Gyroscopic exercise tool: Difference between revisions

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{{about|the gyroscopic exercise tool and toy|the US lottery|Powerball|other "powerballs"|Powerball (disambiguation)}}{{More citations needed|date=August 2024}}

[[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 forces.]]

A '''gyroscopic exercise tool''' is a specialized device used in [[physical therapy]] to improve wrist strength and promote the development of palm, wrist, forearm, and finger muscles. It can also be used as a unique demonstration of certainsome aspects of [[dynamics (physics)|rotational dynamics]]. The device consists of a [[tennis ball]]-sized plastic or metal shell surrounding a free-spinning mass, with aan inner heavy inner core, which can be spun by a short rip string. Once the [[gyroscope]] inside is going fast enough, the person holding the device can increaseaccelerate the spinning mass to high rotation rates by moving theirthe wrist in a circular motion<ref>{{Cite journal |last=Heyda |first=P. G. |date=2002-09-12 |title=Roller ball dynamics revisited |url=https://pubs.aip.org/aapt/ajp/article-abstract/70/10/1049/310601/Roller-ball-dynamics-revisited?redirectedFrom=fulltext |journal=American Journal of Physics |volume=70 |issue=10 |pages=1049–1051 |doi=10.1119/1.1499508 |issn=0002-9505 |archive-url=https://web.archive.org/web/20170808024204id_/http://file.seekpart.com/keywordpdf/2010/12/3/201012316133421.pdf |archive-date=2017-08-08}}</ref>. The force exertedenacted on the user increases as the speed of the inner gyroscope increases.

==Mechanics==

{{Tone|date=March 2024}}

Inside the outer shell, the spinning mass is fixed to a thin metal [[axle]], with 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, allowing for the ball to spin perpendicular to the rotational axis of the ring.

To increase the angular velocity of the ball, the sides of the groove exert forces on the ends of the axle. While normal and axial forces have no effect, the tangential force is provided 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. This tilting results in a shift of the axle ends along the groove. The direction and speed of this shift can be determined using 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. The rate of rotation of the internal ball increases as the total applied torque increases. The direction of the torque does not matter, as long as it is perpendicular to the plane of rotation of the ball. The friction of the ring increases on the side opposite to the plane of rotation. This process is symmetric across the plane perpendicular to the axle. The only restriction is that the relative speed of the surface of the axle and the side of the groove due to precession (ΩP​Rgroove​) must exceed the relative speed due to the rotation of the spinning mass (ωraxle​). The minimum torque required to meet this condition is Iω2(Rgroove​raxle​​), where I is the moment of inertia of the spinning mass, and ω is its angular velocity.

To increase the [[angular velocity]] of the ball, the sides of the groove exert forces on the ends of the axle. WhileThe normal and axial forces will have no effect, so the tangential force ismust be provided 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. ThisThe tilting results in a shift of the axle ends along the groove. The direction and speed of thisthe shift can be determinedfound usingfrom 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. The rate of rotation of the internal ball increases as the total appliedamount of torque increasesapplied is increased. The direction of the torque does not matter, as long as it is perpendicular to the plane of rotation of the ball. The friction of the ring increases on the other side of the side opposite to the plane of rotation. This process isobeys symmetricsymmetry 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 due to precession, (ΩP​Rgroove​)<math>\mathit{\Omega}_{\mathrm{P}} R_{\mathrm{groove}}</math>, must exceed the relative speed due to the rotation of the spinning mass, (ωraxle​)<math>\omega r_{\mathrm{axle}}</math>. The minimum torque required to meet this condition is Iω2<math> I \omega^2 \left(Rgroove​raxle​​ 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|angular velocity.]]

Angular acceleration occurs regardless of the direction of the applied torque, as long as it is sufficient. The device functions without fine-tuning the driving motion, and the tilting of the shell does not need to be in rhythm with the precession or at the same frequency. Since kinetic friction is almost as strong as static friction for the materials typically used, it is not necessary to apply exactly the amount of torque needed for the axle to roll without slipping along the side of the groove. These factors enable beginners to learn to speed up the rotation within a few minutes of practice.

AngularSince [[angular acceleration]] will occursoccur regardless of the direction of the applied torque, as long as it is sufficient.large Theenough, the device functionswill function without any fine-tuning of the driving motion,. and theThe tilting of the shell does not needhave to behave a inparticular rhythm with the precession or ateven have the same frequency. Since [[Friction|kinetic friction]] is usually almost as strong as [[Friction|static friction]] for the materials typically used, it is not necessary to apply exactly the amount of torque needed for the axle to roll without slipping along the side of the groove. These factors enableallow beginners to learn to speed up the rotation withinafter only a few minutes of practice.

Using the proportionality of the kinetic force of friction to the normal force, fk​=μk​Fn​, where μk​ is the kinetic coefficient of friction, it can be shown that the torque spinning up the mass is a factor of μk​(Rgroove​raxle​​) 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 ensure that the friction of the ring can exert a force on the gyro.

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 as 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>

==References==

{{reflist}}