# Gyroscope: what is it and how does it work?

This post’s subject is the gyroscope. An instrument of great utility for electronic devices and for air, naval, and space navigations.

## Operation principle

### Angular momentum A force created to rotate an object produces a perpendicular torque (in N\cdot m), following the right hand rule. Source: Pinterest.

Every rotating object has an angular momentum parallel to the rotation axis.

The formula to calculate the angular momentum vector, in $kg\cdot m^{2}/s$.

$\vec{L}=I\cdot \vec{\omega}$

Where $I$ is the moment of inertia (in $kg\cdot m^{2}$) and $\vec{\omega}$ is the angular speed. This formula shows that faster the body spins, higher will be the angular momentum. Moment of inertia is the capacity of a body to resist the variation of speed rotation. The formula for a rigid body is:

$I=\int r^{2}dm$

Where $m$ is the object’s mass and $r$ is the distance from rotation axis to the rigid body’s edge. This formula, with the symbol $\int$, is an integer and from it, you can find the equation of moment of inertia to any object of any shape.

### Precession

The Law of Angular Momentum Conservation claims that if the resulting torque is null, the angular momentum will be constant in direction and magnitude. When higher the angular momentum, higher the resistance to change of this momentum’s direction, caused by a force that produces a torque. The gravity force causes a torque at 90°, creating the precession motion. This torque can only exist with the variation of angular momentum in a time interval.

$\vec{\tau }=\frac{d\vec{L}}{dt}$

The $d$ means infinitesimal variation, a variation so low that tends to zero, but it isn’t zero. The precession phenomenon explains how a wheel with an axis, sustained by a rod or a wire in the ceiling, continues spinning horizontally for a good time. Source: StackExchange.

The precession rate ($\Omega$) in radians per second (rad/s).

$\Omega=\frac{d \theta }{dt}$

$\theta$ is the angle between the initial angular momentum $L$ and the current angular momentum $L'$. Due to infinitesimal variation, you can consider \Delta L=dL. Source: wikimedia.

Since $\theta$ is an infinitesimal angle or very small, you can consider,

$\theta \simeq sen\theta=tan \theta =\frac{dL}{L}$

Because an angle’s tangent is the opposite side divided by adjacent. The torque is the mass $M$ times gravity acceleration $g$ times rotation axis to the disk $r$.

$dL=\tau dt=(Mgr)dt$

$d\phi =\frac{dL}{L}=\frac{Mgrdt}{I\omega}$

$\Omega =\frac{Mgr}{I\omega}$

## The mechanical gyroscope

In a mechanical gyroscope, when the central wheel spins at a high speed, the central axis will keep in the same direction, doesn’t matter the motion of inner rings and support.

This feature is useful to indicate if the aircraft or submarine is going up, down, or is kept at the same altitude or depth. The attitude indicator, used in aircraft, is a vacuum tube with a gyroscope inside. The air enters to exert force to spin the brass disk, which has grooves. Then the air goes out through a vacuum pump so that the disk does not lose speed due to friction. Source: FSAC Training Academy. Boats use a stabilizer gyroscope to keep them stable at sea. Source: Sea Keeper.

In space, where there are few reference points, the gyroscope’s axis serves as a reference for orientation.

## Other types of gyroscopes

### MEMS gyroscope

This is the type of gyroscope found in many electronic systems. Consist of a resonant silicon ring inside the chip. Whose movements are converted into very small electric signals that are detected by electrodes, but need to be amplified. The ring deforms to an ellipse shape due to Coriolis force, the vectors on diagonal are the resulting radial motion. Source: Electronic Paper. On the left, a vibrating ring without angular acceleration, blue points are fixed. On the right, the ring receives angular velocity, producing a Coriolis force that changes the ring’s vibration direction. Source: Silicon Sensing.

A Silicon Sensing video shows the operation of the gyroscope inside a vehicle.

Exist other types of MEMS gyroscopes, with the same operation principle, use a resonant mass linked to springs and with variable capacitors to detect changes in capacitances when the mass receives angular acceleration.

### Optical gyroscope 