Revision: Mechanical Properties of Solids Physics HSC Science (General) 11th Standard Maharashtra State Board

The change in shape or size or both of a body due to an external force is called deformation.

"The change in shape or size or both of a body due to an external force is called deformation."

"When a force is applied to a solid (which is not free to move), the size or shape, or both, change due to changes in the relative positions of molecules. Such a force is called a deforming force."

A body that does not regain its original shape and size and retains its altered shape or size upon removal of the deforming force is called a plastic body, and the property is called plasticity.

If a body regains its original shape and size after removal of the deforming force, it is called an elastic body and the property is called elasticity.

A body that regains its original shape and size after removal of the deforming force is called an elastic body, and the property is called elasticity.

A body that regains its original shape and size completely and instantaneously upon removal of the deforming force is said to be perfectly elastic.

The internal restoring force per unit area of a body is called stress.

Strain is defined as the ratio of the change in dimensions of the body to its original dimensions.

The strain is defined as the ratio of change in dimensions of the body to its original dimensions.

Strain = `"change in dimensions"/"original dimensions"`

The modulus of elasticity of a material is the ratio of stress to the corresponding strain. It is defined as the slope of the stress-strain curve in the elastic deforming region and depends on the nature of the material.

\[\frac {stress}{strain}\] = Constant

The constant is called the modulus of elasticity.

"Young’s modulus is the ratio of longitudinal stress to longitudinal strain."

"Bulk modulus is defined as the ratio of volume stress to volume strain."

"Shear modulus or modulus of rigidity: It is defined as the ratio of shear stress to shear strain within elastic limits."

Within elastic limit, the ratio of lateral strain to the linear strain is called the Poisson's ratio.

In case of some materials like vulcanized rubber, when the stress applied on a body decreases to zero, the strain does not return to zero immediately. The strain lags behind the stress. This lagging of strain behind the stress is called elastic hysteresis.

A graph drawn by taking tensile strain along the x-axis and tensile stress along the y-axis, obtained by gradually increasing the load on a metal wire suspended vertically from a rigid support until the wire breaks, and measuring the elongation produced during each step.

The elastic potential energy gained by a wire during elongation by a stretching force is called as strain energy.

Hardness is the property of a material that enables it to resist plastic deformation.

The property which resists the relative motion between two surfaces in contact is called friction.

"The property which resists the relative motion between two surfaces in contact is called friction."

"This mechanical force between two solid surfaces in contact with each other is called the frictional force."

The frictional force that balances the applied force when the body is static (or at rest). It prevents sliding motion between two surfaces in contact.

Friction between two surfaces in contact when one body is actually sliding over the other body is called kinetic friction or dynamic friction.

or

The force of friction that comes into play when a body is in a steady state of motion over another surface is called the force of kinetic friction.

"Friction between two bodies in contact when one body is rolling over the other, is called rolling friction."

The mathematical expression for Young's modulus (Y) is:

Y = \[\frac{MgL}{\pi r^2l}\]

Where:

• Y = Young’s Modulus
• M = Mass of the load attached
• g = Acceleration due to gravity
• L = Original length of the wire
• r = Radius of the wire cross-section
• l = Extension or elongation produced in the wire

The mathematical representation of Bulk Modulus (K) is:

K = \[\frac{\text{Volume Stress}}{\text{Volume Strain}}\]

K = \[\frac{dP}{\left(\frac{dV}{V}\right)}\] = V \[\frac {dP}{dV}\]

Where:

• K: Bulk Modulus
• dP: Change in pressure (Volume Stress)
• dV: Change in volume
• V: Original volume

The formula for modulus of rigidity is:

η = \[\frac{\text{Shear Stress}}{\text{Shear Strain}}=\frac{F/A}{\theta}=\frac{F}{A\cdot\theta}\]

Where:

• η = Modulus of rigidity (Pa or N/m²)
• F = Tangential force applied (N)
• A = Cross-sectional area on which force acts (m²)
• θ = Shear strain = Δl/l (in radians)
• Δl = Displacement of the upper surface relative to the lower surface (m)
• l = Original height of the block (m)

SI Unit: Pascal (Pa) or N/m²​

Dimensional Formula: M¹L⁻¹T⁻²

\[\sigma=\frac{\text{Lateral strain}}{\text{Linear strain}}=\frac{\frac{d}{D}}{\frac{\Delta l}{l}}=\frac{d\cdot l}{D\cdot\Delta l}\]

Where:

• σ = Poisson's ratio
• l = original length of the wire
• ∆l = increase or decrease in length of the wire
• D = original diameter of the wire
• d = corresponding change in diameter of the wire

Important Note: Poisson's ratio has no unit. It is dimensionless.

W = \[\frac {1}{2}\]Fl

Where:

• W = Work done (Strain energy)
• F = Stretching force applied
• l = Extension/elongation produced
• Y = Young's modulus
• Stress = Force per unit area = \[\frac {F}{A}\]
• Strain = Change in length per unit length = \[\frac {l}{L}\]

FL = μs N

Where:

• FL = Limiting force of friction (maximum static friction)
• μs = Coefficient of static friction (dimensionless constant)
• N = Normal reaction (normal force between surfaces)

μs = FL / N

Where:

• μs = Coefficient of static friction
• FL = Limiting force of friction
• N = Normal reaction

μₖ = Fₖ/N

The coefficient of kinetic friction is defined as the ratio of force of kinetic friction to the normal reaction between the two surfaces in contact.

Fₖ = μₖ N

Where:

• Fₖ = Force of kinetic friction
• μₖ = Coefficient of kinetic friction (constant of proportionality)
• N = Normal reaction between the two surfaces in contact