The Earth has an angular speed of 7.272·10 -5 rad/s in its rotation. Find the ne

Question

The Earth has an angular speed of 7.272·10-5 rad/s in its rotation. Find the new angular speed if an asteroid (m = 3.91·1022 kg) hits the Earth while traveling at a speed of 3.03·103 m/s (assume the asteroid is a point mass compared to the radius of the Earth) in each of the following cases:

a) The asteroid hits the Earth dead center.

b) The asteroid hits the Earth nearly tangentially in the direction of Earth's rotation.

c) The asteroid hits the Earth nearly tangentially in the direction opposite of Earth's rotation.

Explanation / Answer

the angular momentum of earth:
Io = 2mr² / 5 = 2 * 5.98 * 10^24kg * (6.371 * 10^6m)² / 5 = 9.71 *10^37 kg·m²
o = 2 rads/day * 1day/24hr * 1hr/3600s = 7.272 * 10^-5 rad/s

Initial angular momentum Lo = Io * o = 7.06 * 10^33 kg·m²/s

a) The asteroid adds to the moment of inertia, but does not contribute an angular momentum component of its own. Assuming it remains near the earth's surface,
new I = 9.71*10^37kg·m² + 3.91 * 10^22kg * (6.371 * 10^6m)² = 9.87*10^37 kg·m²
Then = 7.272 *10^-5 rad/s * 9.71 / 9.87 = 7.154 * 10^-5 rad/s

b) Now the asteroid contributes L = mvr = 3.91 *10^22kg * 3030m/s * 6.371 * 10^6m,
or L = 0.755*10^33 kg·m²/s
so Lnew = (7.06 + 0.755) *10^33 kg·m²/s =7.815 * 10^33 kg·m²/s
Then new = Lnew / Inew = 7.815*10^33kg·m²/s / 9.87*10^37kg·m² = 7.92*10^-5 rad/s

c) Now the asteroid contributes -0.755 * 10^33kg·m²/s, so
Lnew = 6.305*10^33 kg·m²/s, so
new = 6.305 *10^33kg·m²/s / 9.87e37kg·m² = 6.388*10^-5 rad/s

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