Tuesday, December 9, 2014

Mechanical Properties of Solids and Young's Modulus

Elasticity is the property of a material by virtue of which it is able to come back to its original position after removing the external force applied on it. Depending on the nature of the material every body has elastic nature up to some extend. External  force applied on the body tries to change the shape of the body and that’s why it is called as deformation force. 

Within the elastic limit, the molecules of the body  pull themselves back to its original  position due to the  force of attraction between molecules of  the body and it  is called as a restoring force. Within the elastic limit the magnitude of the deformation force shall be equal to the restoring force.

The restoring force is there simply because the molecules of the material are having a strong force of attraction among them to keep them together.Theoretically after removing the applied external force all the particles shall come back to their original positions and hence there shall not be any change in the shape of the body.This kind of the body is called as a perfectly elastic body. But practically no body in the nature is perfectly elastic. Because of the external force application the distance between the molecules increases slightly which leads to the increase in the size of the body.

We can say that  more the increase in the size of the body, less the  elastic nature of the body has and vice versa.

No body in the nature is perfectly elastic. Among all the available bodies the best elastic body is Diamond which will try to protect its shape in the best possible way. The molecular force of attraction among the molecules of the body are the basic reason behind the elastic force. The molecules behave as if like connected with a mass less kind of the Springs which are having the restoring force. Thus the Springs always tries to get back to the original positions and hence the body also always tries to come back to original position. If the body is unable to come back to its original position ,after removing the applied external force then the kind of a body is called as a plastic body.

No body in the nature is also perfectly inelastic or plastic. In fact nothing perfect exists in the real life there are only there for the sake of reference in the books.

To study the elastic nature of a given material we shall define two terms called stress and strain. Stress is defined as the restoring force acting on a body per  unit surface area and strain is defined as the ratio of change in the dimensions of a body to its original dimensions.

We have a law  called Hooks law. As per this law, within the elastic limit stress acting on a body is directly proportional to the strain. And hence the ratio of stress to the strain is constant and the ratio is called modulus of elasticity. It depends on the nature of the material but not on the values of stress and strain.

Young's Modules

Depending on how do we apply the stress on a body that are three different types of stress and correspondingly there are three different types of strains also.

If the stress is applied on a thin wire,it is called longitudinal stress which acts along its length.The change in the dimensions also happen in majority along its length and the corresponding strain is called longitudinal strain. As the stress is directly proportional to strain longitudinal stress is also directly proportional to longitudinal strain and the proportionality constant there is called Young's modulus.

The stress applied on a body depends on the load that we have applied at the bottom and in that case in the place of the force we can consider the weight of the body itself. Here area of cross-section depends on the shape of the wire and in general if it is a circular shape we can consider the area of the circle.

When the same forces applied on a steel and rubber wire of similar physical dimensions, it is quite easy to notice that the expansion the rubber wire is more than that of the steel. As elastic modulus is inversely proportional to increase the length of the wire,we can conclude that the steel is having more elastic nature than that of the rubber.

That is the reason why we say still is more elastic than the rubber.

Problem one and solution

A copper wire of length 2.2 m and a steel wire of length 1.6 m are having the same diameter of 3 mm and are connected end-to-end. When a load is applied the net elongation is found to be due 0 .7 mm find the load applied as well as the elongation of the each wire?

We can solve this problem basing on the definition of young's modules of the material itself. It is very clear from the defamation of the wire in length of the wire is directly proportional to the length of the wire as well as inversely proportional to young's modulus the material. As modulus of elasticity is different materials are given in the problem,we can solve the problem becomes simple as shown below.

Suppose instead of the area of the wire if the mass and density of the wire is given in the problem in solving, that is also quite simple because area is the ratio of volume per length . We can further apply it volume as the ratio of mass to the density.

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