Mechanical Properties of Solids - JEE Mindmap PDF

# MIND MAP FOR JEE ASPIRANTS - Physics - Mechanical Properties Of Solids - Elasticity By - Saleem Ahmed Sir - **Physics** - Mechanical Prop. of solids - (Elasticity) ## Today's Targets 1. Elasticity 2. 3. 4. ## Elasticity - Prop. of body by virtue of which it tends to regain its...

# MIND MAP FOR JEE ASPIRANTS - Physics - Mechanical Properties Of Solids - Elasticity By - Saleem Ahmed Sir - **Physics** - Mechanical Prop. of solids - (Elasticity) ## Today's Targets 1. Elasticity 2. 3. 4. ## Elasticity - Prop. of body by virtue of which it tends to regain its original shape & size when applied forces are removed - **longitudinal stress:** $stress = \frac{F}{A}$ - **longitudinal strain:** $strain = \frac{\Delta l}{l}$ ## Stress - `Longitudinal stress = E = Y` - `Shear stress = η` - `Volumetric stress (B)` - `Bulk stress` - `Stress α strain` - `Stress = E strain` - `Strain = change in dim/original dim` ## Stress-Strain Relation - `E = Y = Stress/Strain = (F/A)/(\Delta l/l) = F l / A \Delta l ` - `Y = F l / A \Delta l` ## Elasticity - **Elastic stress:** - $Stress = \frac{internal restoring force}{area of cross section}$ - stress characterizes the strength of the forces causing the deformation, on a "force per unit area" basis. - Stress depends on direction of force as well as area of application, so it is a tensor. - Here we define three types of stress: 1. Tensile or compressive stress 2. Shear stress 3. Hydraulic/volumetric stress: ## Tensile stress - $Tensile stress = \frac{F_{perp}}{area of cross section}$ ## Shear stress - $Shear stress = \frac{F_{||}} {area}$ - (tangential stress) - $Shear strain = \theta = tan\theta$ - $Shear stress = \frac{F\cos\theta}{A}$ - $long shear = \frac{Fsin\theta}{A}$ - $Shear stress / Shear strain = \frac{F/A}{\theta} = \frac{F/A}{x/l}$ ## Volume stress - When stress is due to pressure exerted by a fluid on all sides of body, and the resulting deformation is a volume change. - We define bulk stress (or volume stress) as - $volume stress = \frac{F_{perp}}{area of cross section} = pressure in fluid$ - If $P = \frac{F}{A}$ - `F = Pds` - Then $ds = \frac{F}{P}$ - `Exam pressure` - $B = -\frac{\Delta P}{\frac{\Delta V}{V}}$ ## Bulk Modulus Of Elasticity - $Bulk Modulus Of Elasticity B = \frac{Vol's stress}{Vol's Strain}$ - $B = -\frac{\Delta P}{\frac{\Delta V}{V}} = -\frac{P}{\frac{\Delta V}{V}} = -\frac{dP}{\frac{dV}{V}}$ ## Applications of Elasticity - `Stress α strain` - `Stress = Y strain` - `\frac{mg}{A} = Y \frac{\Delta l}{l}` - `\Delta l = \frac{Fl}{AY} = \frac{mgl}{AY}` - $E = \frac{F}{A} = Y \frac {\Delta l}{l}$ - $E = \frac{F}{A} = Y \frac {\Delta l}{l}$ - `\frac{mg}{\pi\times (r)^2} = Y \frac{\Delta l_1}{l}` - `\frac{mg}{\pi\times(2r)^2} = 5Y \frac{\Delta l_2}{l}` - `(E.P.E) = \frac{1}{2} (stress)^2 \times Vol` - `\frac{1}{2} \times (\frac{mg}{\pi r^2})^2 \times \frac{A\times l}{Y}` ## Work done & Elastic Potential Energy - `F = Y \frac{\Delta l}{l}` - `\frac {m \times g}{A} = Y \frac{d3} {dx}` - `\int d3 = \frac{mg}{LAY} \int_{0}^{l} x dx` - `\Delta l = \sqrt{}` - `du = \frac{1}{2} (\frac{T\times A}{l})^2` - `du = \frac{1}{2} (\frac{m\times g}{l})^2 A \times \frac{1}{Y} dx` - `\int du = \frac{m^2 g^2 A}{2 L^2 A^2 Y} \int_{0}^{l} dx` - `(m \times l)` - `T = \frac{m \times X}{l} \times A = \frac{m \times X}{l} \times F = \frac{F}{l} \times X` - `\frac{FX}{LA} = Y \frac{d3}{dx}` - `\int d3 = \frac{F}{LAY} \int_{0}^{l} x dx` ## Spring Force - `K = \frac{YA}{l}` - `F = KX` - `F = \frac{YA}{l} \Delta l` ## Elastic potential energy - $U = \frac{Elastic P.E.}{Vol^{m}} = \frac{1}{2} \times (stress \times strain)$ - $U = \frac{1}{2} \times Y \times (strain)^2$ - $U = \frac{1}{2} \times (\frac{Stress}{Y})^2 \times Vol$ - $Stress = Y \times strain$ - $E.P.E = \frac{(Stress)^2}{2Y} \times Vol$ ## Stress-Strain graph - **Proportion Limit:** The limit in which Hooke's law is valid and stress is directly proportional to strain is called proportion limit. Stress ∞ Strain - **Elastic limit:** That maximum stress which on removing the deforming force makes the body to recover completely its original state. - **Breaking strength** - **Plastic region** - **Slope = tan0=Y** - **Shear Stress = Y strain** - **Strain** ## Bulk Modulus of Elasticity - Within elastic limit the ratio of the volume stress and the volume strain is called bulk modulus of elasticity. - `K or B = \frac{volume stress}{volume strain} = \frac{F/A}{\Delta V/V} = \frac{\Delta P}{- \Delta V/V}` - The minus sign indicates a decrease in volume with an increase in stress. - Unit of K: N m-² or pascal ## Illustration of Elasticity - `\frac{- \Delta d}{d} = \frac{- \Delta l}{l} = \frac{- \Delta d}{\theta Y l}` - `\theta = \frac{- \Delta d}{\Delta l \times l}` ## Thank You