1) Calculate the torque required to accelerate the Earth in 5 days from rest to its present angular speed about its axis. 2) Calculate the energy required. 3) Calculate the average power required.

a) τ = 4.47746 * 10^25 N-m

b) E = 2.06301 * 10^13 J

c) P = 3.25511*10^21 W

Explanation:

Given:

- The radius of earth r = 6.3781*10^6 m

- The angular speed of earth w = 7.27*10^-5 rad/s

- The time taken to reach above speed t = 5 yrs = 1.57784760 * 10^8 s

- The mass of earth m = 5.972 * 10^24 kg

- The inertia of sphere I = 2/5 * m * r^2

Solution:

- The angular acceleration of the earth from rest to w is given by α:

α = w / t

α = (7.27*10^-5) / (1.57784760 * 10^8)

α = 4.60754*10^-13 rad/s^2

- The required torque τ is given by:

τ = I*α

τ = 2/5 * m * r^2 * α

τ = 2/5 * (5.972 * 10^24) * (6.3781*10^6) ^2 * (4.60754*10^-13)

τ = 4.47746 * 10^25 N-m

- The power required P to turn the earth to the speed w is:

P = τ*w

P = (4.47746 * 10^25) * (7.27*10^-5)

P = 3.25511*10^21 W

- The energy E required is:

E = P / t

E = (3.25511*10^21) / (1.57784760 * 10^8)

E = 2.06301 * 10^13 J