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 31.0 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 4.00 ? 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.)

Explanation / Answer

According to my books, the mean value of Earth Rotational Inertia is 8.023 x 10^37 kg.m^2
The equatorial radius of the earth 6378 km
So the braking Torque of Zorch = Force x radius = 4.00 x 10^7 x 6.378 x 10^6 = 2.5512 x 10^14 N.m

Torque = l*a
Angular deceleration = T / I = 2.5512 x 10^14 / 8.023 x 10^37
a = - 0.318 x 10^ -23 rad /sec^2
Initial angular vel , wo = 1rev /24 hr = 2 pi / 24 x 3600 = 7.272 x 10^ -5 rad /s

Final angular vel, wf = 1rev/31hr = 2 *3.14 / 31 x 3600 = 5.627 x 10^ -5 rad /s

Time = wf - wo / a
= (5.627 x 10^ -5 - 7.272 x 10^ -5) / - 0.318 x 10^ -23
= 5.17 x10^18 sec

1Yr = 365.25 (days) x 24 (hrs) x 3600 (sec) = 3.156 x10^7 sec

Time = 5.17 x10^18 / 3.156 x10^7
= 1.638 x 10^11 years
= 163,814,950,000 Years
That is almost 164 Billion years

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