In: Physics
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.
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