed to the given disc. The disc is spinning about the +k axis at a constant rate of omega rad/s. At the initial time (t = 0 s), theta = 0 rad. At t = 0, an ant starts to move from the center of the disc towards the edge along the i axis at a constant rate of zeta cm/s. Let r_R denote the ant's position in the rotating frame and r_F the position in the inertial (fixed) basis. (a) What is the angular rotation vector Ohm for this problem in the I-J-K basis? (b) As a function of t, what is the instantaneous T^R_F operator such that r_F = T^R_F r_R? (c) What are the position and velocity vectors of the ant in the inertial frame as a function of t? Express the solution in terms of T^R_F. Ignore the possibility that the ant may reach the edge.

Explanation / Answer

the capitals are unit vectors along the frame of reference fixed to the origin, where as small letters are unit vectors in the body fixed frame

for the angular velocity w, and initial theta = 0 deg

also, in body ficed axis, speed of ant = z i cm/s

rR is position of ant in body fixed frame, rF is position of ant in FIxed frame

a) Angular rotation vector in IJK basis is omega k

b) now, rR = zt i

rF = |rR|*[cos(wt)I + sin(wt)J]

so the operator is [cos(wt)I + sin(wt)J]|rR|/rR = [cos(wt)I/i + sin(wt)J/i]

c) so, rF = |rR|*[cos(wt)I + sin(wt)J] = zt[cos(wt)I + sin(wt)J] = T(R,F)rR

let velocity vector be vF

vF = [zcos(wt) - w|rR|sin(wt)]I + [zsin(wt) + w|rR|cos(wt)]J

vF = z[cos(wt)I + sin(wt)J] + w|rR|[-sin(wt)I + cos(wt)J]

vF = z*T(R,F)i + |rR|d(T(R,F))/dt

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