Youngs modulus by uniform bending

Young’s modulus is named after Thomas Young,19th century ,British scientist. In solid mechanics, Young’s modulus is defines as the ratio of the longitudinal  stress over longitudinal strain, in the range of elasticity  the Hook’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

Youngs modulus by uniform bending

Uniform Bending  Using Pin and Microscope Method

In uniform Bending , the Young’s modulus of the material of the bar is given  by

  (1)

Where,

m – Mass at each end of the bar.

p – Distance between the point of suspension of the mass and nearer knife edge.

g – Acceleration due to gravity.

is the length of the bar between the knife edges.

e – Elevation of the midpoint of the bar for a mass m at each end.

I – Geometrical moment of inertia.

For a bar of rectangular cross section,

 (2)

Where b is the breadth and d is the thickness of the bar.

Substituting (2)  in equation (1)

  (3)

Youngs modulus by uniform bending

 From graph, can be calculated.

Uniform Bending

The bar is placed symmetrically on two knife edges. Two weight hangers are suspended at equal distance from the knife edges. The distance l  between knife edges and distance p of the weight hanger from knife edges are measured. A pin is fixed vertically at the midpoint of the bar with its pointed end upwards. The microscope is arranged in front of the pin and focused at the tip of the pin. The slotted weights are added one by one on both the weight hangers and removed one by one a number of times, so that the bar is brought into an elastic mood. With the some “dead load” W0 on each weight hanger, the microscope is adjusted so that the image of the tip of the pin coincides with the point of intersection of cross wires. The reading of the vernier scale and vernier of microscope are taken. Weights are added one by one and corresponding reading are taken. From these readings, the mean elevation (e) of the mid-point of the bar for a given mass is determined. The value of is calculated . The breadth of the bar (b) is measured by using vernier calipers and thickness of the bar (d) is measured by using screw gauge. Hence calculate the Young’s modulus of the material bar.

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.

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