Physics

$i=\frac{v_0}{R}\left(1-e^{\frac{tR}{L}}\right)$

$U=\frac{1}{2}L{i}^2$

$\frac{dU}{dt}=Li\frac{di}{dt}$

$2L*\frac{R}{L}*\frac{v_0^2}{R^2}\left(1-e^{\frac{tR}{L}}\right)e^{\frac{tR}{L}}=\frac{v_0^2}{R}\left(1-e^{\frac{tR}{L}}\right)$

$t=\frac{L}{R}In2$

$y=A\sin ?\frac{2\pi t}{T}$

$t_2-t_1=\frac{T}{2\pi }\left. [{\sin }^{-1}?\frac{x_1}{A}-{\sin }^{-1}?\frac{x_2}{A}\right]$

Deformation energy per unit volume,

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

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

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

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

$=1.5\mu J$

$U=a{P}^4$

$\frac{PV}{\gamma -1}=a{P}^4$

$P^{-3}V=a\left(\gamma -1\right)$

$P{V}^{-1/3}=constant$

$ComparingwithP{V}^x=constant$

$x=-\frac{1}{3}$

$C=\frac{R}{\gamma -1}+\frac{R}{1-x}=\frac{R}{\frac{5}{3}-1}+\frac{R}{1+\frac{1}{3}}=\frac{3R}{2}+\frac{3R}{4}$

$C=\frac{9R}{4}$

$b=a\sqrt[]{1-e^2}$

$\frac{dA}{dt}=\frac{L}{2m}$

$\frac{A}{T}=\frac{L}{2m}$

$T=\frac{2mA}{L}=\frac{2m\pi ab}{L}=\frac{2m\pi a^2\sqrt[]{1-e^2}}{L}$

Angular retardation,

Let S be the distance between Two ends a be the constant acceleration

As we know $v^2-u^2=2aS$

or, $aS=\frac{v^2-u^2}{2}$

Let $v_c$  be velocity at mid point.

Therefore, $v_c^2-u^2=2a\frac{S}{2}$

$v_c^2=u^2+aS$

$v_c^2=u^2+\frac{v^2-u^2}{2}$

$v_c=\sqrt[]{\frac{u^2+v^2}{2}}$

P? V indicator diagram for isobaric

P? V indicator diagram for isochoric process

P? V indicator diagram for isothermal process

When the ring rotates about its axis with a uniform frequency fHz, the current flowing in the ring is

I=q/T=qf

Magnetic field at the centre of the ring is

When Two rods are connected in series

$Q=\frac{A\left(T_1-T_2\right)t}{\frac{d_1}{K_1}+\frac{d_2}{K_1}}=\frac{A\left(T_1-T_2\right)t}{\left(d_1+d_2\right)/K}$

$\frac{d_1+d_2}{K}=\frac{d_1}{K_1}+\frac{d_2}{K_2}$

$K=\frac{\left(d_1+d_2\right)}{\frac{d_1}{K_1}+\frac{d_2}{K_1}}$