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Consider a normal pendulum with a spherical bob oscillating back and forth. Would the period of the pendulum be longer, shorter or unchanged if the bob were to spin around the axis of the string that holds it?

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    $\begingroup$ Do you have any reasoned expectation? Is there a specific conceptual idea you're asking about? What affects the period of a pendulum, to the first order? $\endgroup$ – Bill N Sep 25 '17 at 19:13
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Rotation will create a gyroscopic tilting force perpendicular to the plane of oscillation, as the gyroscope "tries" to remain oriented the same way. This causes precession, giving the gyroscope oscillations of it's own.

Simplistically, It may be modelled as a compound pendulum, one pendulum on the end of another. There will be two periods, and these will combine to give a displacement waveform as two superimposed sine waves. The overall system will have a period that is the period of the beats of the two waveforms, which is

$\frac{1}{T_{sys}}=\frac{1}{T_{main}}-\frac{1}{T_{gyro}}$

There are other effects, however, making the above approximate; and the actual motion is probably chaotic - (meaning it is multi-stable with almsot random transitions).

As the forces the gyroscope experience depend on the position of the main pendulum as well as gravity, this will introduce a further harmonic. Additionally if the motion of the gyroscope increases its height, this could shorten a swing. If its height is lowered it could lengthen a swing. Finally of course practically the motion is damped by air resistance and friction.

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You say the pendulum has a bob "on a string". That's a complicated geometry to analyze, because even the spherical bob that is not spinning will have a complex motion - if you consider the angle of the string to the vertical $\theta_1$ and the angle of the bob to the vertical $\theta_2$, then these two can oscillate either in phase, or out of phase, for the two possible modes.

Alternatively, the bob is rigidly linked to the pendulum. In that case, when you make the bob spin, it will act like a gyroscope and it will slow down the pendulum. In fact, if you look from the top, you need the angular momentum to remain constant - so as the angle of the pendulum drops the entire thing needs to start precessing. The motion will be complicated (look up nutation for more on this, or see this earlier question).

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