The Spinning Platform Rescue · AI Generator #GEN-4FBDC1C0

Physics · mechanics

Problem

A space station's circular observation deck of radius R=12.0  mR = 12.0\;\text{m} rotates about its center with angular velocity ω0=0.50  rad/s\omega_0 = 0.50\;\text{rad/s} to provide artificial gravity. An astronaut of mass m=80  kgm = 80\;\text{kg} stands at the rim and needs to rescue a toolbox that has broken free and is now sliding frictionlessly across the deck. The toolbox has mass M=20  kgM = 20\;\text{kg} and is currently at distance r1=8.0  mr_1 = 8.0\;\text{m} from the center, moving tangentially with the same speed it had when it broke free from the rim. The astronaut jumps radially inward with speed vj=2.0  m/sv_j = 2.0\;\text{m/s} (as measured in the lab frame) and catches the toolbox when their paths intersect. After catching the toolbox, the astronaut uses small thrusters to move radially back to the rim. Find the angular velocity of the observation deck when the astronaut reaches the rim again.

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