Comprehensive Summary of Important Formulae: Mechanical Properties of Solids and Fluids

Objective:

To generate a clear and concise list of key physics formulae from the specified chapters of the NCERT textbook, focusing on fundamental concepts and problem-solving techniques related to the mechanical properties of solids and fluids.

1. Mechanical Properties of Solids

1.1 Stress and Strain

• Stress ($\sigma$): $\sigma =\frac{F}{A}$

  Stress is the restoring force per unit area developed in a body when an external force is applied.

• Strain ($ϵ$): $ϵ=\frac{\Delta L}{L}$

  Strain is the fractional change in dimension (length) due to applied stress.

• Hooke's Law: $\sigma =Y\cdot ϵ$

  Stress is directly proportional to strain within the elastic limit. The constant of proportionality is Young's modulus (Y).

1.2 Elastic Moduli

• Young’s Modulus ($Y$): $Y=\frac{\sigma }{ϵ}=\frac{FL}{A\Delta L}$

  Describes the tensile or compressive elasticity of a material, defined as the ratio of tensile stress to tensile strain.

• Shear Modulus ($G$): $G=\frac{\text{Shearing Stress}}{\text{Shearing Strain}}=\frac{F/A}{\Delta x/L}$

  The ratio of shear stress to shear strain, representing the material's response to shearing forces.

• Bulk Modulus ($B$): $B=-\frac{p}{\Delta V/V}$

  The ratio of hydraulic stress to the corresponding hydraulic strain. It measures a substance's resistance to uniform compression.

1.3 Stress-Strain Curve

• Elastic Region: The region in the stress-strain curve where the material returns to its original shape when the stress is removed.
• Plastic Region: The region beyond the yield point where the material deforms permanently.
• Ultimate Tensile Strength (UTS): The maximum stress a material can withstand.

2. Mechanical Properties of Fluids

2.1 Pressure in Fluids

• Pressure ($P$): $P=\frac{F}{A}$

  Pressure is the force exerted per unit area in a fluid.

2.2 Pascal’s Law

• Pascal’s Law: $P=\text{constant throughout a fluid in equilibrium}$

  In a confined fluid, the pressure is transmitted equally in all directions.

2.3 Buoyancy

• Archimedes’ Principle: $\text{Buoyant Force}=\text{Weight of displaced fluid}$

  An object submerged in a fluid experiences an upward force equal to the weight of the fluid displaced by the object.

2.4 Fluid Dynamics

• Equation of Continuity: $A_1{v}_1=A_2{v}_2$

  The product of cross-sectional area and velocity at any two points in a streamline flow remains constant.

• Bernoulli’s Equation: $P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$

  In a streamline flow, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant.

Example Applications

Example Problem 1:

Given: A steel wire of length 2 m and cross-sectional area 1 mm² is stretched by a force of 100 N.
Find: The stress, strain, and elongation of the wire.
Solution:

• Stress ($\sigma$) = $\frac{F}{A}=\frac{100}{1\times 10^{-6}}=10^8$ N/m²
• Strain ($ϵ$) = $\frac{\Delta L}{L}=\frac{\sigma }{Y}=\frac{10^8}{2\times 10^{11}}=5\times 10^{-4}$
• Elongation ($\Delta L$) = $ϵ\times L=5\times 10^{-4}\times 2=0.001$ m = 1 mm

Common Mistakes

1. Confusing stress with force: Stress is force per unit area, not just force.
2. Neglecting the elastic limit: Remember that Hooke's law only applies within the elastic limit of the material.

Final Review and Formatting

Review all formulae and concepts for clarity. Ensure the content is organized logically, with each formula followed by a brief explanation and example application where applicable.

This combined document now serves as a comprehensive summary of important formulae for both the mechanical properties of solids and fluids, tailored for NEET UG Physics preparation.