The earth, which is not a uniform sphere, has a moment of inertia of 0.3308?MR?2 about an axis through its north and south poles. It takes the earth 86,164 s to spin once about this axis. Use Appendix F to calculate (a) the earth’s kinetic energy due to its rotation about this axis and (b) the earth’s kinetic energy due to its orbital motion around the sun. (c) Explain how the value of the earth’s moment of inertia tells us that the mass of the earth is concentrated toward the planet’s center.

Solution 77P 1 2 The earth’s kinetic energy due to rotation about its axis is = I , wher2 I is the moment of inertia of the earth and is angular velocity. 2 Given that, I = 0.3308MR Time period to spin about its axis = 86.164 s Angular velocity = 2/86,164 rad/s = 7.3 × 10 5 rad/s 24 Now, the mass of earth is M = 5.97 × 10 kg Radius of earth R = 6.37 × 10 m 6 24 6 2 2 So, I = 0.3308 × (5.97 × 10 ) kg × (6.37 × 10 ) m I = 8.01 × 10 37 kg.m 2 (a) Now, substituting the values of I and in the expression for kinetic energy K = I , 2 2 1 37 2 5 2 K = 2 × 8.01 × 10 kg.m × (7.3 × 10 rad/s) 27 K = 214 × 10 J K = 2.14 × 10 29 J This is the earth’s kinetic energy due to its rotation about this axis. (b)...