Question

Assume Earth being a uniform sphere with mass M ≈ 6×10^24 kg and radius R ≈ 6400 km. Suppose that you could harness the Earth’s kinetic energy of rotation. If mankind requires a total power of P = 10^14 W, how long could you supply this amount of power by spinning down the Earth?

Answer 1

5.16×10¹⁵ s

Step-by-step explanation:

Considering the rotational motion of the Earth,

K.E. =
`(1)/(2)I`ω² where, I =
`(2)/(5)`MR²

=
`(2)/(5)`MR²ω² ---------(1) where ω = angular velocity of Earth

To find ω,

ω = 2π/T where T = time period of Earth

= 2π/(60×60×24) = 7.27×10^-5 rad/s

Substituting the values in (1),

K.E. = 0.4×(6×10^24)×(6400000)²×(7.27×10^-5)²

= 5.16×10²⁹ J

To calculate possible time,

Time = Energy/ Power

= 5.16×10²⁹ / 10¹⁴

= 5.16×10¹⁵ s