Zorch, an archenemy of Superman, decides to slow the Earth\'s rotation to once p

Question

Zorch, an archenemy of Superman, decides to slow the Earth's rotation to once per 31.5 h by exerting an opposing force at the equator and parallel to it. Superman is not immediately concerned, because he knows Zorch can only exert a force of 4.80 107 N (comparable to a Saturn V rocket's thrust). Assume the Earth's initial rotation is exactly once per 24.0 h to find how long Zorch must push with this force to accomplish his goal. (This gives Superman time to devote to other villains. The Earth's mass is 5.98 1024 kg and its radius is 6.38 106 m.)
s

Explanation / Answer

initial angular velocity, wi = 2pi/T

wi = 2pi / (24 x 3600 sec ) = 7.272 x 10^-5 rad/s

final angular velocity, wf = 2pi/ (31.5 x 3600)s

wf = 5.541 x 10^-5 rad/s

and torque = I x alpha

4.80 x 10^7 x R = (2 M R^2 / 5) alpha

4.80 x 10^7 = (2 x 5.98 x 10^24 x 6.38 x 10^6 / 5) alpha

alpha = 3.145 x 10^-24 rad/s^2

now using wf = wi + alpha*t

5.541 x 10^-5 = 7.272 x 10^-5   - (3.145 x 10^-24)t

t = 5.503 x 10^18 sec

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