(a) Assume the Earth is a uniform solid sphere. Find the kinetic energy of the Earth due to its rotation about its axis. (b) Suppose we could somehow extract \(1.0 \%\) of the Earth's rotational kinetic energy to use for other purposes. By how much would that change the length of the day? (c) For how many years would \(1.0 \%\) of the Earth's rotational kinetic energy supply the world's energy usage (assume a constant \(1.0 \times 10^{21} \mathrm{J}\) per year)?

First, we need to express the Earth's kinetic energy formula. Kinetic energy (K) for a rotating sphere is given by the formula: K = (1/2) * I * ω² where I is the moment of inertia of the sphere, and ω is the angular velocity. For a uniform solid sphere, the moment of inertia is: I = (2/5) * M * R² where M is the mass of the Earth (approximately 5.972 x 10²¹ kg), R is the Earth's radius (approximately 6.371 x 10⁶ m), and ω is the Earth's angular velocity (2 * π / T radians per second, where T is Earth's rotation period, approximately 86164 seconds). By substituting these values into the kinetic energy formula, we can find the Earth's rotational kinetic energy. K = (1/2) * ((2/5) * (5.972 x 10²¹ kg) * (6.371 x 10⁶ m)²) * (2 * π / 86164 s)²

If we extract 1.0% of the Earth's kinetic energy, we need to determine the new angular velocity (ω') and how it affects the length of the day. First, find 1.0% of the calculated kinetic energy: K_extracted = 0.01 * K Next, we need to find the new total kinetic energy (K'): K' = K - K_extracted Using the kinetic energy formula, we can express the new angular velocity: ω'² = (2 * K') / I Now that we have the new angular velocity, we can find the new length of the day (T'): T' = (2 * π) / ω' Finally, we can find the change in the length of the day by subtracting the original period T from the new period T': ΔT = T' - T

We are given that the world's energy usage is approximately 1.0 x 10²¹ J per year, and we have calculated how much energy we can extract from the Earth's rotation (K_extracted). To find out for how many years this extracted energy would suffice, we can simply divide the total extracted energy by the world's annual energy consumption: Years_supplied = K_extracted / (1.0 x 10²¹ J) Now, perform the numerical calculations for each step to find the answers to the questions.