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-----------------------------1902 - The Moon and Angular Momentum.
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- If the moon’s orbit takes longer than a day the planet’s rotation slows down and the moon drifts further away to compensate due to the Conservation of Angular Momentum. Our Moon is drifting away at 1 ½ inches per year.
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- The orbital angular momentum of the Moon is 29 *10^33 kg^2 / second.
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- The rotational angular momentum of the Earth is 7.1 * 10^33 kg^2 / second .
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- The Moon’s orbit has 4 times the momentum as Earth’s rotation momentum decreasing the Earth’s rotation rate by 23 micro-seconds per year. This effect results in the Earth-Moon distance expanding by 1 ½ inches per year in order to restore the Conservation of Energy, the Conservation of Angular Momentum.
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----------------------- Momentum = mass * velocity = mass * distance / time
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--------------------- Angular Momentum = mass * radius ^2 * angular velocity.
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- This formula requires that we define the composition of the distance, or radius of the rotation. In this case we define it as a “uniform sphere”. Angular Velocity is in radians / second , or degrees of rotation per second.
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- To calculate the Angular Momentum of the spinning Earth:
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----------------------- Mass = 6 * 10^24 kilograms
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---------------------- Radius^2 = ( 6.4*10^6 meters )^2
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---------------------- Angular Velocity = 2*pi radians / 24 hours
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---------------- Angular Velocity = 2*pi radians / 86,400 seconds = 7.3 *10^-5 radians per second
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- The rotational inertia of a uniform sphere is:
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----------------------- Inertia - 2/5 mass * radius^2
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------------------------ I = 2/5 m*r^2
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- Of course, the Earth is not a uniform sphere so this calculation is only an approximation. Our calculated value will be 20% too high. But here it is:
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------------ Angular Momentum = 0.4 * (6 * 10^24) * ( 6.4*10^6)^2 * ( 7.3*10^-5)
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------------ Angular Momentum = 7.3 * 10^33 kilograms * meters^2 / second.
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- The more accurate calculation for the Earth’s rotational angular momentum is reduced by 20%:
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------------- Angular Momentum = 7.1 * 10^33 kilograms * meters^2 / second.
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- We now make the same calculation for the Moon’s rotational angular momentum;
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--------------------- Momentum = 2/5 * m * r^2 * w
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---------------------- Mass of the Moon = m = 7.35*10^22 kilograms
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--------------------- Radius = r = 1.74*10^6 meters
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--------------------- Angular Velocity = w = 2*pi / 27 days
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--------------------- Angular Velocity = w = 6.24 radians / 2,332,800 seconds
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-------------------- w = 2.7*10^-6 radians per second
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------------------- Momentum = 0.4 * ( 7.35*10^22 ) * (1.74*10^6 )^2 * ( 2.7*10^-6 )
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-------------------- Rotational Momentum = 2.4 *10^29 kg * m^2 /sec
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- Earth - Moon rotational momentum ratio = 7.1 *10^33 / 2.4 * 10^29 = 30,000. So, the Moon’s rotational momentum is not much of a factor for the Earth-Moon system, 1/30,000th is very small. Let’s try calculating the Moon’s orbital momentum.
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--------------------- Orbital Momentum = m * r^2 * w
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--------------------- m = 7.35 * 10^22 kilograms
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--------------------- r = 385,000 kilometers = 3.85 *10^8 meters
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---------------------- w = 2.7*10^-6 radians per second. It is the same because the Moon makes only one rotation with each orbit around the Earth.
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----------- Orbital Momentum = ( 7.35 * 10^22 ) * ( 3.85 *10^8 )^2 * ( 2.7*10^-6 )
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----------- Orbital Momentum = 29.4 *10^33 kg*m^2/sec
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- The Moon’s orbital angular momentum is 4 times greater than the Earth’s rotational momentum. The Earth’s rotation momentum is decreasing , loosing 23 micro-seconds (23*10^-6 sec) per year. The Moon’s orbital angular momentum is increasing to compensate and maintaining the Conservation of Angular Momentum a constant. The Earth-Moon distance is increasing 1 ½ inches per year.
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- The tidal friction increases the length of a day by only 1 second in 50,000 years. This effect is small compared to the effects of earthquakes and slight changes in the Earth’s internal mass distribution. These mass distribution changes can change Earth’s rotation by as much as one second per year. Thus the introduction of the “leap second” in our time standards.
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- Other reviews available on this subject:
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- #1899, #1900, #1901 about moons in our Solar System.
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--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
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----- 707-536-3272 ---------------- Friday, August 26, 2016 -----
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