OBJECT: To verify the relation C = I.ω.ωp for gyroscope.
THEORY:
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, e.g. locomotive, Automobile and
aeroplane taking 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.
Z
OX-Axis of Spin
OY-Axis of Couple
OZ-Axis of Precession
Y
X
O
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 about axis OX in a clockwise
direction viewing from front. Therefore, the angular momentum of disc is I. 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 δθ about axis OZ and in the plane
XOY, from OX to OX` in time δt. 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 δt.
Change of angular momentum OX − ⃗
= ⃗ OX =⃗
XX
Angular Displacement XX
or rate of change of angular momentum = =
Time δt
But rate of change of angular momentum = Couple applied, C
where, XX` = OX x δθ in direction of XX` = (I.ω).δθ
δθ
C = I .ω.
δt
and in the limit, when δt is very small,
dθ
C = I .ω.
dt
Let dθ / dt = ωp, the angular velocity of precession of yoke, which is uniform and is about
axis OZ.
Thus, we get C = I.ω.ωp
The direction of the couple applied on the body is clockwise when looking in the direction
XX` and in the limit this is perpendicular to the axis of ω and ωp.
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.
EXPERIMENTAL SET-UP: Schematic arrangement of the gyroscope is shown in figure.
The motorized gyroscope consists of a disc rotor mounted on a horizontal shaft rotates about
XX axis in two ball bearings of one frame number-2. This frame can swing about YY axis in
bearings provided in the yoke type frame. The rotor shaft is coupled to a motor mounted on a
trunnion frame having bearings in a yoke frame, which is free to rotate about vertical axis ZZ.
Thus freedom of rotation about three perpendicular axes is given to the rotor (or the disc be
rotated about three perpendicular axis). Angular scale and pointer fitted to frame helps to
measure precession rate. In steady position, frame number-1 is balanced by providing a
weight pan on the opposite side of the motor.
EXPERIMENTAL PROCEDURE:
Step-1: Balance the rotor position on the horizontal frame.
Step-2: Start the motor by increasing the voltage with the autotransformer and wait till the
disc attains constant speed. Note down the speed.
Step-3: Put weight (0.5 kg, 1kg or 2 kg) in the weight pan, and start the stop watch to note the
time in seconds required for precession through 30° or 45° etc.
Step-4: The vertical yoke precesses about OZ axis.
Step-5: Speed may be varied by the autotransformer provided on the control panel.
OBSERVATION: Given data of experiment
Weight of Rotor, kg = 6.7
Rotor Diameter, mm = 300
Rotor Thickness, mm = 10.5
Moment of inertia of the disc, coupling and motor rotor
2 W D2
about central axis, I, kg cm sec = x = 0.768
g 8
Distance of bolt of weight pan from disc centre, L, cm = 17.6
OBSERVATION TABLE:
Time required for
Weight, Speed, Angle of precession,
S. No. precession,
W (kg) N (rpm) dθ (degree)
dt (sec)
CALCULATION:
Angular velocity of disc in rad/sec, ω =
Angular velocity of precession of yoke in rad/sec, ωp =
Gyroscopic Torque in kg cm, C = I.ω. ωp =
Cactual = W x L = 0.5 x 17.6 = 8.8 kg cm
Outcome: The gyroscopic relation C = I.ω. ωp is verified.
The calculated value of gyroscopic couple is = .............
The actual value of gyroscopic couple is = .............
Comments: We observed time and angular displacement manually that may be the cause of
the difference observed between the calculated and actual values of gyroscopic couple.
PRECAUTIONS:
1. ωp is to be calculated for short duration of time, as the balance of rotation of disc about
the horizontal axis YY due to application of torque, because of which ω p goes on
reducing gradually.
2. Avoid using the tachometer while taking the reading of time as it will reduce the time
taken for precession.
3. Autotransformer should be varied gradually.
