The axis of Earth makes a 23.5° angle with a direction perpendicular to the plan

Question

The axis of Earth makes a 23.5° angle with a direction perpendicular to the plane of Earth's orbit. As shown in the figure below, this axis precesses, making one complete rotation in 25,780 y. (Assume L = 7.07 1033 kg · m2/s.)

Calculate the change in angular momentum in half this time. kg · m2/s

(b) What is the average torque producing this change in angular momentum? N · m

(c) If this torque were created by a single force (it is not) acting at the most effective point on the equator, what would its magnitude be? N

Explanation / Answer

dL = 2*L*sin 23.5 deg = 0.7974*L

angular momentum of earth

L = I*w

L = 2*M*R^2*w/5

L = 0.4*5.98*10^24*(6.38*10^6)^2*2*pi/86400 = 7.07*10^33 kg-m^2/sec

change in angular momentum will be

dL = 0.7974*7.07*10^33 = 5.64*10^33 kg-m^2/sec

B.

torque = dL/dt

dt = 25780*365*24*3600/2

torque = 5.64*10^33*2/(25780*365*24*3600)

torque = 1.38*10^22 N-m

C.

torque = R*F

F = torque/R

F = 1.38*10^22/(6.38*10^6) = 2.16*10^15 N

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