UG Physics Lab

Aim :

To find the moment of inertia of gyroscope by measuring the precession frequency as a function of spin frequency of gyroscope.

Theory :

Gyroscopes are used in compasses, in the steering mechanism of torpedoes and in inertial guidance systems. The objective is to find the moment of inertia of the gyroscope by measuring the precession frequency, as a function of the spin frequency of the gyroscope. The gyroscope that is free to rotate about all the three axes is balanced in horizontal position with the help of a counter weight C as shown in Fig. 7.1. As soon as a small weight is added on the left hand side of counter weight C, the gyroscope destabilizes and falls down. Remove the extra small weight, balance it again as before and spin the disk of the gyroscope with some angular velocity. Now hang the small weight again on the left hand side as before. The gyroscope now shows a completely new behavior and starts rotating in a direction perpendicular to the previous plane. This movement is known as precession. That is how a gyroscope increases the stability of a system. Try to explain this behavior using laws of mechanics. Can you cite some example from our everyday life where you see actual demonstration of this gyroscopic phenomenon?

If I is the moment of inertia of the gyroscope about its symmetric axis, the angular momentum L is given by,

(6.1)

L = Iωg

where ωg is the angular velocity of the spinning gyroscope. Now, the addition of an additional weight m, at a distance r from the pivot point P, introduces a supplementary torque T

(6.2)

The gyroscope now starts precising with frequency ωp under the influence of T . Since T is perpendicular to L its effect is to change the direction of L. In a time dt, L will rotate by

Figure 6.1: Gyroscope experimental setup

dΦ, given by

(6.3)

(6.4)

where we have used Eqns. 6.1, 6.2 and 6.3. If Tp is the time for one precession revolution and T is the time taken by the gyroscope to spin about its axis (one rotation) then

(6.5)

(6.6)

Therefore from Eqn. 6.4,

(6.7)

Thus a plot of 1/Tg vs Tp should yield a straight line for a fixed m, from which the moment of inertia I of the gyroscope, can be obtained.

Apparatus Required :

Gyroscope, tachometer, stopwatch, weights.

Procedure :

Balance the gyroscope horizontally, using the counterweight C, without any weight m.

1. Give a spin to the horizontal balanced gyroscope and measure the time (Tg) required to complete one revolution using the given light barrier counter.

2.Immediately after this, hand a mass m into the groove at the longer end of the gyroscope. This is at a distance r = 27cm. The gyroscope will precess. Using the stop watch, measure the duration of half the rotation Tp/2.

3.Without any delay, remove the mass m, so that gyroscope stops processing, and measure Tg again, using the light barrier counter.

4.The average of Tg measured in steps (1) and (3) above is to be used in Eq. (6.7).

5.Repeat for several different initial spins of the gyroscope and fixed m and plot 1/Tg vs Tp and find the slope. Find I using Eq. (6.7).

6. Find I for another value of m.

Observation :

For mass m = ...... gm.

(6.8)

where, ωgr = (ωg1 + ωg2)/2

(6.9)

ωg = 2π(ωgr/60)

(6.10)

Tg = 2π/ωg

(6.10)

Repeat your observations for another mass.

Calculation :

Plot a graph of 1/Tg vs Tp for both masses on the same graph paper using the same axes. The graphs should be a straight line passing through origin. Find the slope (s) and calculate the moment of inertia of the gyroscope from the obtained slope values.

(6.11)

I = mgr/4π2s

Results :

Moment of inertia of the gyroscope

1.For mass m1 = ......... kg m2.

2. For mass m2 = ........ kg m2.

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-Check Yourself-

1. The general equation that relates the torque to the rate of change of angular momentum is:

てdt=dL
てdL=dt
て=dt.dL
None of the above

Correct Answer is A

2. In torque-free precession

Angular momentum and angular velocity is constant
Angular velocity is constant and angular momentum vector changes orientation with time
Angular momentum is constant and angular velocity vector changes orientation with time
Both vector changes orientation with time

Correct Answer is C

3. The angular velocity of the precession is given by:

Ω =Mgω/Ir
Ω =dt/dӨ
Ω =Mgr/L
Ω =Mg/Iω

Correct Answer is C

4. Uses of Gyroscopes

Navigation
Satellites
Racing cars
All of the above

Correct Answer is D

5. A student sits on a spinning stool with her arms folded. When she extends her arms away from her body, her kinetic energy

Increases
Decreases
Stays the same
Depends on her mass

Correct Answer is B

6. Gyroscopic effect is not observed in which of the following actions performed by the ships?

Rolling
Pitching
Steering
All of the above

Correct Answer is A

7. What is the effect of reactive gyroscopic couple when an aeroplane takes a right turn and propeller rotates in anticlockwise direction?

The nose and tail of the aeroplane are in horizontal plane
The nose of the aeroplane is raised and tail is dipped
The nose of the aeroplane is dipped and tail is raised
None of the above

Correct Answer is C

8. The axis of precession is ____________ to the plane in which the axis of spin is going to rotate

parallel
Perpendicular
spiral
none of the mentioned

Correct Answer is B

Gyroscope

Aim :

Theory :

Apparatus Required :

Procedure :

Observation :

Calculation :

Results :

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