A gyroscopic exercise tool is a specialized device used in

physical therapy Physical therapy (PT), also known as physiotherapy, is a healthcare profession, as well as the care provided by physical therapists who promote, maintain, or restore health through patient education, physical intervention, disease preventio ...

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 some aspects of rotational dynamics. The device consists of a

tennis ball A tennis ball is a small, hollow ball used in games of tennis and real tennis. Tennis balls are fluorescent yellow in Professional sports, professional competitions, but in Amateur sports, recreational play other colors are also used. Tennis bal ...

-sized plastic or metal shell surrounding a free-spinning mass, with an inner heavy core, which can be spun by a short rip string or using a self-start mechanism by means of rewinding it against a spring to give it

potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...

. Once the

gyroscope A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining Orientation (geometry), orientation and angular velocity. It is a spinning wheel or disc in ...

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. The force enacted on the user increases as the speed of the inner gyroscope increases.

Mechanics

Inside the outer shell, the spinning mass is fixed to a thin metal

axle An axle or axletree is a central shaft for a rotation, rotating wheel and axle, 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 ...

, 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 the ball to spin perpendicular to the rotational axis of the ring. To increase the

angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...

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 provided by the

friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...

of the ring acting on the axle. The user can apply a

torque In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...

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 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

: 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 ...

of the

of precession and the

angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...

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 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,

\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/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. It is the ratio between ...

of the spinning mass, and ω is its

. Since

angular acceleration In physics, angular acceleration (symbol α, alpha) is the time rate of change of angular velocity. Following the two types of angular velocity, ''spin angular velocity'' and ''orbital angular velocity'', the respective types of angular accele ...

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 rhythm with the precession or even have the same frequency. Since

kinetic friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...

is usually almost as strong as

static friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...

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 allow beginners to learn to speed up the rotation after only a few minutes of practice. By applying the proportionality of the kinetic force of friction to the

normal force In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts. In this instance '' normal'' is used in the geometric sense and means perpendicular, as opposed to the meanin ...

f_\mathrm = \mu_\mathrm F_\mathrm

, where

\mu_\mathrm

is the kinetic coefficient of friction, it can be shown that the

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 to allow for the friction of the ring to enact a force on the gyro.

References

{{Commons category, Gyroscopic exercise tools Exercise equipment Physical activity and dexterity toys