Here students can locate TS Inter 1st Year Physics Notes 10th Lesson Mechanical Properties of Solids to prepare for their exam.

## TS Inter 1st Year Physics Notes 10th Lesson Mechanical Properties of Solids

→ Elasticity: The property of a body by virtue of which it tends to regain its original size (or) shape when applied force is removed.

→ Plasticity: It is the inability of a body not to regain its original size (or) shape when external force is removed.

Note : Substances which will exhibit plasticity are called plastic substances.

→ Stress (σ): The restoring force per unit area is called stress (σ).

Stress (σ) = Force/area

Unit Nm^{-2} (or) pascal; D.F : ML^{-1} T^{-2}

→ Tensile Stress: When applied force is normal to area of cross section of the body then restoring force per unit area is called tensile stress.

→ Tangential (or) shearing stress: The restor-ing force developed per unit area of cross section when a tangential force is applied is known as shear stress or tangential stress.

→ Hydraulic stress (Volumetric stress): For a body in a fluid force is applied on it in all directions perpendicular to its surface.

“The restoring force developed in the body per unit surface area under hydraulic compression is called hydraulic stress.”

→ Strain: Change produced per unit dimension is called strain. It is a ratio.

Types:

- Longitudinal strain : The ratio of increase in length to original length is called as longitudinal strain. Longitudinal strain = \(\frac{\Delta \mathrm{L}}{\mathrm{L}}\)
- Tangential (or) shear strain : The ratio of relative displacement of faces Ax to the perpendicular distance between the faces is called shear strain.

Shear strain = \(\frac{\Delta \mathrm{L}}{\mathrm{L}}\) = tan θ - Volume strain: The ratio of change in volume AV to the original volume (V) is called volume strain.

Volume strain = \(\frac{\Delta \mathrm{V}}{\mathrm{V}}\)

→ Hooke’s Law : For small deformations the stress is proportional to strain.

Stress ∝ strain ⇒ stress/strain = constant.

This proportional constant is called modulus of elasticity.

→ Elastic constant: The ratio of stress to strain is called “elastic constant”. Unit: Newton/m^{2}. D.F: ML^{-1}T^{-2}

Note : Elastic constants are three types.

→ Young’s modulus (Y): The ratio of tensile stress (or) compressive stress to longitudinal strain or compressive strain is called Young’s modulus.

Y = \(\frac{\sigma}{\varepsilon}=\frac{\text { Tensile or compressive stress }(\sigma)}{\text { Tensile or compressive strain }(\varepsilon)}\)

→ Shear modulus (G) : The ratio of shearing stress to the corresponding shearing strain is called shear modulus.

Shear modulus (G) = \(\frac{\text { Shear stress }\left(\sigma_{\mathrm{s}}\right)}{\text { Shearstrain }(\theta)}\)

→ Bulk modulus (B): The ratio of hydraulic stress to the corresponding hydraulic strain is called Bulk modulus.

Bulk modulus (B) = \(=-\frac{\text { Hydraulic pressure (F/A) }}{\text { Hydraulic strain }(\Delta \mathrm{V} / \mathrm{V})}\)

→ Compressibility (K): The reciprocal of bulk modulus is called compressibility. Compressibility K = 1/B

→ Poisson’s ratio: In a stretched wire the ratio of lateral contraction strain to longitudinal elongation strain is called Poisson’s ratio.

Poisson’s ratio σ = \(\frac{\Delta \mathrm{d} / \mathrm{d}}{\Delta \mathrm{L} / \mathrm{L}}\)

Poissons ratio is a ratio of two strains so it has only numbers.

Note : For steel Poisson’s ratio is 0.28 to 0.30, for aluminium alloys it is upto 0.33.

→ Elastic potential energy (u) : When a wire is under tensile stress, work is done against the inter atomic forces. This work is stored in the wire in the form of elastic potential energy.

(or)

Work done to stretch a wire against inter atomic forces is termed as “elastic potential energy”.

Elastic potential energy (u) = \(\frac{1}{2} \frac{\mathrm{YA} l^2}{\mathrm{~L}}\) = \(\frac{1}{2}\)σs

or u = \(\frac{1}{2}\) stress × strain × volume of wire.

→ Ductile materials : If the stress difference between ultimate tensile strength and fracture point is high then it is called ductile material. Ex : Silver, Gold.

→ Brittle material : If the stress difference between ultimate tensile strength and fracture point is very less then that substance is called

brittle material. Ex : Cast iron.

→ Elastomers : Substances which can be stretched to cause large strains are called elastomers. Ex : Rubber, Tissues of aorta.

→ Stress = \(\frac{\text { Force }}{\text { Area }}=\frac{F}{A}\); Unit: N/m2 (or) Pascal.

→ Strain = \(\frac{\text { elongation }}{\text { original length }}=\frac{\Delta \mathrm{L}}{\mathrm{L}}\); No units

→ Hooke’s Law: Within elastic limit, stress

stress oc strain (or) \(\frac{\text { stress }}{\text { strain }}\) = constant (Elastic constant).

→ Young’s modulus (Y) = \(\frac{\text { longitudinal stress }}{\text { longitudinal strain }}\)

= \(\frac{\mathrm{F} / \mathrm{A}}{\Delta \mathrm{e} / \mathrm{L}}=\frac{\mathrm{FL}}{\mathrm{A} \Delta \mathrm{e}}\). In Searle’s apparatus Y = \(\frac{g L}{\pi r^2} \cdot \frac{M}{e}\)

- If two wires are made with same materials have lengths l
_{1}l_{2}and radii r_{1}, r_{2}then ratio of elongations \(\frac{\mathrm{e}_1}{\mathrm{e}_2}=\frac{l_1}{l_2} \cdot \frac{\mathrm{r}_2^2}{\mathrm{r}_1^2}\)\ (∵ e ∝ l / r) - If two wires are made with same material and same volume has areas Aj and A2 are subjected to same force then ratio of elongations \(\frac{e_1}{e_2}\) = r
_{2}^{4}/r_{1}^{4}(∵ e ∝ \(\frac{1}{\mathrm{r}^4}\)) - If two wires of same length and area of cross section are subjected to same force then ratio of elongations e
_{1}/e_{2}= \(\frac{y_2}{y_1}\) (∵ e ∝ \(\frac{1}{y}\)) - If two wires are made with same material have lengths /j and 1% and masses m} and m2 are subjected to same force then ratio of elongations \(\frac{\mathrm{e}_1}{\mathrm{e}_2}=\frac{l_1^2}{l_2^2} \times \frac{\mathrm{m}_2}{\mathrm{~m}_1}\) (∵ e ∝ \(\frac{l^2}{\mathrm{~m}}\))

→ Rigidity modilus, G = \(=\frac{\text { shear stress }}{\text { shear strain }}=\frac{\mathrm{F}}{\mathrm{A} \theta}=\frac{\mathrm{FL}}{\mathrm{Al}}\)

→ Shear strain,

θ = \(\frac{\text { relative displacement of upper layer }}{\text { perpendicular distance between layers }}=\frac{\Delta x}{z}=\frac{l}{L}\)

→ Bulk modulus (B) = \(\frac{\text { volumetric stress }}{\text { volumetric strain }}=\frac{\mathrm{F} / \mathrm{A}}{\Delta \mathrm{V} / \mathrm{V}}=\frac{\mathrm{PV}}{\Delta \mathrm{V}}\)

→ \(\frac{1}{B}\) is called “coefficient of compressibility” (K).

→ Poisson’s ratio,

σ = \(\frac{\text { lateral contraction strain }}{\text { longitudinal elongation }}=-\frac{\Delta D / D}{e / L}=-\frac{L \Delta D}{D \cdot e}\)

→ Theoretical limits of poisson’s ratio σ is – 1 to 0.5

Practical limits of poisson’s ratio σ is 0 to 0.5

→ Relation between Y, G, B and σ are

B = \(\frac{Y}{3(1-2 \sigma)}\)

η = \(\frac{Y}{2(1+\sigma)}\)

σ = \(\frac{3 B-2 G}{2(3 B+G)}\)

Y = \(\frac{9 \mathrm{~GB}}{3 \mathrm{~B}+\mathrm{G}}\)

→ Relation between volume stress and linear stress is \(\frac{\Delta V}{V}=\frac{\Delta L}{L}\)(1 – 2σ)

→ Strain energy = \(\frac{1}{2}\) × load × extension = \(\frac{1}{2}\) × F × e

→ Strain energy per unit volume = \(\frac{1}{2}\) × stress × strain or \(\frac{\text { stress }^2}{2 \mathrm{Y}}\) or \(\frac{\left({strain}^2\right) Y}{2}\)

→ When a body is heated and expansion is prevented then thermal stress will develop in the body.

- Thermal stress Y ∝ Δt
- Thermal force = YA ∝ Δt

→ When a wire of natural length L is elongated by tensions say T_{1} and T_{2} has final lengths l_{1} and l_{2} then Natural length of wire = L

= \(\frac{l_2 T_1-l_1 T_2}{\left(T_1-T_2\right)}\)

→ When a wire is stretched by a load has an elongation ‘e’. If the load is completely immersed in water then decrease In elongation e^{1} = Vρgl/(AY) where V is volume of load, ρ = density of water.