A runner of mass \"m\" runs around the edge of a horizontal turntable (disk) mou

Question

A runner of mass "m" runs around the edge of a horizontal turntable (disk) mounted on a vertical, frictionless axis through its center. The runner's velocity relative to the earth is "v". The turntable is rotating in the opposite direction with an angular velocity "w1" relative to the earth. The radius of the turntable is R and its mass is "6m". The runner may be treated as a point. Given [m, v, R, w1] Determine:

a. The final velocity of the system if the runner comes to rest relative to the turntable.

Explanation / Answer

mass of the runner is m

velocity of the runner relative to the earth is v

angular velocity of turn table is w ( rotating opposite direction)

radius of the turn table is R

mass of the turn table is 6m

moment of inertia of the only turn table is, I=6(m*R^2/2)

I=3m*R^2/2

here,

by uing law of angular momentum,

total initial angular momentum=total final angular momentum=L1=Iw1-m*v*R

(I*w1-m*v*R)=(I+m*R^2)*w2

((3m*R^2/2)*w1-m*v*R))=((3m*R^2/2)+m*R^2))*w2

((3R/2)*w1-v))=((3R/2)+R))*w2

(3/2)*w1-v/R=(3/2+1)*w2

(3/2)*w1-v/R=(5/2)*w2

===>

w2=(1/5)*(3*w1-2*v/R) ---------is answer

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