The Earth can be approximated as a sphere of uniform density, rotating on its ax

Question

The Earth can be approximated as a sphere of uniform density, rotating on its axis once a day. The mass of the Earth is 5.97 times 10^24 kg. the radius of the Earth is 6.38 times 10^6 m. and the period of rotation for the Earth is 24.0 hrs. The moment of inertia of the Earth is smaller than predicted by the assumptions of the problem because the Earth does not have a uniform density. When the mass is concentrated near the center of a spinning object, the moment of inertia is smaller than if the mass is concentrated near the rim. Engineers keep this in mind as they design wheels for cars. A good car wheel will concentrate as much mass as possible near the axle, which gives it the smallest moment of inertia possible. A small moment of inertia makes the wheel easier to speed up or slow down, thereby increasing the fuel efficiency of a car. What is the rotational kinetic energy of the Earth? Use the moment of inertia you calculated in Part A rather than the actual moment of inertia given in Part B. Express your answer in joules to three significant figures. Recall that while energy can change forms, it is always conserved. In other words, if you start with a certain amount of energy, you must end with the same amount of energy. In Part C, you calculated how much rotational kinetic energy the Earth now has. By conservation of energy, that energy had to come from somewhere. Where did the rotational kinetic energy of the Earth come from? Select the option that best explains where the Earth's rotational kinetic energy came from.

Explanation / Answer

C) Moment of inertia of Earth, I = (2/5)*M*R^2

= (2/5)*5.98*10^24*(6.38*10^6)^2

= 9.736*10^37 kg.m^2

angular speed, w = 2*pi/T

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

= 7.27*10^-5 rad/s

KE_rotational = 0.5*I*w^2

= 0.5*9.736*10^37*(7.27*10^-5)^2

= 2.57*10^29 J

D) This energy came when the Earth was formed.

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