Experiment: (a) to find the radius of gyration of objects of different geometrical
shapes but of same mass by noting the period of oscillation.
(b) To show that the time period is independent of mass so long as the shape and size
remain the same.
Apparatus used: one thin rectangular plate of same mass, an annular disc of same
mass, one circular disc having the same mass as the other two objects, two knife edges,
stop watch and telescope.
Theory:
The moment of inertia of any object is given by the formula
I = MK2
Where I = moment of inertia
M = mass of object
K = radius of gyration
Procedure:
a) Note the position of C.G. of rectangular plate, annular disc and the circular disc.
See that the centre of hole in each case is situated at the same distance l from the
C.G.
b) Check the mass of rectangular plate, the annular disc and one circular disc either
by a sensitive physical balance or by a spring balance and see that each of them
has the same mass.
c) Fix the knife edges into the two holes respectively in rectangular plate and draw a
reference line or mark at the lower edge.
d) Suspend the plate on two smooth glass plates with the help of the knife edges.
Focus the reference vertical mark through the telescope placed at a distance about
two meters.
e) Set the plate into uniform vibrations of small amplitude and with the help of a
stop watch find the time taken for 20 vibrations.
f) Invert the plate and fix it at B. again find the time period for 20 vibrations.
g) Similarly find the time period for 20 vibrations for the annular disc and the
circular disc.
Observation:
Acceleration due to gravity = 9.8 ms-2
Distance of hole from C.G.
Rectangular plate (i) hole A, l = 0.047 m (ii) hole B, l = 0.045 m
Annular disc (i) hole A, l = 0.053 m (ii) hole B, l = 0.047 m
Circular disc (i) hole A, l = 0.046 m (ii) hole B, l = 0.047 m
Objects Time for 20 vibrations (in Time T2 Mass (in
s) period Kg)
Hole A Hole B mean T (in
1 2 1 2 s)
Rectangular 24 23 24 23 23.5 1.175 1.38 0.260
plate
Annular 20 19 20 19 19.5 0.975 0.950 0.260
disc
Circular 19 19 20 19 19.25 0.9625 0.926 0.260
disc
Calculations and verifications
(i) rectangular plates
Length of rectangular disc (a) = 0.438 m
Breadth of rectangular plate (b) = 0.140m
Moment of inertia I = M (a2 + b2)
12
Or K2 = a2 + b2
12
= 0.04 m2
» K = 0.2 m
(ii) annular disc
Internal radius of annular disc (r) = 0.041 m
External disc of annular disc (R) = 0.145 m
Moment of inertia I = M (R2 + r2)
2
Or K = R + r2
2 2
2
= 0.0116 m2
» K = 0.11 m
(iii) circular disc
Radius of circular disc (R) = 0.139 m
Moment of inertia I = MR2
2
Or K2 = R2
2
= 0.0097 m2
» K = 0.01 m
Conclusion
1. As the time period is different of each body, the moment of inertia does not only
depend upon mass as other wise it would have been equal.
2. As K is different in all the cases and M.I. depends upon K being equal to MK2, we
conclude that the M.I. of an object depends upon the distribution of mass of the
body about the axis of rotation i.e. on its shape and size.
Precautions
1. The knife edges should be sharp and the point of suspension should not slip.
2. The amplitude should be small.
3. The time period should be calculated by observing the accurately the time of 20
vibrations.
4. The telescope should be focused from a distance of about 2 meters.
Sources of error
1. There is always some air resistance to the motion, which is different shape of the
oscillating objects.
2. The masses of the oscillating objects may not be exactly equal.
3. The distance of the point of the oscillation from the C.G. of each object may not
be exactly the same.
