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## What is Modulus of Rigidity and How to Measure it by Static Method?

Modulus of rigidity, also known as shear modulus, is a property of a solid that measures its resistance to deformation when a force is applied perpendicular to its surface. It is defined as the ratio of shear stress to shear strain within the elastic limit. The SI unit of modulus of rigidity is Pascal or N/m and its dimensional formula is [MLT].

There are different methods to measure the modulus of rigidity of a material, such as dynamic method, torsion method, plate shear method, and panel shear method. In this article, we will focus on the static method using Barton's apparatus. This method involves applying a known torque to a cylindrical rod or wire and measuring the resulting angular twist.

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## Barton's Apparatus

Barton's apparatus consists of a horizontal cylindrical rod or wire fixed at one end and attached to a pulley at the other end. A weight hanger is suspended from the pulley by a thin string. A circular scale is fixed at the center of the rod or wire and a pointer is attached to it. A vernier scale is mounted on a stand near the circular scale.

Figure 1: Barton's apparatus

## Procedure

Measure the length and diameter of the rod or wire using a meter scale and a screw gauge.

Fix one end of the rod or wire to the clamp and attach the other end to the pulley.

Adjust the pointer and the vernier scale such that they are in line with each other.

Add some weights to the weight hanger and note down the reading of the circular scale and the vernier scale.

Repeat step 4 for different values of weights and record the readings in a table.

Calculate the angular twist for each value of weight using the formula:

where Î is the angular twist in radians, C is the reading of the circular scale in degrees, V is the reading of the vernier scale in minutes, and n is the number of divisions on the circular scale.

## Calculation

To calculate the modulus of rigidity, we use the formula:

where G is the modulus of rigidity, T is the torque applied by the weight, L is the length of the rod or wire, J is the polar moment of inertia of the rod or wire, and Î is the angular twist.

The torque applied by the weight can be calculated by:

where W is the weight, r is the radius of the pulley, and g is the acceleration due to gravity.

The polar moment of inertia of the rod or wire can be calculated by:

where d is the diameter of the rod or wire.

Using these formulas, we can plot a graph between T and Î and find the slope of the graph. The slope will give us the value of G.

## Precautions

The rod or wire should be free from any bends or kinks.

The string should be thin and inextensible.

The weights should be added gently without jerks.

The readings should be taken when there is no oscillation in the system.

The zero error of the screw gauge and vernier scale should be noted and corrected 0efd9a6b88

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