22 CHAPTER 1. PROPERTIES OF MATTER
1.7.1 Torsion Pendulum - Determination of Rigidity Modulus
of Suspension Wire
Consider the suspension fibre of a torsion pendulum as a thin cylinder of lengthland
radiusrfixed at the top end and twisted at the free end by an angle◊as shown in Figure
1.21. Due to the elasticity of the wire, an internal restoring force is developed in the wire
P x
Q
A
B
C
φ
θ
dx
r
l
Fixed End
Free End
(twisted)
Figure 1.21: Couple per unit twist
and it is said to be in torsion.
The Rigidity Modulus of the wire (n) = (shear stress) / (shear strain).
Hence, shear stress,=n◊(shear strain) (1.7)
We first consider an imaginary cylindrical shell of outer radiusxand thickness dxas
shown in Figure1.21and calculate the shear stress at this shell. A vertical line AB drawn
on the shell get shifted to the position AC due to the torsional deformation, resulting in
an angle of twist\BQC=◊ and an angle of shear\BAC=„.Itcanbeseenfrom
Figure1.21thatBC=l„=x◊. Therefore,
shear strain,„=
x◊
l
(1.8)
Therefore from equations (1.7) and (1.8), at the imaginary shell,
shear stress,=n◊
x◊
l