Gyroscope: A gyroscope consists of a mass in the form of a rotor or wheel which suspended in such a way that it is free to spin about an axis passing through its centre of mass and perpendicular to the plane of the rotor. Thus, it is a mass in the shape of wheel or so, symmetrical about an axis & can freely rotate about that axis. If such rotor is so suspended that it is free to point its spin axis in any direction. This is achieved as follows:
1. Freedom to spin about its own axis.
2. Freedom to tilt about a horizontal axis.
3. Freedom to turn in azimuth about a vertical axis.
Properties of a free gyroscope
A free gyroscope has two properties that are made use of, when constructing a gyro compass.
Gyroscope inertia: A free gyroscope, once set spinning will always remain pointed in the same initial direction relative to a point in space i.e. the spin axis will point in a constant direction with respect to space regardless of how the gyro support system has been moved.
The spin axis can therefore be considered to be continuously pointing at a fixed star in the celestial sphere. In fact both the axles if extended till celestial sphere would meet some set of stars. Now imagine the rotor, axles with stars are fixed and only the earth is spinning. It is this property which is known as gyroscopic inertia or rigidity in space. This causes the relative motion of axle wrt.
1. Meridians
2. Horizon.
In the relative motion as witnessed by an observer, the axle will follow the movement of that star on the celestial sphere.
In linear motion, momentum is the product of mass and the velocity of the object. Any object moving with mass possesses momentum. In case of rotating or spinning objects however, the angular momentum comes into play. So we may say that the angular momentum is the rotatory equivalent of linear momentum?
Let us try and understand rigidity of a free gyroscope with a simple day to day observation. A person if tries to sit on a bicycle and tries to balance without the stand he will fall. But once he starts pedaling, wheels pick up angular momentum. The axes of wheels will resist change, thereby the rider is able to balance.
The angular momentum of a particle spinning about an external axis is the product of its linear momentum and the perpendicular distance of the particle from the axle is given by the expression: angular momentum = mv x r or linear momentum x radius of turn. or mr2, where ω is the angular velocity and is equal to v/r. Thus, the earth revolving round the sun is an example.
Angular momentum is defined as the property of any rotating object given by moment of inertia times angular velocity. The SI unit of angular momentum is kg.m².s-¹.
When an ice-skater goes for a spin, her spinning speed reduces when she stretches her hands. To increase his/her spinning speed he/ she brings hands closer. Thus, by reducing the radius there is increase of angular velocity.
In case of a gyro rotor or the earth sinning about own axis, the amount of inertia is dependent on:
1. mass of the wheel / earth;
2. the distribution of mass around spin axis; and
3. the angular velocity of the rotor (faster the spin greater the inertia).
The angular momentum is related to moment of inertia as follows:
H = IΩ
H = angular momentum.
ω = angular velocity.
I = moment of inertia about the spin axis.
Conversely it can be stated that gyroscopic inertia depends upon the momentum of the spinning rotor. The angular momentum is now proportional to mr2. If one or more of these is changed, the rotor’s inertia will change. Thus, to get best results as to stability, the rotor mass is kept on the higher side with most of it concentrated at its outer edge. Normally, the radius also is kept large. The rotor must be perfectly balanced and friction at pivots must be as low as possible. Thus, the mass as well as radius is constant. To have a steady gyroscopic inertia the speed of the rotor must be constant and accordingly controlled.
Tilt and tilting: the spin axis of a free gyro is said to be tilting when it moves in a vertical plane either in an upward direction (when east of meridian) or in a downward direction (when west of meridian). The tilt is equivalent of altitude whereas rate of change of altitude refers to ‘tilting’. The movement is similar to a star changing its altitude. The tilting or rate of change of altitude we know is zero on meridian. Any star, weather in lower meridian area or upper meridian area, has upward tilting (Tg) while it is to the east of meridian and downward while it is to the west of meridian. This rule will follow in all the drawings, showing the trace of axle. Since, rate of tilting per hour = 15o Sin azimuth x Cos latitude, the rate of tilting in a given latitude increases with the azimuth and in a trace the Tg is maximum on the east or west extremity.
The angle by which, the spin axis is elevated above or below the horizontal is the tilt. The maximum angle of tilt occurs when the spin axis is in the meridian, just as the maximum altitude of a star occurs when it is on the meridian. Minimum altitude occurs at lower meridian passage.
Azimuth and Drifting: The spin axis is said to be drifting when it changes its azimuth, which means it turns about a vertical axis. Rate of drifting means the rate at which, the azimuth is changing at any given time. Since, rate of drifting per hour = 15o Cos altitude x Sin latitude, the drifting is maximum at poles. In an ellipse or a spiral trace the maximum tilt is of just a small value only, hence Cos altitude is taken as unity. This means that for a given latitude the rate of drifting should be shown constant throughout the trace. In the trace, drift refers to the angular distance from the meridian to the east or west and is equivalent to the azimuth.
Direction of drifting: Drifting means change of azimuth of the north axle. It the observer’s RH diagram, drawn for north latitude, the north axle follows a star on the inferior meridian. This star moves from west of observer to east. Thus, the direction of movement is eastwards. Due to this the drift of north axle is eastwards. The, axle follows star on the upper meridian in south latitude, hence the north axle drifts westwards.
Gyroscope Precession: Precession causes axle of a free gyroscope to move when a torque is applied. Torque is a force or a couple of force applied at a distance from the reference axis.
Precession will always caused in a direction right angles to the plane of torque. If a torque is applied about the horizontal axis trying to move the spin axis up or down the gyro spin axis will precess sideways i.e. about the vertical axis. If a torque is applied about the vertical axis the gyro will precess up or down i.e. about a horizontal axis.
Rate of precession of the gyro axle is proportional to the applied torque. It is inversely proportional to the gyroscopic inertia of the rotor and hence to the angular momentum possessed by the rotor.
Direction of precession depends on
1. Direction of applied torque.
2. Direction of rotation of wheel (clock wise or anticlockwise).
In the diagram below, at end A a vertically downward force (// to plane of paper) is applied. To find the direction of precession paste a parallel arrow on the disc in radially outward direction. After a 90º turn this arrow will point in a direction that emerges out of plane of paper. This is also the direction in which end A will precess.
Note:
1. If direction of applied force is reversed at A, the direction of precession will also reverse.
2. If rotational direction of disc is reversed the direction of precession will be reversed.
3. By choosing the top heavy / bottom heavy arrangement or clockwise / anticlockwise rotation we can achieve the precession in desired direction.
4. End B will precess in a direction opposite as end A.
Transforming a free gyroscope into a marine gyro compass (North seeking).
Due to the earth rotation the spin axis of a free gyroscope will trace out a circular path as it remains pointed in a fixed direction in space. The requirement of a gyro compass is that the spin axis should point in a fixed direction on earth and the chosen direction is true north.
To convert a free gyroscope into a gyroscope the free gyroscope must be made.
1. North seeking i.e. seek the meridian.
2. North settling i.e. must point in or near the meridian.
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