a gyroscope is a device that depending on its composition can fulfill two functions to provide information about the variation of the orientation of a system with respect to a reference axis or to provide information about the rate at which the orientation of a system varies when it is rotating that is its angular velocity gyroscopes are used in a wide range of applications including our phones aeronautics video game consoles and robotics moreover our own body has a built-in gyroscope known as the vestibular system which gives us information about our orientation and helps us maintain our balance that is why in this video we will see how a gyroscope works including mechanical coriolis effect vibratory and optical gyroscopes let's start by looking at how mechanical gyroscopes work and to do that we need to understand what torque and angular momentum are in a system with an axis of rotation when we apply a force at a point away from the axis a torque is generated which rotates the system and is represented as a vector parallel to the axis of rotation in addition to this when a system is rotating it has an angular momentum as well which is also represented by a vector parallel to the axis of rotation and is determined by two factors first the moment of inertia of the system which depends on its shape and mass distribution and second the angular velocity which tells us how many degrees the system rotates during a defined period of time understanding this we can rely on the operation of an accelerometer to understand the operation of this first type of gyroscope in the previous video we saw how some of newton's laws could be used to know the linear acceleration of a system newton's first law tells us that when the net force applied on an object is equal to 0 its acceleration is also zero and consequently its velocity will remain constant similarly if we have a system which is capable of rotating on an axis and the net external torque acting on it is zero the total angular momentum of the system will also remain constant on the other hand newton's second law tells us that the force applied on an object is equal to its mass multiplied by the acceleration generated as a result of the applied force similarly for rotating systems the torque of the net force acting on a system is equal to the rate of change of its angular momentum which in this case is also equal to the moment of inertia multiplied by the angular acceleration having clarified this we can finally focus on the practical applications of these principles in mechanical gyroscopes the first application makes use of the spatial rigidity of a rotating object if we rotate a disk at a certain angular velocity and there are no other forces that generate any torque on the disc its angular momentum will be conserved because of this it will continue to rotate on the same axis and therefore maintain its orientation this was used by leon foucault who mounted a disc on a card and suspension or gimbal which allows free rotation of the centerpiece in this system by rolling the center disc at a high speed or in technical terms applying a large angular momentum to it its orientation will not change even if the rest of the system does in ancient times when gps did not exist they were extremely useful as an alternative method of orientation because unlike compasses which use the magnetic fields of the planet to orient themselves and indicate north this type of gyroscope could be oriented in any direction desired and its accuracy would not be affected by variations in magnetic fields although of course its limitation is that the friction of the axis however small would eventually reduce the angular momentum of the disk and similarly the torque transmitted by the suspension however small would eventually change the original orientation anyway in spite of all this it is still a valid method for particular cases at present an example of this is gravity probe b a satellite used to test albert einstein's theory of relativity and whose gyroscopes could theoretically rotate for up to fifteen thousand years which by the way is extremely complex and a perfect subject for a future video now going back to the topic the second practical application of mechanical gyroscopes is to determine the speed of a system and to do this they take advantage of a physical phenomenon called precession in simple terms precession is a circular motion that is generated when a rotating object is affected by a force that causes it to change its orientation an example of this would be a bicycle wheel hanging at one end of its axle which if it were at rest would fall due to gravity however by possessing an angular velocity this counter-intuitive phenomenon known as precession occurs first it is able to stay in its original orientation without falling and second it begins to rotate around the supporting point to understand why precession occurs let's take a closer look at what happens to the wheel respect to its supporting point the weight will generate a perpendicular torque that would cause the wheel to fall to the ground however according to the second law the torque will generate a small change of angular momentum in its direction which in this case has no vertical component by adding the initial angular momentum with the small change due to the torque the resulting angular momentum and thus the wheel's axis of rotation will have slightly changed its orientation horizontally without falling due to gravity moreover since it is always true that the torque generated by the weight will be perpendicular to the angular momentum of the wheel then the wheel will roll continuously forming a circumference this relationship between the torque applied to a rotating object and the rotation resulting from precession is what we can use to determine the angular velocity of a system more specifically we can have a system like this with a rotating disc at the center which in turn will be suspended by torsion bars these will allow the suspension supporting the disc to rotate but will also impose a resistance that will increase proportionally to the torque which means they allow us to use the angle of torsion to calculate the applied torque similar to how springs allow us to use elongation to calculate the applied force this particular system is made to detect movements in the z-axis if the central disk is rotating and the whole system is rotated about the z-axis the precession will generate a torque on the torsion bar and the frame will rotate slightly marking a value on a conversion scale allowing the angular velocity of the system to be known it's not the most intuitive system in the world but that makes it even more impressive from a design standpoint in addition these systems have one major limitation since their operation originally depends on the moment of inertia of the rotating disc which as i mentioned at the beginning depends on the mass of the disc it is impossible to reduce the size of these systems without affecting their accuracy or rotation time but we had to start from something the next type of gyroscope we will discuss is the coriolis effect vibrating gyroscope this type of gyroscope is one of the most widely used nowadays as they can be manufactured in really small sizes at a low cost and therefore can be integrated into all kinds of electronic devices such as your phones to understand the principle of their operation we must understand what coriolis acceleration is imagine a particle rotating around a point at a constant angular velocity the particle's trajectory will form a circle with a radius r1 and the particle will have a tangential velocity of one now if that particle moves radially to a distance t2 the size of the circle defining its trajectory will increase and therefore its tangential velocity will also have to increase to continue rotating at the same angular velocity in other words if the tangential velocity increases then there is an acceleration which is known as coriolis acceleration in honor of its discover this acceleration by the way can be calculated as minus two times the angular velocity of the particle multiplied by the speed at which it moves radially values that we can then replace in the classic formula of force equals mass times acceleration thus if we have a system in which we know the value of the mass the velocity perpendicular to the axis of rotation and the applied force we can calculate its angular velocity an example of a device with such characteristics would look something like this in this configuration a mass is forced to oscillate with a frequency of several kilohertz because of this when the system has been rotated the oscillating mass will experience a coriolis force that will move it to the left or to the right depending on the direction of the vibration and similar to accelerometers this displacement in turn can be used to calculate the force experienced by the mass so we would have all the necessary elements to calculate the angular velocity of the system while the characteristics of these gyroscopes make them ideal for a large number of applications they have a disadvantage although they are designed to measure angular velocities linear accelerations will also exert a force on the oscillating mass and therefore if the system is exposed to large accelerations their accuracy would be compromised but fortunately this is where the third type of gyroscope we will discuss comes in optical gyroscopes which can detect angular velocities completely independently of the linear accelerations of the system these types of gyroscopes work on the basis of the sagnac effect let's consider a ring composed of fiber optics and suppose that two beams of light generated by a laser propagate in opposite directions inside the ring if the system is static both beams will travel around the perimeter of the ring in the same amount of time however if the system is rotating this will no longer be the case the beam emitted in the same direction as the rotation of the system will have to travel a longer distance before reaching the end of the path since the end point will basically be moving away from it on the contrary the beam traveling in the opposite direction of the rotation of the system will travel a shorter distance because the end point will be getting closer to it this difference in the distance traveled by the light beams is the key to calculating the angular velocity of the system at this point you may be wondering if a variation in travel time was already generated with each beam individually then why do we need two beams to calculate the angular velocity the reason is that since we are dealing with the speed of light which remember is approximately 300 000 kilometers per second it would be extremely difficult to make a system precise enough to accurately measure the time from the time that light is emitted until it reaches the end point since this occurs virtually instantaneously since light is an electromagnetic wave with a certain frequency and wavelength by having two beams of light traveling in opposite directions they interfere with each other generating a resulting beam with new characteristics the characteristics of this new beam are related to the phase difference between the beams that produced it and therefore to the difference in distances traveled ultimately allowing us to calculate the angular velocity this type of gyroscope is not only highly accurate but also quite reliable because unlike the previous ones it is able to operate completely without moving parts moreover despite the fact that in the beginning they used to be of a large size due to their technical requirements nowadays this is no longer a limitation since in 2018 scientists at caltech were able to build an optical gyroscope of just two millimeters square i hope you liked this video remember to subscribe and if you think what i do is worthwhile you can also support me on patreon to make more and better videos that's all for now and see you in the next video