Thursday, December 7, 2023

MOTORISED GYROSCOPE

 

1.0     THEORY:

             


 
(A)      DEFINITIONS:

 (a)       Axis of Spin:

 

If a body is revolving about an axis, the latter is known as axis of spin.

 

(b)       Gyroscopic Effect:

 

To a body revolving (or spinning) about an axis say OX, if a couple represented by a vector OY perpendicular to OX is applied, then the body tries to precess about an axis OZ which is perpendicular both to OX and OY.  Thus the plane of spin, plane of precession and plane of gyroscopic couple are mutually perpendicular. The above combined effect is known as precessional or gyroscopic effect.

 

(c)       Precession:

 

Precession means the rotation about the third axis OZ, which is perpendicular to both the axis of spin OX and that of couple OY.

 

(d)       Axis of Precession:

 

The third axis OZ is perpendicular to both the axis of spin OX and that of couple OY is known as axis of precession.

 

(e)       Gyroscope:

 

Gyroscope is a body while spinning about an axis is free to rotate in other directions under the action of external forces. For example locomotive, automobile and aero plane making a turn. In certain cases the gyroscope forces are undesirable whereas in other cases the gyroscopic effect may be utilized in developing desirable forces. For minimizing rolling, yawing and pitching of ship or air-craft Gyroscope is used. Balloons use Gyroscope for controlling direction.

 

(B)       GYROSCOPIC COUPLE OF A PLANE DISC:

 




Let a disc of weight W and having a moment of inertia I be spinning with an angular velocity w about axis OX in an anti clockwise direction viewing from front. Therefore, the angular momentum of disc is Iw. Applying right hand screw rule, the sense of vector representing the angular momentum of disc which is also a vector quantity will be in the direction OX as shown. A couple, whose axis is OY perpendicular to OX and is in the plane XOZ, is now applied to precess the axis OX.

 

Let axis OX turn through a small angular displacement dq about axis OZ and in the plane XOY, from OX to OX’ in time dt. The couple applied produces a change in the direction of angular velocity, the magnitude remaining constant. This change is due to the velocity of precession. Therefore, ‘OX’ represents the angular momentum after time dt.

 

\ Change of angular momentum = OX’ – OX = XX’

 

                                                                                    Angular Displacement

            or rate of change of angular momentum =   ¾¾¾¾¾¾¾¾¾¾

                                                                                                Time

                                                                                   

            XX’

                                                    =   ¾¾¾  

                                                                                     dt       

But rate of change of angular momentum = Couple applied, C

 

            Where, XX’ = OX x dq in direction of XX’

                                = (Iw)dq

 

                           dq

            \ C= Iw ¾    

                           dt

 

            and in the limit, when dt is very small,

 

           dq

            C= Iw ¾        

                       dt

 

Let dq / dt = wp, the angular velocity of precession of yoke, which is uniform and is about axis OZ.

 

            Thus, we get   C = Iw.wp

 

The direction of the couple applied on the body is anticlockwise when looking in the direction XX’ and in the limit this is perpendicular to the axis of w and wp.

 

In the supplied apparatus, the reaction couple exerted by the body on its frame is equal in magnitude to that C, but opposite in direction.

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