Effect of Asteroid Impact on Earth's Rotation
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Exercise:
An asteroid of mass mO travelling at a speed of vO relative to the Earth hits the Earth at the equator tangentially and in the direction of Earth's rotation. Asing the collision is perfectly inelastic use conservation of angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.
Solution:
Angular momentum is conserved in the Earth--asteroid system. Asing the asteroid becomes embedded in the Earth at the surface both will share the same angular velocity after the collision. Modelling the Earth as a uniform solid sphere and the asteroid as a po mass at the surface we have I_EarthIndex fracm_EarthIndex r_EarthIndex^ sscIA sscmA r_EarthIndex^ Conservation of angular momentum gives: L_ L_ I_EarthIndexomega_EarthIndex + I_aomega_a biglI_EarthIndex + I_abigromega Solving for the final angular velocity omega fracI_EarthIndexomega_EarthIndex + I_aomega_a I_EarthIndex + I_a and the change in angular velocity omega - omega_EarthIndex fracI_EarthIndexomega_EarthIndex + I_aomega_a I_EarthIndex + I_a - omega_EarthIndex fracI_aomega_a - omega_EarthIndex I_EarthIndex + I_a we get for the change as a percentage: eta fracomega - omega_EarthIndexomega_EarthIndex fracI_aomega_a - omega_EarthIndexI_EarthIndex + I_aomega_EarthIndex And since I_a ll I_EarthIndex and omega_a gg omega_EarthIndex it follows that eta &approx fracI_aomega_aI_EarthIndexomega_EarthIndex &approx fracm_a r_EarthIndex^ dfracv_ar_EarthIndexdfracm_EarthIndexr_EarthIndex^ omega_EarthIndex &approx fracm_a v_am_EarthIndexr_EarthIndexomega_EarthIndex &approx numpr. which is negligibly small.
An asteroid of mass mO travelling at a speed of vO relative to the Earth hits the Earth at the equator tangentially and in the direction of Earth's rotation. Asing the collision is perfectly inelastic use conservation of angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.
Solution:
Angular momentum is conserved in the Earth--asteroid system. Asing the asteroid becomes embedded in the Earth at the surface both will share the same angular velocity after the collision. Modelling the Earth as a uniform solid sphere and the asteroid as a po mass at the surface we have I_EarthIndex fracm_EarthIndex r_EarthIndex^ sscIA sscmA r_EarthIndex^ Conservation of angular momentum gives: L_ L_ I_EarthIndexomega_EarthIndex + I_aomega_a biglI_EarthIndex + I_abigromega Solving for the final angular velocity omega fracI_EarthIndexomega_EarthIndex + I_aomega_a I_EarthIndex + I_a and the change in angular velocity omega - omega_EarthIndex fracI_EarthIndexomega_EarthIndex + I_aomega_a I_EarthIndex + I_a - omega_EarthIndex fracI_aomega_a - omega_EarthIndex I_EarthIndex + I_a we get for the change as a percentage: eta fracomega - omega_EarthIndexomega_EarthIndex fracI_aomega_a - omega_EarthIndexI_EarthIndex + I_aomega_EarthIndex And since I_a ll I_EarthIndex and omega_a gg omega_EarthIndex it follows that eta &approx fracI_aomega_aI_EarthIndexomega_EarthIndex &approx fracm_a r_EarthIndex^ dfracv_ar_EarthIndexdfracm_EarthIndexr_EarthIndex^ omega_EarthIndex &approx fracm_a v_am_EarthIndexr_EarthIndexomega_EarthIndex &approx numpr. which is negligibly small.