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Analysis of possible existence of a 'gyroscopic' resistance to change in swing plane.


Homer sees a 'gyroscopic' action for a whirling weight which should resist any attempt to change its plane of rotation. We will analyze this view below.


Fig1 shows a point mass M whirling around the vertical axis.  Re to Fig1 one can readily derive the relations (1), (2) and (3).

The centrifugal force Fc,

Fc = M r Cos[θ[t]] ϕ^′[t]^2 ------------------------------------ (1)

The centrifugal restoring torque Tc,

Tc = M r^2  Cos[θ[t]] Sin[θ[t]] ϕ^′[t]^2 --------------------------- ---(2)

The gravitational torque Tg,

Tg = M r g Cos[θ[t]] ----------------------------   ---------- (3)



The two governing equations derived to describe in general the motion of the point mass M in Fig1 are given by relations (4) and (5).

T1 = M r^2θ^′′[t] +   M r g Cos[θ[t]] +   M r^2 Cos[θ[t]] Sin[θ[t]] ϕ^′[t]^2 -- ------ - ---(4)

T2 = M r^2 Cos[θ[t]] Cos[θ[t]] ϕ^′′[t] - 2 M r^2 Cos[θ[t]] Sin[θ[t]] θ^′[t] ϕ^′[t] ---------- (5)


If we look at equation (4), which deals with the out-of-swing-plane vertical motion of the point mass M, one notices again, in addition to the inertial term, exactly the same centrifugal and gravitational torques terms, derived above directly and given by the relations (2) and (3).

The two different approaches both cleary show that except for gravity or centrifugal force no other mechanism is at work such a 'gyroscopic action".



The mass M has a constant angular velocity when at some time a torque T1 is applied perpendicular to the swingplane. The motion of the mass M is analyzed solving the two governing equations (1) and (2).

The resulting motion is shown in Fig2. 'Red' indicates constant velocity. 'Black' indicates that disturbing torque T1 is applied. The result is shown in Fig2 and simply shows a gradual vertical out of plane motion, in accordance witht the torque T1. For simplicity gravity has been ignored.

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Fig2

Conclusion:

From above it is clear that no similarity, no like behavour, exists between a golf swing and a gyroscope. If bbftx was correct expecting such a behavior to exist for a golf swing it should have shown up in the governing equations derived for the point mass M. The restoring mechanism for the out of plane motion is however clearly shown to be caused by the centrifugal force, as I have indicated in various posts.

mandrin

PS.: For those who always prefer precison and exactitude I like to mention that Fc in Fig1 is not acting on the point mass M but pulling on the end of the rigid rod. The centripetal force, not shown, is the force acting on M.