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.) (a) Calculate the change in angular momentum (in kg·m2/s) in half this time. kg·m2/s (b) What is the average torque (in N·m) 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 (in N) be?

Explanation / Answer

a) L = I w = 2 M R2 w / 5

= 0.4 * 5.98 * 1024 * (6.38 * 106)2 * 2 pi / 86400

= 7.08 * 1033 kg.m2/s

change in angular momentum delta L = 2 L * sin 23.5

= 2 * sin 23.5 * 7.08 * 1033

=  5.65 * 1033 kg.m2/s

b) torque = delta L / delta t

= (5.65 * 1033 ) / (25780 * 365 * 24 * 3600 / 2)

= 1.39 * 1022 N.m

c) magnitude of force = Torque / R

= (1.39 * 1022) / (6.38 * 106)

= 2.17 * 1015 N

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