A woman of mass m stands at the edge of a solid cylindrical platform of mass M a

Question

A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity 0 about a vertical axis through its center, and the woman begins walking with speed v (relative to the platform) toward the center of the platform.

Part A) Determine the angular velocity of the system as a function of time. Express your answer in terms of the variables m, M, R, v, 0, and t.

Part B) What will be the angular velocity when the woman reaches the center? Express your answer in terms of the variables m, M, and 0.

Explanation / Answer

Moment of inertia of solid cyliner =(MR2)/2

initial moment of inertia of system Io = (MR2)/2 +mR2

Initial angular momentum Lo= Iowo =[ (MR2)/2 +mR2 ]wo

Part A:

distance travelled by women d =vt

d+r =R

r =R-d =R -vt

Final angular momentum Lf= Lc+Lw

Lf =[(MR2)wf/2] +mvr]

Final angular momentum Lf= Ifwf =[ (MR2)/2 +mv(R-vt) ]

From conservation of angular momentum Lf =Lo

[(MR2)wf/2] +mv(R-vt)]=[ (MR2)/2 +mR2 ]wo

(MR2)wf/2 = [ (MR2)/2 +mR2 ]wo -mv(R-vt)

Final angular velocity wf = (2/MR2) {[(MR2)/2 +mR2 ]wo -mv(R-vt) }

Part B : r =0

Final angular momnetum of system Lf = [MR2/2]wf

From conservation of angular momnetum Lf +Lo

[MR2/2]wf =[ (MR2)/2 +mR2 ]wo

wf = (M+2m)wo/M

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