Question

The apparent rotation of the stars is due to Earth’s actual rotation, and the “north star” moves very little because the rotational axis of the Earth points almost directly at it. However, this has not always been the case, and will not be so in the future, because Earth’s axis precesses with a period of 25,770 years; we will have a new north star in the future (or no north star at all!). Modeling the Earth as a uniform sphere, find the magnitude of the torque that is responsible for this precession.

Answer 1

Tp = 25770 years

= 25770*365*24*60*60

= 8.127*10^11 s

angular speed of precession,
wp = 2*pi/Tp

= 2*pi/(8.127*10^11)

= 7.731*10^-12 rad/s

moment of inertia of the earth about its axis,

I = (2/5)*Me*Re^2

= (2/5)*5.97*10^24*(6.37*10^6)^2

= 9.69*10^37 kg/m^2

angular speed of the earth, w = 2*pi/T

= 2*pi/(24*60*60)

= 7.27*10^-5 rad/s

angular momentum of the Earth, L = I*w

= 9.69*10^37*7.27*10^-5

= 7.04*10^33 kg.m^2/s

now use, wp = Torque/angular momentum

= T/L

==> T = wp*L

= 7.731*10^-12*7.04*10^33

= 5.44*10^22 N.m