##

## Aim

To find the Young's modulus of the given material bar by non uniform bending using pin and microscope method.

## Apparatus

Pin and Microscope arrangement, Scale ,Vernier calipers, Screw gauge, Weight hanger, Material bar or rod.

## Theory

YoungÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢s modulus is named after Thomas Young,19th century ,British scientist. In solid mechanics, YoungÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢s modulus is defines as the ratio of the longitudinal stress over longitudinal strain, in the range of elasticity the HookÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢s law holds (stress is directly proportional to strain). It is a measure of stiffness of elastic material.

If a wire of length L and area of cross-section 'a' be stretched by a force F and if a change (increase) of length 'l' is produced, then

## Non Uniform Bending Using Pin and Microscope

Here the given beam(meter scale) is supoorted symmetrically on two knife edges and loaded at its centre. The maximum depression is produced at its centre. Since the load is applied only one point of the beam, the bending is not uniform through out the beam and the bending of the beam is called non- uniform bending.

In non-uniform bending (central loading), the Young's modulus of the material of the bar is given by

(1)

I is the moment of inertia of the bar.

For a rectangular bar**,**

** **

Substituting (4) in (3)

In non uniform bending, the youngÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢s modulus of the material of the bar is given by,

(2) ** **

m - Mass loaded for depression.

g - Acceleration due to gravity.

*l* - Length between knife edges.

b - Breadth of the bar using vernier calipers.

d - Thickness of the bar using screw gauge.

e - Depression of the bar.** **

From graph can be calculated.* *** **

### Applications

1. Thin film applications.

2.It helps to predict the directional and orientation properties of metals and has application in ceramics.

3. Measurement of soft tissues -early detection, elasticity imaging, etc.

4. It is used to test equipments like ultrasonic transducers, ultrasonic sensors.