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How to Calculate Precession Cycles

If you have watched a top whirl, you might have noticed the peg makes small circles as the top spins. Watch long enough, and those circles will get larger and faster as the top slows and nears the tipping point. The path traveled by the peg is called a precession cycle. Deducing the mathematics behind this physical problem might sound like a mere intellectual curiosity, but the axis of our planet spins in a similar pattern. Understanding the relationship between Earth's axis and the sun can aid in making predictions about future climate change.

Instructions

Spin Angular Momentum
- 1
  Determine as many physical measurements as possible about your model. If nothing else, you should have the mass of the object and the radius of the object along the axis that it spins (i.e., half the distance from top to bottom).
- 2
  Establish the object's moment of inertia by multiplying the mass times the square of the radius. I = mr^2, where I is moment of inertia, m is mass and r is radius.
- 3
  Determine the angular velocity by multiplying pi times 2 times the frequency of the revolutions. w = 2(3.14)f, where w is the angular velocity and f is the number of revolutions per unit of time.
- 4
  Calculate the spin angular momentum. L = Iw, where L is the angular momentum result, I is the moment of inertia and w equals angular velocity. You will need this value to plug into the formula for the next step.
Precession Cycles
- 5
  Multiply the mass of the object by the gravitational constant (assuming that your object is Earth-bound). The gravitational constant is 6.67300 --- 10^-11 m^3 kg^-1 s^-2.
- 6
  Take the product of Step 1 and multiply it times the radius of the object, which should have the same value as r used in the moment of inertia. (Do not square it here.)
- 7
  Divide the result of the first two steps by the spin angular momentum. Your result will be the precessional cycle rate. The above steps combine to give the formula W = mgr / L
  
  where W is the precession cycle rate; m is mass, g is the gravitational constant and r equals the radius of the object; and L is the spin angular momentum.

Astronomy