Gyroscopic exercise tool: Difference between revisions

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{{Short description|Device used in physical therapy}}

{{about|the gyroscopic exercise tool and toy|the US lottery|Powerball|other "powerballs"|Powerball (disambiguation)}}

[[Image:Gyrotwister.jpg|thumb|A gyroscopic wrist exerciser.]]

[[File:Video of a complete use session with a gyroscopic exercise tool.webm|thumb|Video showing the use - from starting the rotation with a 'shoestring' over various movements with the holding hand until stopping the rotor with the second hand. The demonstrated speeds are, in part, very high and not recommended for normal exercise due to the high resulting high forces.]]

A '''gyroscopic exercise tool''' is a specialized device used to exercise thein [[wristphysical therapy]] asto partimprove ofwrist [[physicalstrength therapy]]and orpromote inthe orderdevelopment to buildof palm, wrist, forearm, and finger strengthmuscles. It can also be used as a unique demonstration of some aspects of [[dynamics (physics)|rotational dynamics]]. The device consists of a [[tennis ball]]-sized plastic or metal shell aroundsurrounding a free-spinning mass, with an inner heavy core, which iscan startedbe withspun by a short rip string or using a self-start mechanism by means of rewinding it against a spring to give it [[potential energy]]. Once the [[gyroscope]] inside is going fast enough, athe person holding the device can accelerate the spinning mass to high rotation rates by moving the wrist in a circular motion. The force enacted on the user increases as the speed of the inner gyroscope increases.

== History ==

{{empty section|date=May 2020}}

==Mechanics==

{{Tone|Disputed section|Friction based explanation is wrong|date =June 2011March 2025}}

{{Tone|date=March 2024}}

The device consists of a spinning mass inside an outer shell. The shell almost wholly covers the mass inside, with only a small round opening allowing the gyroscope to be manually started. The spinning mass is fixed to a thin metal [[axle]], each end trapped in a circular, equatorial groove in the outer shell. A lightweight ring with two notches for the axle ends rests in the groove. This ring can slip in the groove; it centres the spinning gyroscope in the shell, preventing the two from coming into contact (which would slow the gyro down) while still allowing the orientation of the axle to change.

TheInside device consists of a spinning mass inside anthe outer shell. The shell almost wholly covers the mass inside, with only a small round opening allowing the gyroscope to be manually started. The spinning mass is fixed to a thin metal [[axle]], each end trapped in a circular, equatorial groove in the outer shell. A lightweight ring with two notches for the axle ends rests in the groove. This ring can slip in the groove; it centres the spinning gyroscope in the shell, preventingallowing the twoball fromto comingspin intoperpendicular contact (which would slowto the gyrorotational down) while still allowing the orientationaxis of the axlering. to change.

Since the spinning mass is balanced, the only possibility to speed up the rotation is for the sides of the groove to exert forces on the ends of the axle. Furthermore, the normal and axial forces will have no effect, so tangential force must be provided by [[friction]]. If the axle is stationary, the friction will only act to slow down the rotation, but the situation is very different if the axle is turned by applying a [[torque]].

ThisTo canincrease the [[angular velocity]] of the ball, the sides of the groove exert forces on the ends of the axle. The normal and axial forces will have no effect, so the tangential force must be accomplishedprovided by the [[friction]] of the ring acting on the axle. The user can apply a [[torque]] on the ball by tilting the shell in any direction except in the plane of the groove or around an axis aligned with the axle. The tilting results in a shift of the axle ends along the groove. The direction and speed of the shift can be found from the formula for the [[precession]] of a [[gyroscope]]: the applied torque is equal to the [[cross product]] of the [[angular velocity]] of precession and the [[angular momentum]] of the spinning mass. ItThe israte observedof hererotation thatof the directioninternal ball increases as the total amount of torque applied is suchincreased. thatThe ifdirection of the torque isdoes largenot enoughmatter, as long as it is perpendicular to the plane of rotation of the ball. The friction betweenof the ring increases on the side opposite to the plane of rotation. This process obeys symmetry across the plane perpendicular to the axle. The only restriction to this process is that the relative speed of the surface of the axle and the side of the groove surfacedue willto precession, <math>\mathit{\Omega}_{\mathrm{P}} R_{\mathrm{groove}}</math>, must exceed the relative speed updue to the rotation of the spinning mass, <math>\omega r_{\mathrm{axle}}</math>. The minimum torque required to meet this condition is <math> I \omega^2 \left( r_{\mathrm{axle}} / R_{\mathrm{groove}} \right) </math>, where '''I''' is the [[moment of inertia]] of the spinning mass, and '''ω''' is its [[angular velocity]].

Since the[[angular acceleration of the rotation]] will occur regardless of the direction of the applied torque, as long as it is large enough, the device will function without any fine-tuning of the driving motion. The tilting of the shell does not have to have a particular phase relationshiprhythm with the precession or even to have the same frequency. Since sliding ([[Friction|kinetic) friction]] is usually nearlyalmost as strong as [[Friction|static (sticking) friction]] for the materials typically used, it is not necessary to apply preciselyexactly the valueamount of torque (whichneeded will result infor the axle rollingto roll without slipping along the side of the groove). These factors allow beginners to learn to speed up the rotation after only a few minutes of practice.

Usually, if the axle were shifting in a horizontal groove, the friction on one end that speeds up the rotation would be cancelled by the friction at the other end, operating in the opposite direction. But, here, the difference is that a torque is being applied, so one end of the axle is pushed against one side of the groove, while the other is pushed against the other. It does not matter in which direction the torque is applied. If the torque is reversed, each end of the axle will be pressed against the opposite side of the groove, but the direction of precession will also be reversed. The only restriction is that the relative speed of the surface of the axle and the side of the groove due to precession, <math>\mathit{\Omega}_{\mathrm{P}} R_{\mathrm{groove}}</math>, must exceed the relative speed due to the rotation of the spinning mass, <math>\omega r_{\mathrm{axle}}</math>. The minimum torque required to meet this condition is <math> I \omega^2 \left( r_{\mathrm{axle}} / R_{\mathrm{groove}} \right) </math>, where '''I''' is the [[moment of inertia]] of the spinning mass, and '''ω''' is its angular velocity.

By applying the proportionality of the [[Friction|kinetic force of friction]] to the [[normal force]], <math>f_\mathrm{k} = \mu_\mathrm{k} F_\mathrm{n}</math>, where <math>\mu_\mathrm{k}</math> is the [[Friction#Coefficient_of_friction|kinetic coefficient of friction]], it can be shown that the [[torque]] spinning up the mass is a factor of <math>\mu_\mathrm{k} \left( r_{\mathrm{axle}} / R_{\mathrm{groove}} \right)</math> smaller than the torque applied to the shell. Since frictional force is essential for the device's operation, the groove must not be lubricated to allow for the friction of the ring to enact a force on the gyro.<ref>{{cite journal |title=The Physics of the ''Dyna Bee'' |date=February 1, 1980 |issn=0031-921X |doi=10.1119/1.2340452 |issue=2 |volume=18 |pages=147–8 |journal=The Physics Teacher |first=J. |last=Higbie|bibcode=1980PhTea..18..147H}} {{closed access}}</ref><ref>{{cite journal |title=Roller Ball Dynamics |date=2000 |issue=9 |volume=36 |journal=Mathematics Today |first=P. G. |last=Heyda}}</ref><ref>{{cite journal |title=Roller Ball Dynamics Revisited |date=October 1, 2002 |issn=0002-9505 |doi=10.1119/1.1499508 |issue=10 |volume=70 |pages=1049–51 |journal=American Journal of Physics |first=P. G. |last=Heyda|bibcode=2002AmJPh..70.1049H}}</ref><ref>{{cite journal |title=On the Dynamics of the Dynabee |date=June 1, 2000 |issn=0021-8936 |doi=10.1115/1.1304914 |issue=2 |volume=67 |pages=321–5 |journal=Journal of Applied Mechanics |first1=D. W. |last1=Gulick |first2=O. M. |last2=O’Reilly|bibcode=2000JAM....67..321G}}</ref><ref>{{cite journal |title=Modelling of the Robotic Powerball®: A Nonholonomic, Underactuated and Variable Structure-Type System |date=June 1, 2010 |doi=10.1080/13873954.2010.484237 |first1=Tadej |last1=Petrič |first2=Boris |last2=Curk |first3=Peter |last3=Cafuta |first4=Leon |last4=Žlajpah |journal=Mathematical and Computer Modelling of Dynamical Systems|volume=16|issue=4 |pages=327–346 |hdl=10.1080/13873954.2010.484237 |s2cid=120513329 |hdl-access=free}}</ref>

Since the acceleration of the rotation will occur regardless of the direction of the applied torque, as long as it is large enough, the device will function without any fine-tuning of the driving motion. The tilting of the shell does not have to have a particular phase relationship with the precession or even to have the same frequency. Since sliding (kinetic) friction is usually nearly as strong as static (sticking) friction, it is not necessary to apply precisely the value of torque (which will result in the axle rolling without slipping along the side of the groove). These factors allow beginners to learn to speed up the rotation after only a few minutes of practice.

== References ==

By applying the proportionality of the force of friction to the normal force, <math>F_\mathrm{f} = \mu_\mathrm{k} F_\mathrm{n}</math>, where <math>\mu_\mathrm{k}</math> is the [[Friction#Coefficient_of_friction|kinetic coefficient of friction]], it can be shown that the torque spinning up the mass is a factor of <math>\mu_\mathrm{k} \left( r_{\mathrm{axle}} / R_{\mathrm{groove}} \right)</math> smaller than the torque applied to the shell. Since frictional force is essential for the device's operation, the groove must not be lubricated.<ref>Articles on the physics of the device (in approximately increasing order of sophistication):

{{reflist}}

* {{cite journal |title=The Physics of the ''Dyna Bee'' |date=February 1, 1980 |issn=0031-921X |doi=10.1119/1.2340452 |issue=2 |volume=18 |pages=147–8 |journal=The Physics Teacher |first=J. |last=Higbie|bibcode=1980PhTea..18..147H }} {{closed access}}

* {{cite journal |title=Roller Ball Dynamics |date=2000 |issue=9 |volume=36 |journal=Mathematics Today |first=P. G. |last=Heyda}}

* {{cite journal |title=Roller Ball Dynamics Revisited |date=October 1, 2002 |issn=0002-9505 |doi=10.1119/1.1499508 |issue=10 |volume=70 |pages=1049–51 |journal=American Journal of Physics |first=P. G. |last=Heyda|bibcode=2002AmJPh..70.1049H }}

* {{cite journal |title=On the Dynamics of the Dynabee |date=June 1, 2000 |issn=0021-8936 |doi=10.1115/1.1304914 |issue=2 |volume=67 |pages=321–5 |journal=Journal of Applied Mechanics |first1=D. W. |last1=Gulick |first2=O. M. |last2=O’Reilly|bibcode=2000JAM....67..321G }}

* {{cite journal |title=Modelling of the Robotic Powerball®: A Nonholonomic, Underactuated and Variable Structure-Type System |date=June 1, 2010 |doi=10.1080/13873954.2010.484237 |first1=Tadej |last1=Petrič |first2=Boris |last2=Curk |first3=Peter |last3=Cafuta |first4=Leon |last4=Žlajpah |journal=Mathematical and Computer Modelling of Dynamical Systems|volume = 16|issue=4 |pages=327–346 |hdl=10.1080/13873954.2010.484237 |s2cid=120513329 |hdl-access=free }}

</ref>

<br />{{Commons category|Gyroscopic exercise tools}}

As explained, the energy supplied by the external momentum (the hand and arm muscles) can not be directly converted into the rotational energy of the gyroscope around its own axis. Still, it is converted into the energy of precession rotation. Due to the friction between the gyroscope shaft and the sides of the groove, part of this energy is converted into energy of rotation around the gyroscope's axis, accelerating it. This happens when there is friction and a round object, like a bowling ball, is thrown along a horizontal surface. A part of the ball's kinetic energy is converted to rotational energy due to friction.

== References ==

<br />{{Commons category|Gyroscopic exercise tools}}

{{reflist}}

[[Category:Exercise equipment]]