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 29.0 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 4.50 107 N (comparable to a Saturn V rocket's thrust). Assume the earth's initial rotation is exactly once per 24.0 h and use energy considerations to find how long Zorch must push with this force to accomplish his goal. (This gives Superman time to devote to other villains.) Answer in seconds

Explanation / Answer

Moment of inertia of earth,  Isolid sphere = 2/5*Me*Re^2

wi = 1 rev / 24 hrs = 1 rev/24 hrs * (2*pi rad / 1 rev) * (1 hr / 3600 s)
wi = 7.27 * 10^-5 rad/s

wf = 1 rev / 29 hrs = 1 rev/29 hrs * (2*pi rad / 1 rev) * (1 hr / 3600 s)
wf = 6.02 * 10^-5 rad/s

Torque, t = I*
Torque, t = F*r
Equating the two,
F*r = I*
4.5* 10^7 * Re = 2/5*Me*Re^2 *
4.5* 10^7 = 2/5*Me*Re * (wi - wf)/t
4.5* 10^7 = 2/5*(5.98 * 10^24) * (6.37*10^6) * (7.27 * 10^-5 - 6.02 * 10^-5 )/t
t = 4.23 * 10^18 s
t = 1.34 * 10^11 Years

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