11th Physics Notes Elasticity | NCERT Solutions for Physics Class 11
NCERT Physics Class 11 : Elasticity is that property of a Body by which it experiences a change in size or shape whenever a deforming force acts on the body. When the force is removed the body returns to its original size and shape. Most people are familiar with the stretching of a rubber band. All materials, however, have this same elastic property, but in most materials, it is not so pronounced. The explanation of the elastic property of solids is found in an atomic description of a solid. Most solids are composed of a very large number of atoms or molecules arranged in a fixed pattern called the lattice structure of a solid and shown schematically in figure 1.
Fig.1 Lattice structure of solid.
These atoms or molecules are held in their positions by electrical forces. The electrical force between the molecules is attractive and tends to pull the molecules together. Thus, the solid resist being pulled apart. Any one molecule in figure 1 has an attractive force pulling it to the right and an equal attractive force pulling it to the left. There are also equal attractive forces pulling the molecule up and down, and in and out. A repulsive force between the molecules also tends to repel the molecules if they get too close together. This is why solids are difficult to compress. To explain this repulsive force we would need to invoke the Pauli exclusion principle of quantum mechanics. but here we simply refer to all these forces as molecular forces.
Fig. 2 Actual picture of an atom in a solar cell
The net result of all these molecular forces is that each molecule is in a position of equilibrium. If we try to pull one side of a solid material to the right, let us say, then we are in effect pulling all these molecules slightly away from their equilibrium position. The displacement of any one molecule from its equilibrium position is quite small, but since there are billions of molecules, the total molecular displacements are directly measurable as a change in length of the material. When the applied force is removed, the attractive molecular forces pull all the molecules back to their original positions, and the material returns to its original length. If we now exert a force on the material in order to compress it, we cause the molecules to be again displaced from their equilibrium position, but this time they are pushed closer together. The repulsive molecular force prevents them from getting too close together, but the total molecular displacement is directly measurable as a reduction in the size of the original material. When the compressive force is removed, the repulsive molecular force causes the atoms to return to their equilibrium position and the solid returns to its original size. Hence, the elastic properties of matter are a manifestation of the molecular forces that hold solids together. Figure 2 shows a typical lattice structure of atoms in a solar cell analyzed with a scanning tunneling microscope.
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ELASTICITY OF SHEAR :
In addition to being stretched or compressed, a body can be deformed by changing the shape of the body. If the body returns to its original shape when the distorting stress is removed, the body exhibits the property of elasticity of shape, sometimes called shear.
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As an example, consider the cube fixed to the surface in figure 3. A tangential force F11 is applied at the top of the cube, a distance h above the bottom.
The magnitude of this force Ft times the height h of the cube would normally cause a torque to act on the cube to rotate it. However, since the cube is not free to rotate, the body instead becomes deformed and changes its shape, as shown in figure 3 The tangential force applied to the body causes the layers of atoms to be displaced sideways; one layer of the lattice structure slides over another. The tangential force thus causes a change in the shape of the body that is measured by the angle φ, called the angle of shear. We can also relate φ to the linear change from the original position of the body by noting from figure 3 that
tan φ = x / h
Because the deformations are usually quite small, as a first approximation the tan φ can be replaced by the angle φ itself, expressed in radians. Thus,
φ = x / h
This equation represents the shearing strain of the body. The tangential force F11 causes a deformation φ of the body and we find experimentally that
φ ∝ F11
That is the angle of shear is directly proportional to the magnitude of the applied tangential force F11. We also find the deformation of the cube experimentally to be inversely proportional to the area of the top of the cube. With a larger area, the distorting force is spread over more molecules and hence the corresponding deformation is less. Thus,
φ ∝ 1 / A
these two equations can be combined into the single equation
φ ∝ F11
A Note that F11/A has the dimensions of a stress and it is now defined as the shearing stress:
Shearing stress = F11 / A
Since φ is the shearing strain, above equations the familiar proportionality that stress is directly proportional to the strain. Introducing a constant of proportionality S, called the shear modulus, Hooke’s law for the elasticity of shear is given by
F11 / A =Sφ
The larger the value of S, the greater the resistance to shear. Note that the shear modulus is smaller than Young’s modulus Y. This implies that it is easier to slide layers of molecules over each other than it is to compress or stretch them. The shear modulus is also known as the torsion modulus and the modulus of rigidity.
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