PH8151 Engineering Physics Chapter 1

24 CHAPTER 1. PROPERTIES OF MATTER

Squaring,

T 02 =4fi^2

I

c

(1.18)

Substitute forcfrom equation (1.14):

T 02 =4fi^2

1 I

finr^4

2 l

(^2) =^8 fiIl
nr^4
Therefore, Rigidity Modulus can be written as,
n=
8 fiIl
T 02 r^4
(1.19)
The moment of inertia of a circular disc of massM and radiusR, about an axis passing
through its centre and perpendicular to the circular face, is
I=
1
2
MR^2 (1.20)
Substituting from equation (1.20) into (1.19) we obtain:
n=
4 filMR^2
T 02 r^4
(1.21)
Hence by experimentally measuring the length (l) and radius (r) of suspension wire ,
mass (M) and radius (R) of the suspended disc and the period of oscillation (T 0 ) of the
pendulum, the Rigidity Modulus of the wire can be obtained from (1.21).
Note: A detailed discussion of the theory of simple harmonic motion, of which torsional
oscillation is an example, can be found in Chapter 2 - Waves and Fiber Optics.
Rigidity Modulus - Determination by Dynamic Torsion Method
It is possible to obtain an expression for the Rigidity Modulus of the suspension wire
which does not explicitly require the expression (1.20) for the moment of inertia of the
disc. For this we adopt an experimental procedure known as dynamic torsion method
which is explained in the following.
The periods of oscillation of the given torsion pendulum are determined in three dier-
ent configurations as shown in Figure3.29by adding extra weights (usually of cylindrical
shape) to the suspended disc. In configuration (i), the period of oscillationT 0 is measured
for the bare disc with no extra masses. In configuration (ii), the period of oscillationT 1
is measured with two identical bodies (each having massm) placed symmetrically on
the disc at distancesd 1 from the suspension wire. In configuration (iii), the period of
oscillationT 2 is measured with the identical bodies placed symmetrically on the disc at
distancesd 2 from the suspension wire (withd 2 >d 1 ).
As mentioned before,I = the moment of inertia of the suspended disc about an axis
passing through its centre and perpendicular to the circular face (i.e., about the axis co-
inciding with the suspension wire).
LetI 1 = the moment of inertia of the configuration-ii
andI 2 = the moment of inertia of the configuration-iiiabout the axis coinciding with the
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