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So, I've been led to believe that the frequency of nutation of a gyroscope can be calculated using the formula

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In which the I's are the moments of inertia around the principal axes and omega-3 is the angular velocity of the disk of the gyroscope. But the moments of inertia around the principal axes of a cilinder are

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and working this out for a gyroscope with a disc of 1.5 kg, a radius of 11.5 cm and a height of 2,5 cm I get:

enter image description here

(There was a math error here, it's gone now) Which would mean the frequency of nutation is 2 times higher than the angular velocity, which does not match observation. What am I doing wrong? Thank you in advance.

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The gyroscope used.

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  • $\begingroup$ You have misunderstood the formula. The correct version relates the nutation and precession frequencies, not the angular velocity. Your interpretation of what the three moments of inertia are also seems to be wrong. $\endgroup$
    – alephzero
    Jun 10, 2019 at 21:18
  • $\begingroup$ Could you perhaps direct me to some place where this is explained further? I got that formula from Morin's "Introduction to Classical Mechanics" and this is the best I can make of it. $\endgroup$
    – wasneeplus
    Jun 10, 2019 at 21:23
  • $\begingroup$ The moments of inertia that you are currently using are the ones for the case of a body that is in free fall. The torque-free precession in free fall is with respect to the center of mass of the body. (The spinning disk in free fall case is also known as Feynman's wobbling plate, which indeed displays that ratio of 2:1) However, the picture of your setup shows a wheel mounted on an axis, and the pivot point is far away from the wheel. You need to use the moment of inertia of that entire assembly: wheel, axis and counterweight. $\endgroup$
    – Cleonis
    Jun 12, 2019 at 18:54

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I can confirm that the nutation frequency is proportional to the angular velocity of the spinning gyroscope.

Article:
precession and nutation of a gyroscope, by Eugene Butikov
In the above article it is discussed that the nutation frequency is the same frequency as the frequency of torque-free precession because it is in fact the torque-free precession. (The torque-free precession of the entire precessing assembly.)

Also:
Documentation for an educational set containing a gimbal mounted gyroscope and weights that can be added to the gimbal mounting, altering a specific moment of inertia, thus altering the nutation frequency:
Laws of gyroscopes

My understanding is that if the gyroscope wheel is mounted on a long axis that axis has a significant effect on the nutation frequency.

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