Two astronauts of equal mass 84 kg are holding on to opposite ends of a straight

Question

Two astronauts of equal mass 84 kg are holding on to opposite ends of a straight rope in space. The length of the rope between them is 16 meters. They are rotating about an axis that passes through the center of mass midway between the two astronauts with angular velocity 0.8 radian/sec. One of the astronauts pulls on the rope, decreasing the distance separating them to 16/2 meters. (In the following, take the rope to be massless.) 1. What is the new angular velocity of the two astronauts given in rads/s? 2. How much work did the astronaut who pulled on the rope do given in Joules? Thank you!

Explanation / Answer

(1) initial moment of inertia i1 = 2mR^2 = 2 x 84 x (16/2)^2 = 10752 kg m^2. final moment of inertia i2 = 2mr^2 = 2 x 84 x (8/2)^2 = 2688 kg m^2. initial angular velocity = 0.8 rad/s. final angular velocity = ? conserving angular momentum, i1 x w1 = i2 x w2, so w2 = i1 x w1 / i2 = 10752 x 0.8/2688 = 3.2 rad/s Ans (2) work = change in kinetic energy = 0.5 i2w2 ^2 - 0.5 i1w1 ^2 = 0.5 (2688 x 3.2^2 - 10752 x 0.8^2) = 10321.92J Ans

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