# Young's Modulus

When materials are stretched or squeezed, they can resist changes in their length. This is called Young's Modulus or elastic modulus. Engineers use this measurement to understand how stretchy and stiff materials are. It can also tell them how strong a material is when it's pulled in different directions. So, if you're into engineering, knowing about Young's Modulus is pretty important!

## Defining Young's modulus

Young's Modulus is a measurement used to understand how stretchy and stiff materials are. It's calculated by dividing the longitudinal stress the strain. When you apply tension to a metal object, it will stretch and its cross-sectional area will decrease. The stress is measured in newtons and the cross-sectional area is measured in square meters. The resulting unit of stress is N/m². The strain is the change in length divided by the original length L0. It's measured in meters and is dimensionless. You can calculate the strain using a formula.

## Young's modulus formula

Young's modulus, or elastic modulus, is a measure of how stretchy and stiff materials are. It's calculated by dividing stress by strain. The units of Young's modulus are the same as stress, which is measured in N/m² or Pa. Since the elastic modulus is usually a very large number, it's often expressed in Giga Pascals (GPa). To find the elastic modulus of a metal bar, we need to find the stress and strain first. If a metal bar with a cross-sectional area of 0.02 mm² experiences a load of 60 N at the end and increases in length by 0.30%, we can calculate the stress and strain using formulas. Then we can divide stress by strain to find the Young's modulus.

### How is Young's modulus related to Hooke's Law?

Hooke's law is a principle that states the force acting on a body or spring is linearly proportional to the displacement created, as shown in the equation F = kΔx, where k is a constant that relates to the spring's stiffness. This law applies to situations where a body deforms elastically. Similarly, the stress applied to a body is linearly proportional to Young's modulus E, which is expressed in the equation σ = Eε, where σ is the stress, ε is the strain, and E is the elastic modulus. This relationship is similar to Hooke's law, where the extension or compression of a spring is linearly proportional to the force applied.

## Graphically calculate Young's modulus

Young's modulus, or the elastic modulus, can also be calculated from a stress-strain graph. The stress-strain graph plots the stress on the y-axis and the strain on the x-axis. If the material is following Hooke's law, the stress-strain graph will produce a linear relationship between stress and strain. In this linear region, the slope of the graph represents the Young's modulus. Therefore, we can calculate the Young's modulus by finding the slope of the linear region of the stress-strain graph. This method provides an alternative way to determine the elastic modulus of a material, especially when direct measurements of stress and strain are challenging or not feasible.

Young's modulus calculation from stress-strain graph

## Experiments to show Young's modulus

Yes, that's correct. To measure the Young's modulus of a metal, several experiments can be conducted. One such experiment involves applying varying loads to a copper wire and measuring the resulting extension, which can then be used to construct a stress-strain graph. To conduct this experiment, the following equipment would be required:

1. Copper wire
2. Micrometer to measure the diameter of the wire
3. Pulley
4. Meter ruler or tape to measure the length of the wire
5. Caliper to measure the diameter of the weights
6. Weights of different masses
7. Clamp to hold the wire in place
8. Wooden block to support the pulley
9. Bench to hold the apparatus in place

By carefully applying loads and measuring the resulting extension, the stress and strain equations can be used to determine the Young's modulus of the metal. This experiment is a common method used to measure the elastic modulus of materials in engineering and material science.

### Methodology

To measure the Young's modulus of a metal using the experiment I described earlier, we would follow the steps you mentioned.

First, we would measure the initial length of the wire using a ruler. Then, we would use a micrometer to measure the diameter of the wire at three points along its length to calculate the average diameter. This is important because the diameter of the wire affects the stress calculation.

Next, we would connect one end of the wire to a pulley that is clamped to a bench and the other end to a clamped wooden block. We would then add weights of different masses to the wire and measure the resulting extension using a ruler or a meter tape. It is recommended to take 5 to 10 measurements with various weights to reduce errors.

Using the difference between the new length and the original length prior to extension, we would calculate the strain. We would then use the stress and strain equations to calculate the stress and plot stress vs strain. The elastic modulus can be found by taking the gradient of the linear region of the stress-strain graph.

Overall, this experiment is a reliable method to determine the Young's modulus of a metal and is commonly used in material science and engineering.

### Results analysis

The aim of the experiment is to estimate the Young's modulus of a material, which is its ability to resist changes in length under tension or compression. To find stress and strain, we would need to calculate the area of the wire using its diameter and the equation A=πr^2, where r is the radius of the wire.

We would then rearrange the Young's modulus formula to solve for force, which would give us F=((E⋅A)/L)⋅ΔL. By graphing the force vs. extension points and finding the slope of the linear region, we can estimate the Young's modulus of the material.

It's also important to note the characteristics of materials shown in the stress-strain graph. The red region indicates the elastic region where the material follows Hooke's law and stress and strain are proportional. The red point indicates the elastic limit or yield point, which is the point to which a material can still hold its original length after the load is applied.

The green region indicates the plastic region where the material is unable to return to its initial state and has undergone permanent deformation. The green point indicates the tensile strength point, until which the material can withstand the maximum load per unit without breaking.

Finally, the blue point indicates the breaking point or breaking stress, where the material will break. By examining a material's stress-strain graph, we can also determine its tensile strength and breaking point.

## Young's Modulus

How do you determine Young's modulus experimentally?

You can determine Young's modulus experimentally by applying a load to a material, then creating a load vs change in length graph, and calculating the slope of the graph under study.

What is Young's modulus?

Young’s modulus is the ability of materials to retain their original length/shape under load.

Why does aluminium have a lower Young's modulus than steel?

Because steel is stiffer than aluminium, hence it will be more likely to retain its shape under load.

What does a higher Young's modulus indicate?

A higher Young's modulus indicates that the material is stiffer and requires higher force to be deformed.