Zorch, an archenemy of Superman, decides to slow earth’s rotation to once per 29
ID: 1428655 • Letter: Z
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. The rocket can exert a thrust of F= 4.05x10^7 N. If the original angular velocity of Earth is Wo and Zorch is trying to get the earth to an angular velocity of w, how much time will it take him? Use M and R for the mass and radium of the Earth.
How long, in seconds, will it take for him to do this?
Explanation / Answer
Moment of Inertia of the Earth:
I_earth = I_solid sphere = 2/5*M*R^2
Convert angular velocity to rad/s
wi = 1 rev / 24 hrs = 1 rev/24 hrs * (2*pi rad / 1 rev) * (1 hr / 3600 s) = 7.27 * 10^-5 rad/s
wf = 1 rev / 29 hrs = 1 rev/29 hrs * (2*pi rad / 1 rev) * (1 hr / 3600 s) = 6.0166 * 10^-5 rad/s
Define the angular accleration
alpha = (wf - wi)/t
From the definition of torque
torque = I*alpha
The torque is simply the force of Zorch * radius of the earth. Sub that, I, and alpha into the torque equation.
One more thing, torque and the angular acceleration are vectors. Since the force is slowing the Earth, the torque is negative if we take the direction of Earth's spin as positive. (wf-wi) will be a negative number, so we don't need to add a negative sign for that term.
-F*Re = 2/5*M*R^2*(wf-wi)/t
Solve for t
t = 2/5*M*R*(wi-wf)/F
Plug in numbers
t = 2/5 * (5.98 * 10^24 kg)*(6.37*10^6 m) * (7.27 * 10^-5 rad/s - 6.0166*10^-5 rad/s)/(4.05*10^7 N) = 4.716*10^18 s
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