Physics Mechanical Properties of Solids

This refers to the highest limit of pressure which the body can tolerate in order to regain its original shape after removing the pressure. If the pressure exceeds the elastic limit, the body will never be able to regain its original shape agian. example: rubber stretching.

This is the case of fluids or gases. In such scenarios, the material does not have any limit to its flexibility and can be deformed to any extent. Such materials do not possess any stiffness and cannot regain their original form as they show no resistance towards any kind of pressure.

A diamond crystal with young modulus of more than 1050 Gpa is known to be the element which has the highest elastic moduli found in nature. This is possible due to the high electron density as well as strong covalent bonds between the carbon atoms.

$dm=\left(\frac{m}{L}\right)dx$

$\therefore T=\frac{m{\omega }^2}{2L}\left(L^2-x^2\right)$

$\therefore ?L={\int }_0^L\frac{m{\omega }^2}{2L\pi r^{2}Y}\left(L^2-x^2\right)dx$

$=?L=\frac{m{\omega }^2{L}^2}{3\pi r^{2}Y}$

Deformation energy per unit volume,

$U=\int dU=\int \frac{P^2}{2B}Ady$

$=\int \frac{{\rho }^2{g}^{2}y}{2B}Ady$

$U=\frac{{\rho }^2{g}^2}{2B}\frac{h^3}{3}A$

$=\frac{{\rho }^2{g}^2{h}^2}{6B}\left(volume\right)$

$=1.5\mu J$

If $\mu$  is Poisson's ratio,

Y = 3K (1 - 2 $\mu$ )      ……… (1)

and Y = 2 $n\left(1+\mu \right)$  ……… (2)

With the help of equations (1) and (2), we can write

  $\frac{3}{Y}=\frac{1}{\eta }+\frac{1}{3k}\Rightarrow K=\frac{\eta Y}{9\eta -3Y}$

If   $\mu$ is Poisson's ratio,

Y = 3K (1 - 2  $\mu$ )      ……… (1)

and Y = 2  $n\left(1+\mu \right)$ ……… (2)

With the help of equations (1) and (2), we can write

$\frac{3}{Y}=\frac{1}{\eta }+\frac{1}{3k}\Rightarrow K=\frac{\eta Y}{9\eta -3Y}$            

dm = (m/L)dx
∴ T = (mω²/2L) (L² - x²)
∴ ΔL = ∫? (mω²/2Lπr²Y) (L² - x²)dx
= ΔL = mω²L²/3πr²Y

Stress $=\frac{\mathrm{F}}{\mathrm{A}}=\frac{\mathrm{T}}{\mathrm{A}}=\frac{\mathrm{W}}{\mathrm{A}}$

Resistance of P & Q should be approximately equal as it decreases error in experiment.