This chapter explores the mechanical properties of solids, focusing on how they deform under external forces and the importance of these properties in engineering applications.
Mechanical Properties of Solids – Formula & Equation Sheet
Essential formulas and equations from Physics Part - II, tailored for Class 11 in Physics.
This one-pager compiles key formulas and equations from the Mechanical Properties of Solids chapter of Physics Part - II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Stress (σ) = F / A
σ (stress) is the restoring force per unit area, where F is the applied force (in newtons) and A is the cross-sectional area (in m²). This formula is fundamental in mechanics of materials.
Strain (ε) = ΔL / L₀
ε (strain) is the ratio of change in length (ΔL) to the original length (L₀). Strain is dimensionless, representing the deformation of a material under stress.
Hooke's Law: σ = Y × ε
This law states that stress (σ) is proportional to strain (ε) for small deformations, where Y is Young's modulus (in N/m²), a material's ability to deform elastically.
Young's Modulus (Y) = σ / ε
Y represents Young's modulus, the ratio of tensile or compressive stress to longitudinal strain. It provides insight into a material's elasticity.
Shear Modulus (G) = σ_s / ε_s
G is the shear modulus, defined as the ratio of shearing stress (σ_s) to the shearing strain (ε_s). It measures a material's response to shear forces.
Bulk Modulus (B) = -p / (ΔV / V)
B indicates the material's response to uniform compression, where p is applied pressure and ΔV/V is the volume strain. The negative sign reflects that increased pressure reduces volume.
Hydraulic Stress (p) = F / A
Hydraulic stress is calculated similarly to normal stress, with F as the force exerted by fluid and A as the area in contact with the fluid.
Volumetric Strain = ΔV / V₀
This measure compares the change in volume (ΔV) to the original volume (V₀) of a material, quantifying deformation caused by external pressure.
Poisson's Ratio (ν) = - (Δd / d₀) / (ΔL / L₀)
ν (Poisson's ratio) relates the lateral strain (Δd/d₀) to the longitudinal strain (ΔL/L₀). It helps characterize a material's expansion in dimensions.
Elastic Potential Energy (U) = 1/2 × σ × ε × V
U is the elastic potential energy stored per unit volume, with σ as stress and ε as strain. It reflects energy stored in an elastic material when deformed.
Equations
ΔL = (F × L₀) / (Y × A)
This equation derives elongation (ΔL) under tensile stress, where L₀ is original length, F is the force applied, A is the area, and Y is Young's modulus.
σ_s = G × ε_s
This equation indicates that shear stress (σ_s) is proportional to shear strain (ε_s), linked by the shear modulus (G).
p = B(ΔV / V)
This expresses the relationship between hydraulic stress (p) and volumetric strain, supporting the concept of bulk modulus (B) under pressure.
δ = WL³ / (4bd³Y)
This formula gives the deflection (δ) of a beam under load (W), depending on its length (L), breadth (b), depth (d), and Young’s modulus (Y).
A ≥ Mg / σ_y
This ensures safety in designing structural ropes, where A is cross-sectional area, M is mass, g is acceleration due to gravity, and σ_y is yield strength.
E = 1/2 × σ × ε
E signifies the energy density of elastic potential energy in a material, representing energy per unit volume due to stress and strain.
ΔV = (B × V₀ × Δp) / p
This shows how the change in volume (ΔV) of a body relates to its initial volume (V₀) and bulk modulus (B) as pressure (p) increases.
θ = tan(Δx / L)
This relates the angular displacement (θ) of a cylindrical object under shear stress through the tangent of lateral displacement (Δx) over its length (L).
F/A = Y(ΔL/L₀)
This reformulation of Hooke's Law relates stress to strain in terms of Young's modulus (Y), applicable for determining the behavior of materials under axial loads.
Δx = (Stress × L) / G
This calculates the displacement (Δx) in a material under shear stress through shear modulus (G), useful in engineering applications.
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