A gyroscopic exercise tool is a device used to exercise the

wrist In human anatomy, the wrist is variously defined as (1) the carpus or carpal bones, the complex of eight bones forming the proximal skeletal segment of the hand; "The wrist contains eight bones, roughly aligned in two rows, known as the carpal ...

as part of

physical therapy Physical therapy (PT), also known as physiotherapy, is one of the allied health professions. It is provided by physical therapists who promote, maintain, or restore health through physical examination, diagnosis, management, prognosis, patie ...

in order to build palm, forearm and finger strength. It can also be used as a unique demonstration of some aspects of

rotational dynamics Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rota ...

. The device consists of a

tennis ball A tennis ball is a ball designed for the sport of tennis. Tennis balls are fluorescent yellow in organised competitions, but in recreational play can be virtually any color. Tennis balls are covered in a fibrous felt which modifies their aerody ...

-sized plastic or metal shell around a free-spinning mass, which is started with a short rip string. Once the gyroscope inside is going fast enough, the person holding the device can accelerate the spinning mass to high rotation rates by moving the wrist in a circular motion.

History

Mechanics

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 An axle or axletree is a central shaft for a rotating wheel or gear. On wheeled vehicles, the axle may be fixed to the wheels, rotating with them, or fixed to the vehicle, with the wheels rotating around the axle. In the former case, beari ...

, 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. 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 Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...

. 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 In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of t ...

. This can be accomplished 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 Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...

of a gyroscope: the applied torque is equal to the

cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...

of the

angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

of precession and the

angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...

of the spinning mass. It is observed here that the direction is such that if the torque is large enough, the friction between the axle and the groove surface will speed up the rotation. 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,

\mathit_ R_

, must exceed the relative speed due to the rotation of the spinning mass,

\omega r_

. The minimum torque required to meet this condition is

I \omega^2 \left( r_ / R_ \right)

, where I is the

moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...

of the spinning mass, and ω is its angular velocity. 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. By applying the proportionality of the force of friction to the normal force,

F_\mathrm = \mu_\mathrm F_\mathrm

, where

\mu_\mathrm

is the kinetic coefficient of friction, it can be shown that the torque spinning up the mass is a factor of

\mu_\mathrm \left( r_ / R_ \right)

smaller than the torque applied to the shell. Since frictional force is essential for the device's operation, the groove must not be lubricated.Articles on the physics of the device (in approximately increasing order of sophistication): * * * * * 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

{{reflist Exercise equipment Physical activity and dexterity toys