Determination of Rigidity Modulus of Material

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BACKGROUND

The Modulus of Rigidity of a material is the relation between shear stress and shear strain of a particular substance and defined as the ratio of shear stress to displacement per unit of sample length (shear strain).¹ It is also known as Shear Rigidity, and it is valid in material only till it shows the elastic limit like it return to its original position after unloading the load.

Similar to the modulus of elasticity, modulus of rigidity is also generalized for Hooke’s Law. This property of a specimen depends upon the material. For example, cast steel possess a modulus of rigidity of 78 GPa, while concrete and Aluminium have the modulus of rigidity of 21 Gpa and 28 Gpa respectively.²

• Unit of Modulus of rigidity = G

• S.I unit of Modulus of rigidity = N/m²

• C.G.s unit of Modulus of rigidity = dyne/cm²

• Dimension of Modulus of rigidity = ML^-1T^-2

In this experiment, a metal wire fixed at one end with an attached mass rotates torsionally in SHM with a period T depending on cross-sectional shape and dimensions, the rigidity of wire, Moment of Inertia I of an attached block, length L of wire.³ The shear modulus describes the material’s response to shear stress.

The central objective of this experiment is to determine the “Modulus of Rigidity” of a material.

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REQUIREMENTS

Materials:   A metal wire of length about 90 cm

A metal block fixed at the end of the metal wire

Ruler 100 cm long

Vernier scale

Micrometer

Stopwatch

Cylindrical weights

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PROCEDURE

First of all, measure the radius of the wire ‘a’ using a micrometer. Now measure the radius of the cylindrical block, ‘R’ using a vernier scale. Now, note down the mass of the metal block. After this, fix the wire in such a manner so that the free length L be 45 cm. To measure the period of oscillations, pull the attached mass slightly to rotate in a torsional motion. Be cautious, to keep the small amplitude of vibration. Otherwise, the SHM approximation s no longer valid. Use a stopwatch to measure the time it will take to complete 20 periods of oscillation. After dividing the recorded time by 20, an accurate measure for the time period, T is calculated in seconds. If the vibrating wire does not complete 20 cycles, you can also measure ten periods to calculate an approximate period of oscillation. Now, repeat the previous steps using wires of different lengths (55 cm, 65 cm, 75 cm, 85 cm) and find out the modulus of rigidity by using formula {(G= 8πIL)/T2×R4}.

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CONCLUSION

The process of finding the time period of oscillations and Modulus of Rigidity has been discussed. The Modulus of Rigidity varies from one material to another.

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REFERENCES

1. R.Hill, “Theory of mechanical properties of fibre-strengthened materials—III. self-consistent model,” Journal of the Mechanics and Physics of Solids, vol. 13, no. 4, 1965.
   
2. R. K. Hiroshi Hoshikawaa, “Elastic properties of bacterial flagellar filaments: II. Determination of the modulus of rigidity,” Biophysical Chemistry, vol. 22, no. 3, pp. 62-78, 1985.
   
3. Francis Birch, “The Effect of Pressure Upon the Elastic Parameters of Isotropic Solids, According to Murnaghan’s Theory of Finite Strain,” Journal of Applied Physics, vol. 9, no. 4, pp. 53-65, 2004.