The Earth and the torque?

Question

Calculate a) the torque, b) the energy, and c) the average power required to accelerate the Earth in 1 day from rest to its present angular speed about its axis.

Answer 1

Let me get you started and to the point where you can plug numbers in...

a) To find the torque on the Earth, you need to know the moment of inertia of the Earth and the angular acceleration needed to get the Earth from rest to its current angular speed.

The moment of inertia of a sphere is 2/5 MR^2 (M is the mass of the Earth and R the radius); the angular acceleration is equal to change in angular velocity/time, so the angular acceleration here is just the current angular velocity divided by 86,400 s.  

To find the angular velocity, remember that the earth covers an angular displacement of 2 pi radians per day; combine all these numbers and you should get the torque.

b) Use the work energy theorem to find the amount of energy used (or work done) to spin the earth up from zero to current rates.  The work energy theorem tells you that the amount of work done equals the change in kinetic energy; the kinetic energy of a rotating sphere is 1/2 I w^2, where I is the moment of inertia and w is the angular velocity. You have already calculated these in part a) so the calculation is easy. This gives you the amount of work needed.

c) For this, use the fact that power=work/time; take the value of work found in part b) and divide by 86,400 s/day

good luck

Answer 2

1 day = 24*60*60 = 86400 s

a) w = 2*pi/86400 = 7.27 *10^-5

alpha =  w / t = 8.4*10^-10

Consider the Earth as a solid body

>>>> the rotational inertia is 2/5 m * r^2

you already know the mass and the radius of the earth

torque = 2/5*m*r^2*alpha

b) K= 1/2 *m*w^2

c)K=W

P=W/t

Answer 3

Assuming the earth is a homogeneous sphere of mass M and radius R,

I = .4MR²

Further assuming that the current ω is 2π rad/day(1d/86400sec), and that the acceleration α must be dω/dt = 2Θ/t² = 4π/86400² = .1683E-8 rad/s²

a) Q = I*α = .4*5.98E24*6.376E6²*.168E-8 = 1.6337E29 N∙m

b) E = ½Iω² = ½*.4*5.98E24*6.376E6²*(2π/86400)² = 2.57E29 J

c)  Pav = E/t = 2.57E29/86400 = 2.976E24 W