Angular Momentum

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Discussion

the idea

Translational and Rotational Quantities Compared
  translational connections rotational
momentum
p =
m v
L =
r × p = m r × v
L =
I ω
impulse-momentum ∫ F · dt

F =
 = Δp

dp / dt
     ∫ τ · dt

τ
 = ΔL 

dL / dt

text

L =  r × p = r × m v
L =  m r × (ω × r) = m ω(r·r) − mr(r · ω) = mr2 ω − 0 = mr2 ω
L =  I ω

Moment of inertia again

I = ∑ r2m =  ⌠
⌡		  r2 dm = ρ  ⌠
⌡		  r2 dV 

Conservation of angular momentum

∑ τ =  d  L = 0
dt

Thus

L = constant

the earth

The earth is not the world's best clock.


[magnify]


[magnify]

types of LOD change … horrible mess …

  1. a linear increase owing to tidal dissipation

    The tidal friction between the oceans and the Earth's surface causes the Earth's rotation to slow by approximately 0.002 seconds every century. However, ignoring energy lost to heat generated by the tides, the angular momentum of the Earth-Moon system must remain constant. The Earth's angular momentum is decreasing, so the Moon's must increase. The only way it can do this is by moving into a higher orbit around the Earth. Thus, the distance to the farthest point of the lunar orbit is increasing by about 3.8 centimeters per year.

  2. Larger, irregular variations, on the scale of decades, owing to core-mantle interactions
  3. Solar flares are known to abruptly alter the Earth's rotation. For example, the great flares of 1959 and 1972 brought abrupt changes in the LOD. Both long-term and short-term changes in solar activity alter the Earth's rotation. The 11-year, 22-year and 56-year solar cycles are conspicuous in the LOD data. Long-term (secular) changes were noted at times of fluctuations in solar activity in the past, and the evidence indicates that these changes were very likely abrupt. Observations of the LOD, like so many other geophysical phenomena, reveal the solar-FEM linkage.
  4. Seasonal changes from the angular momentum exchange between crust and atmosphere.

The rate of rotation of the earth is high in summer and low in winter (northern hemisphere).

At time scales of hours to days, tidally driven? ocean currents and variations in the ocean height dominate

Summary

Problems

practice

  1. Write something.
    • Answer it.
  2. Write something different.
    • Answer it.
  3. Write something different.
    • Answer it.
  4. Write something completely different.
    • Answer it.

numerical

  1. Sun to a white dwarf. 27 days to … ?

Resources

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