Angular Acceleration of the Earth
Constants
Garvitational Constant:
[1]
[2]
Angle that the tidal bulge leads the moon by (given):
[3]
Height of the tidal bulge (given):
[4]
Mass of the Earth:
[5]
Density of water:
[6]
Calculations
Tidal acceleration
[7]
Density of the bulges:
[8]
Volume in the bulges:
[9]
Mass in the bulges:
[10]
Torque of the bulges:
[11]
Rotational inertia of the earth:
[12]
Angular acceleration of the Earth:
[13]
[14]
Constants
Mass of the Moon:
[15]
Mean distance between the Earth and Moon now:
[16]
Rotational period of the Earth:
[17]
[18]
Orbital period of the moon:
[19]
Calculations
Orbital velocity of the moon:
[20]
Angular velocity of the moon:
[21]
Angular velocity of the Earth:
[22]
Rotational velocity of the Earth:
[23]
Rotational inertia of the moon:
[24]
Ratio of the moon's roatational inertia to the Earth's rotational inertia:
[25]
Period of the moon's orbit:
[26]
Derivative of the total angular
momentum w/ respect to time:
[27a]
[27b]
[28]
Extrapolating Forward (and backward) in Time
Calculations
Orbital kinetic energy of the moon:
[29]
Gravitational potential energy of the moon:
[30]
Rotational kinetic energy of the moon:
[31]
Rotational kinetic energy of the Earth:
[32]
Total energy of the system today:
[33]
Total angular momentum today:
[34]
Difference in total angular energy from today as a function of mean distance
(I switch from w to W in this section to avoid confusion since I am now talking about angular velocities that are changing with distance)
[35]
Solving for Wearth:
[36]
[37]
Solving for Ediff
[38]
Figure 2
Visual guess of the equilibrium distance:
[39]
Solving for the angular velocity of the moon at equilibrium:
[40]
guess value for solve block:
Solve Block:
[41]
Equilibrium:
Length of a day at equilibrium:
[42]
Mean distance of moon at equilibrium:
[43]
Roche Limit:
[44]
[45]
[46]
The Roche limit for the moon in Earth radii:
[47]
The Roche limit for the moon in current Earth-moon distances:
Rate of energy loss:
Rate of energy loss today (given):
[48]
Combined angular velocity of the system:
[49]
Experimentally determining a function for the rate of energy loss:
[50]
Guess function for rate of energy loss:
[51]
Figure 3
o - Today
Time between two distances:
[52]
[53]
Figure 4