Tuesday, December 16, 2014

Problems and Solutions on Elasticity

Problem and solution

A metal ring of radius r and cross-section is a has to be fixed than a wooden circular disc of radius R. Assume that the radius of the disc is greater than that of the ring. What is the force acting on the ring ?

To fix the ring of the disc, we shall increase the size of the ring that is at least equal to the size of the disk. That is why we shall apply a force on the ring therefore its radius can be increased.

Problem and solution

A light rod of length 2 meter is suspended from the ceiling horizontally by means of the 2 vertical wires of the same length as shown.

One of the wire is made up of the steel and other one is made up of the brass and their respective areas of cross-sections are  given. Find the position along the broad at which a weight has to be suspended so that there may be equal stress on the both the wires and also find the location of the weight so that there will be equal strains on both the wires?

Density of the compressed liquid

Compressibility can be defined as the reciprocal of the bulk modulus. We can derive the equation for the density that how it is being effected if an external force is applied as shown below. We know as per the concept of the bulk modulus in the place of the change of the volume even we can write the change in the density and derive the equation as shown below.

Poisons ratio can be defined as the ratio of lateral strain to that of the longitudinal strain. Lateral strain is defined as the ratio of decreasing the thickness of the wire with respect to its original thickness. Longitudinal strain is defined as the ratio of increase the length of the wire to its original length.

These two strains are perpendicular to each other. While the longitudinal strain increases the length of the wire, the lateral strain decreases the thickness of the wire and hence it is shown with a negative sign.

Problem and solution

In the 1st problem volume of the material is given for you and it is also given that it is subjected to how much of the pressure if the corresponding change in the volume is also given in the problem we have to calculate the bulk modulus?

There are two  more different problems in the given now diagram and it can analyse them and comment back if you are having any  doubts.

We have two  more pages of the problems and solutions as attached below. You can go through the diagram by zooming in and you can comment at the end of the page ,if you are having any of the doubts regarding them.

While solving the first problem law of conservation of energy is taken into consideration.When the wire is stretched it has some energy called strain energy and this is converted into kinetic energy.

In solving the second problem we have taken into count that the increase in the length is directly proportional to  the applied force on the wire when all other terms are constant.

In solving the first problem we shall understand that the volume of the wire is constant.When its area changes,its length will change but total volume remains constant.

While solving the last problem, we are dealing with the weight of the body suspended.We can express that mass in terms of volume and density of the body and hence extension is directly proportional to its cube of the radius of the suspended body in spherical shape.

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