Mechanical Properties of Solids Notes for Class 11

Class 11 Physics Chapter 9 Notes – Mechanical Properties of Solids

Class 11 Physics Revision Notes for Chapter 9 – Mechanical Properties of Solids explores the concepts around rigid bodies and has many important numerical questions associated with it. Conceptualising every topic can benefit students in many ways for their exam preparation. Extramarks CBSE Class 11 Physics Chapter 9 Notes are presented with various examples that help students to answer questions related to Class 11 Physics Chapter 9 – Mechanical Properties of Solids. These notes are created by subject-matter experts and are easily accessible for students’ convenience and all-rounded exam preparation. 

1. Introduction:

While applying laws of motion, we considered bodies to be rigid, i.e., they have a definite size and shape which doesn’t change when force is applied. This isn’t the case in reality, wherein the bodies can change their shape when acted upon by a force. This applies to the hardest of substances on earth, such as metals. In this chapter, we deal with the various types of forces that cause an object to change its dimensions.

2. Deforming Force:

A force that results in the change in configuration of an object such as its size and/or shape is termed a deforming force.

3. Elasticity:

Elasticity is the quality of an object that allows it to return to its original size and shape when a deforming force is removed. 

4. Perfectly Elastic Body:

Regaining their original dimensions completely without any deformation upon the removal of the deforming force are characteristics of Perfectly Elastic Bodies. Such bodies are rare in nature. Quartz fibre is an example of a perfectly elastic body.

5. Plasticity: 

Objects such as clay and mud do not retain their dimensions when a deforming force acts upon them. They are permanently deformed and would not go back to their original shape by themselves. Such substances are termed plastic, and this property is called plasticity.

6. Perfectly Plastic Body:

Objects that do not regain their original form at all by removing the deforming force are perfectly plastic. They do not have the tendency to revert to the action of the deforming force. Paraffin wax can be considered perfectly plastic.

7. Stress:

On subjecting an object to deforming force, a restoring force is developed in the object; which is equal in magnitude but opposite in direction. Stress is the restoration force per unit area. This is consistent with Newton’s third law of motion.

7.1 Definition:

Formally, the internal restoring force per unit cross-section area developed in the object in reaction to the deforming force applied is termed as stress.

7.2  Mathematical Form:

Stress =  Applied ForceCross-section Area of the body

SI Unit of stress is N/m2 or pascal (Pa).

The dimensional formula is: [ML-1T-2]

7.3 Types of Stress:

There are three ways in which a solid object may change its dimensions when acted upon by an external force. Correspondingly, the restoring forces (stress) developed in the body are of three types: tensile, compressional and tangential. Tensile and compressional stresses come under longitudinal stress.

1. Longitudinal Stress:

The term “longitudinal stress” or “normal stress” refers to the stress that results when a deforming force is applied normally to a cross-sectional region. It is further classified into two types:

1. Tensile Stress: The restoring force developed when an object is stretched by the external force.
2. Compressional Stress: The restoring force developed when an object is compressed by the external force.
3. Tangential or Shearing Stress: 

The restoring force developed in the body in reaction to a tangential force is known as the shearing force. The tangential deforming force and the stress both act parallel to the surface of the body.

3. Hydraulic Stress:

The internal restoring forces developed when a body is immersed in a fluid is called hydraulic stress. The deforming force acts because of the pressure from the fluid on all sides and seeks to compress the object, and the hydraulic stress developed opposes this.

8. Strain:

As a result of the deforming force, there is a relative displacement x as compared to the original shape. This change, expressed in a relative fraction form, is known as strain.

8.1  Mathematical Equation:

Strain = change in dimensionoriginal dimension = xL

Strain is a dimensionless quantity and has no unit.

9. Hooke’s Law 

According to Hooke’s Law, within the elastic limit, stress and strain are proportional to one another. The application of this concept is not universal; for instance, human muscles do not follow Hooke’s law. 

stress ∝ strain

stress = k * strain

Stress ∝ Strain Stress = K. Strain ; 

Where K is the proportionality constant and is known as the ‘Elastic Modulus’ of the material.

9.1 Types of Modulus of Rigidity:

9.1.1  Young’s Modulus of Rigidity (Y)

Young’s Modulus of Rigidity (Y) is the measure of longitudinal stress to longitudinal strain within the elastic limit.

Y = longitudinal stresslongitudinal strain = = FALL = F * LA * L 

High values of Y correspond to high elasticity, as seen in metals.

Since strain has no units, the unit of Y is N/m2 or Pa.

9.1.2  Bulk Modulus of Rigidity:

The bulk modulus is defined as the ratio of hydraulic stress to the corresponding hydraulic strain. 

B = -p(VV) = -pVV

The negative sign indicates that when pressure rises, volume decreases. The value for a system in equilibrium, the value of B is always positive.

The SI unit of the bulk modulus is the same as that of pressure, N/m2 or Pa.

Compressibility:

The reciprocal of the bulk modulus is termed as compressibility and is defined as the fractional change in volume per unit increase in pressure.

k = 1B =–VpV

Its SI unit is N-1m2 and CGS unit is dyne-1cm2.

9.1.3  Modulus of Rigidity or Shear Modulus ():

The ratio of shear stress to that of shear strain within the elastic limit is termed as Shear Modulus.

= shear stressshear strain = F/AY= FAY

Its SI unit is N/m2.

The Young’s modulus is always considerably larger than the shear modulus.

10. Limit of Elasticity:

The maximum value of the deforming force, upon whose removal the body still exhibits elasticity, is termed the limit of elasticity.

11. Stress-Strain Curve:

A typical stress-strain curve for a metal is shown in the figure.

1. The region OA is linear, showing that stress is proportional to strain and thus Hooke’s Law is obeyed in this region. Here the material is perfectly elastic. Point A is called the proportional limit.
2. After A, the proportionality is lost and the curve AB is generated. On removal of the load at this point, the material retraces the curve BAO and regains its original shape. Point B is the farthest point to where the material remains elastic and thus is called the elastic limit or yield point. The stress at the yield point is called yield strength. The portion OB is called the elastic region.
3. Beyond point B, there is a large change in the value of strain for small increases in stress. On removal of the load at any point C, the material follows the dashed line and cannot regain its original form. Even on decreasing stress to zero, residual stress remains in the material. Thus, the material acquires a permanent set. This phenomenon is called elastic hysteresis.
4. On increasing the load beyond point C, the strain increases rapidly and results in constrictions of the material. This continues until the material eventually breaks at point D which is the Fracture Point.
5. From B to D, the material keeps on adding to its length without the addition of load. This region is termed as plastic region and the material undergoes plastic deformation. The stress at the breaking point is known as the tensile strength of the material.

12. Elastic after Effect:

The delayed time interval in regaining the original shape and size by an object on the removal of the deforming force is given the name Elastic After Effect.

13. Elastic Fatigue:

On repeated application of alternating deforming force, an elastic body becomes less elastic. This property is called elastic fatigue. That is why rubber bands elongate the longer you use them.

14. Ductile Materials:

These materials can be drawn into long wires owing to their large plastic range (BD). They undergo permanent elongation and snap at their fracture point. Examples include copper, silver and aluminium.

15. Brittle Materials:

These materials have a small plastic range (BD) and thus snap quickly once the stress is increased beyond the elastic limit. Examples of brittle materials are ceramics and glass.

16. Elastomers:

In elastomers, the strain produced in the material is much larger than the stress we apply within the elastic limit. Examples include rubber and elastic tissue of the aorta. These materials have no plastic range.

17. Elastic Potential Energy of Stretched Wire:

The interatomic forces of attraction work against the deformation of a material. Therefore, in order to lengthen or compress a wire, we must work against these restoring forces. This work against the restoring forces is then stored as potential energy in the wire. An example of this is raising a body up against the pull of gravity, where the effort required to defy gravity is stored as the body’s potential energy. 

18. Poisson’s Ratio:

A wire’s cross-sectional area slightly decreases along with its lengthening as a wire is stretched. As a result, a strain that is perpendicular to the applied force forms to counter this. It is known as lateral strain.

Poisson noted that in the elastic limit, lateral strain is inversely proportional to longitudinal strain. The Poisson’s ratio is the name given to this ratio.

Poisson’s ratio = lateral strainlongitudinal strain = –d/dL/L = –d * LL * d

The negative sign signifies the opposite nature of lateral and longitudinal strains.

Poisson’s ratio has no units. Theoretical values: -1 to 0.5. Practical Values: 0 to 0.5.

19. Applications of Elasticity:

Engineering inventions require an in-depth understanding of material elasticity. Knowing the strength of the materials used is essential for the structural design of the columns, beams and supports in a building.

Imagine that we must create a bridge that can support its weight, wind, and moving vehicles. The issue of the beam bending under load is a top priority. The beam must not shatter or bend excessively. Consider a beam that is supported at its ends and loaded at its centre. A bar with the dimensions l, b, and d depicted sags at the centre by the amount given by when it is loaded at the centre with weight W.

= W l34bd3Y 

To reduce the bending , we need a material with a large value of Y. Reduction in bending can be effectively achieved by increasing the depth rather than its breadth since it depends on d-3. But the beam may suffer from bending under the effect of flowing traffic. Thus an I-shaped cross-section is better. It provides a large load-bearing surface and depth to prevent bending. This shape reduces the weight of the beam without sacrificing strength, reducing the cost of construction.