Torsion Pendulum Test on Metal Wire

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Modulus of Rigidity is the coefficient of elasticity of wire for a shearing force. In simple words, rigidity modulus of a metal wire is a measurement of the capability of a material to resist deformation when external tangential (parallel to the surface) force is applied to the metal wire.

Aim of the Test

To determine the rigidity modulus of the suspension wire using torsion pendulum.

Apparatus Required

1. Torsion Pendulum
2. Cylindrical wire
3. Stopwatch
4. Vernier Caliper
5. Screw gauge
6. Meter scale
7. Test Specimens - Steel and Brass Wire.

Torsion Pendulum Principle

For small oscillations of the disc, it is in simple harmonic motion and the formula for simple pendulum holds good.

Where,
T = period of oscillation in sec.
I = Mass moment of inertia of the rotating system about the longitudinal axis of wire.
L = Length of the wire between its grips.
N = Modulus of rigidity (Shear modulus).

d = diameter of the given wire in the test

Equations 1 and 2 refer to conditions when no cylindrical weight is added on to the disc and when known cylindrical weights are added,
We have,

From which it follows-

(I₂ -I₁) is the mass moment of inertia of the cylindrical weights about the axis of rotation of the disc. It is given by,

Where,
W = Total weight of cylinders added to the disc.
g = Acceleration due to gravity.
r = Radius of cylindrical weights.
R = distance from center of the cylinder of the cylindrical weight to center of the wire.

Thus from equations (5) and (6)

Test Procedure

Part 1: Determination of Rigidity modulus using Torsion pendulum alone

1. The radius of the suspension wire is measured using a screw gauge.
2. The length of the suspension wire is adjusted to suitable values.
3. The wire for the test is tightened at its bottom to the disc and its top to the bracket.
4. The disc is turned and released without the cylindrical weights on it.
5. Time for a number oscillation (Say 20) is measured with a stopwatch and the mean period of oscillation ' T₀' is determined.

Part 2:  Determination of rigidity modulus and moment of inertia using torsion pendulum with identical masses

1. The two identical masses are placed symmetrically on either side of the suspension wire as close as possible to the center of the disc and and the distance d1 is measured which is the distance between the centers of the disc and one of the identical masses.
2. Find the time for 20 oscillations twice and determine the mean period of oscillation T₁.
3. The two identical masses are placed symmetrically on either side of the suspension wire as far as possible to the center of the disc and measure d₂ which is the distance between the centers of the disc and one of the identical masses.
4. Find the time for 20 oscillations twice and determine the mean period of oscillation T₂.
5. Find the moment of inertia of the disc and rigidity modulus of the suspension wire using the given formulae.

Observation and Calculation

For Part 1 -

Rigidity modulus of the suspension wire =

For Part 2 -

Rigidity modulus of the suspension wire =

Applications of Torsional Pendulum

1. The working of "Torsion pendulum clocks" (shortly torsion clocks or pendulum clocks), is based on torsional oscillation.
2. The freely decaying oscillation of  Torsion pendulum in medium(like polymers), helps to determine their characteristic properties.
3. New researches promising the determination of frictional forces between solid surfaces and flowing liquid environments using forced torsion pendulums.