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# What Remains of the Day

Looking at NASA's Astronomy Picture of the Day for 3-2-05 I see that the big earthquake shortened the Earth's day by about 3 microseconds. Pretty awe inspiring. I did some rough calculations, becuase I do that kind of stuff. I did them in my head so everything is only first order and rounded to simply the math.

Given:
mass of earth (m_e) =~ 6 x 10^24 kg
radius of earth (r_e) =~ 6 x 10^3 km
1 microsecond = 1 x 10^-6 seconds

Calculations:
Circumphrance of earth (c_e) = 2(pi)(r_e) = 2(3)(6 x 10^3 km)
c_e = 4 x 10^4 km = 4 x 10^7 m

Day in seconds: (24 hour/day)(60 min/hour)(60 sec/min) = 8 x 10^4 sec/day

Velocity of Earth (v_e) = (c_e)/(day in seconds)
= (4 x 10^7 m)/(8 x 10^4 sec)
= .5 x 10^3 = 5 x 10^2 m/s

Kinetic Energy of earth pre-earthquake (T_ei)
= (1/2)(m_e)(v_e^2)
= (1/2)(6 x 10^24 kg)(5 x 10^2 m/s)^2
~ 7 x 10^29 J

Kinetic Energy lost (T_e_lost)
= (T_ei)(ratio of second/3microseconds)
= (7 x 10^29 J) (3 x 10^-6)
~ 2 x 10^24 J

Momentum of Earth pre-earthquake (p_ei)
= (m_e)(v_e)
= (6 x 10^24 kg)(5 x 10^2 m/s)
~ 3 x 10^27 kg m/s

Momentum lost (p_e_lost)
= (p_ei)(ratio of second/3microseconds)
= (3 x 10^27)(3 x 10^-6)
~ 1 x 10^21 kg m/s

Conclusions:
So the quake stripped from the Earth system about 2 x 10^24 J and about 1 x 10^21 kg m/s. That is amazing. Sure it doesn't really factor in in any significant way (after all it is just 3 microseconds for crying out loud) 2 x 10^24 J still is a lot of energy to loose by mere mortal standards!
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