Fixed minor spelling and grammatical errors. Used formal and correct scientific/physics terms for coherency and better understanding. Reduced some awkwardness in the structure of phrases in the article.
Inside the outer shell, 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, allowing for the ball to spin perpendicular to the rotational axis of the ring.
To increase the rate[[angular of rotationvelocity]] of the ball, the sides of the groove to exert forces on the ends of the axle. The normal and axial forces will have no effect, so the tangential force must 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. 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. The rate of rotation of the internal ball increases as the total amount of torque applied 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 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 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|angular velocity.]]
Since the[[angular rotationacceleration]] will accelerateoccur 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 rhythm with the precession or even have the same frequency. Since sliding (kinetic) [[Friction|kinetic friction]] is usually nearlyalmost as strong as [[StaticFriction|static friction|static]] (sticking) friction for the materials usuallytypically 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.
By applying the proportionality of the [[Friction|kinetic force of friction]] to the [[normal force]], <math>F_f_\mathrm{fk} = \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>
