Zorch, an archenemy of Superman, decides to slow Earth\'s rotation to once per 2

Question

Zorch, an archenemy of Superman, decides to slow Earth's rotation to once per 29 h by exerting a force parallel to the equator, opposing the rotation. He uses an old Saturn V rocket, stolen from NASA, which was originally used to send astronauts to the Moon. This rocket can exert a thrust of F = 4.15 times 10^7 N. If the original angular velocity of Earth is omega_0 and Zorch is trying to get the Earth to an angular velocity of omega, how much time will it take him? Use M and R for the mass and radius of the Earth. How long, in seconds, will it take him to do this? (Notice that this period gives Superman time to devote to thwarting other villains.)

Explanation / Answer

F = -4.15*10^7 N

M = 5.98*10^24 kg

Re = 6.37*10^6 m

w0 (for 24 h) = (1/24)*(2*pi/3600) = 7.27*10^-5 rad/sec

w (for 29 h) =  (1/29)*(2*pi/3600) = 6.02*10^-5 rad/sec

tau = I*alpha

I = 2Mr^2/5,

tau = rF

alpha = 5F/2Mr

w = w0 + alpha*t

t = (w - w0)/alpha = (w - w0)*2*M*r/5F

t = 5.98*10^24*6.37*10^6*2*(6.02 - 7.27)*10^-5/(5*(-4.15)*10^7) = 4.58*10^18 sec(B)

time = 1.45*10^11 years (A)

Let me know if you have any doubt.

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