Kerala Plus One Physics Notes Chapter 9 Mechanical Properties of Solids
A rigid body means a hard body with a fixed shape and size. But we can bend, twist and stretch solids. So they are not perfectly rigid. To change the shape of a body, a force should be applied. When this force is removed, the body regains its original shape, provided the force is small. This property is called elasticity Springs and rubber tyres are examples of common elastic things. Some substances don’t have a tendency to regain their original shape when the deforming force is removed. They are called plastic substances. Putty and clay are examples of common plastic substances.
Do not confuse the word elasticity used in daily life with that in physics. A material which stretches more permanently cannot be considered elastic. In fact it is just the opposite
Elastic Behavior of Solids :
Imagine a solid magnified very much that we can see the atoms. They are connected by springs.
When a force deforms an atom (displaces it from its position) the springs (inter atomic forces) bring it back after the force is removed. In fact the particles will oscillate about this position. In some time, the oscillations will be damped out and the particle will regain its original position.
No body is perfectly elastic or perfectly plastic. All the bodies in nature lie between these two links.
It is the restoring force developed per unit area of cross section of the deformed body.
S.l unit of stress is Nm2 or Pascal (Pa) and its dimensional formula is [ML-1T2].The restoring force is equal and opposite to the external applied force.
Stress and pressure are both forces per unit area. Then in what respect does stress differ from pressure?
Pressure is the external force per unit area, while stress is the internal restoring force which comes into play in a deformed body acting transversely per unit area of the body.
The effect of stress on a body is called strain. Strain is measured as the ratio of the change in dimension produced to the original dimension.
It has no dimensions and units.
|Tensile stress or Compressive stress or Longitudinal stress||Longitudinal Stain or Linear stain|
|Tangential or shearing stress||Shearing Strain|
|Hydraulic stress||Volume strain|
Different types of Stress and Strain:
There are three ways in which a solid’ may change its dimensions when an external force acts on it. Hence there are three types of stress and corresponding strain.
1. Tensile stress and Longitudinal strain:
When the forces acting on the body produces an elongation along its length, then we call the stress as tensile stress
(see the above figure).
When the forces acting on the body produces an compression along its length, then we call the stress as compressive stress. The tensile stress produces a change in the length (∆L). Hence the longitudinal strain can be written as,
where ‘L’ is the original length.
2. Tangential or Shearing stress and Shearing strain:
If two equal and opposite deforming forces are applied parallel to cross-sectional area of the cylinder as shown in figure, there is a relative displacement between the opposite faces of the cylinder. The restoring force per unit area developed due to the applied tangential force a known as tangential or shearing stress.
As a result of applied tangential force, there is a relative displacement between opposite forces of the cylinder (see figure). The strain so produced is known a shearing strain.
Shear strain =θ = tan θ (as θ is small)
3. Hydraulic Stress and Volume strain:
force applied by the fluid acts in perpendicular direction at each point of the surface. This leads to decrease in its volume without any change of its geometrical shape.
The internal restoring force per unit area in this case is known hydraulic stress. The strain produced by a hydraulic pressure is called volume strain. The volume strain can be written
Note: Hydraulic stress is equal to the hydraulic pressure.
Hooke’s Law :
If the deformation is small, the stress in a body is proportional to the corresponding strain. This fact is known as Hooke’s law.
stress α strain
stress = constant x strain
This constant of proportionality is called the modulus of elasticity of the material. The nature of the material and the manner in which it is deformed determines value of modulus of elasticity.
- When the strain is small the stress is proportional to the strain. This is the region where Hooke’s law is valid. The point a is called the proportional limit. Till this point the wire recovers its original dimesions when the stress is removed.
- If the strain is increased a little bit, the stress is not proportional to the strain. However, the wire remains elastic. This means, if the stretching force
is removed, the wire regains its original length. This behaviour is shown up to a point b which is known as the elastic limit or yeild point.
- If the wire is stretched beyond the elastic limit, the strain increases much more rapidly. If the stretching force is removed the wire does not come back to its natural length. Even on reducing the stress to zero, a residual strain equal to PQ is left in the wire. The material is said to have aquired a permanent set.
- If the stress is increased beyond the point c there is large increase in the strain. The wire breaks at a point d known as the fracture point. The stress corresponding to this point is called breaking stress.
If large deformation takes place between the elastic limit and the fracture point, the material is called ductile. If it breaks soon after the elastic limit is crossed, it is called brittle.
The materials which can be elastically stretched to large values of strain are called elastomers, eg., rubber.
Different types of Modulus of elasticity:
Within the elastic limit, the ratio of longitudinal stress to the longitudinal strain is called Young’s modulus of a wire.
Metals have large values of F Young’s modulus. A material with large Y requires a large force to produce small changes is length.
A wire of length L and radius r is rigidly fixed at one end. On stretching the other end of the wire with a force F, the increase in its length is I. If another wire of same material but of length 2L and radius 2r is stretched with a force of 2F, the Increase in its length will be
Within elastic limit, the ratio of normal stress (volume) to the volume strain is called the bulk modulus of elasticity.
Here the negative sign indicates that the volume decreases with increase in stress.
Note: Compressibility is the reciprocal of the bulk modulus of a material.
Compressibility = 1/K
Bulk modulus is different from modulus of elasticity. A material with large bulk modulus is difficult to compress but it does not need to be stiff. For example, water.
The ratio of shearing stress to the corresponding shearing strain is called the shear modulus of the material (represented by T). It is also called the modulus of rigidity.
A metal cube of side 10 cm is subjected to a shearing stress of K? N/m1. Calculate the modulus of rigidity if the top of the cube is displaced by 0.05 cm with respect to its bottom.
When a rod or a wire is subjected to a tensile stress, its length increases in the direction of the tensile force. At the same time the length perpendicular to the tensile force decreases. For a cylindircal rod, the length increases and the diameter decreases when the rod is stretched.
The fractional change in the transverse length is proportional to the fractional change in the longitudinal length. The constant of proportionality is called Poisson ratio (a).
The strain-stress curves of three wires of different materials are shown in the figure. P, Q and Rare the elastic limits of the wires. The figures & shows that:
As stress is shown on
x-axis and strain on y-axis
So we can say that
So the elasticity of wire P is minimum and of wire R is maximum.
Determination of young’s modulus of a wire:
The arrangement consists of two long straight wires of same length and equal radius suspended side by side suspended by a fixed support. The wire A carries a fixed graduated scale and below it a heavy fixed load which keeps the wire free from kinks. This wire is called the reference wire.
The wire B is called the experimental wire and carries a vernier scale at its bottom. The vernier scale can slide against the main scale attached to the reference wire. A hanger is attached at the lower end of the vernier scale. Slotted half kg or 1 kg weights may be slipped into the hanger. Using a screw we measure the average radius r of the experimental wire. Let L be the length.
The vernier scale reading is noted. A half kg weight is added to the hanger. The wire is allowed to elongate for a minute. The reading on the scale is noted. The difference of the scale readings gives the extension due to the extra weight put. The weight is gradually increased in few steps and every time we note the extension produced. From the data, extension versus load curve is plotted.
All the quantites on the right hand side are known and hence the Young’s modulus, Y is calculated.
Which is more elastic – rubber or steel? Explain
Consider two rods of steel and rubber, each having length I and area of cross-section A. If they are subjected to the same deforming force F, then the extension Δls produced in the steel rod will be less than the extension Δlr in the rubber rod, i.e., Δls <Δlr. Now