Physics

  $k_{eq}=\frac{2k\cdot k}{3k}+k=\frac{5k}{3}$

Angular frequency of oscillation $\left(\omega \right)=\sqrt[]{\frac{k_{eq}}{m}}$  

$\omega =\sqrt[]{\frac{5k}{3m}}$            

Period of oscillation (t) = $\frac{2\pi }{\omega }=2\pi \sqrt[]{\frac{3m}{5k}}$  

$=\pi \sqrt[]{\frac{12m}{5k}}$                                                   

$T=2\pi \sqrt[]{\frac{I}{g}}$

$g=\frac{4{\pi }^{2}I}{T^2}$

$\frac{\Delta g}{g}=\frac{\Delta I}{I}+\frac{2\Delta T}{T}$

$=\frac{0.2}{20}+2\left(\frac{1}{40}\right)$

= 6%

$\frac{1}{2}{\epsilon }_0{E}^2=\frac{B^2}{2{\mu }_0}$

$?E=CBandC=\frac{1}{{\mu }_0{\epsilon }_0}$          

The magnitude of resultant vector (R)

$=\sqrt[]{a^2+b^2+2abcos\theta }$            

here a = b = A

then R = $\sqrt[]{A^2+A^2+2A^{2}cos\theta }$  

$=A\sqrt[]{2}\sqrt[]{1+cos\theta }$

$=\sqrt[]{2}A\sqrt[]{2co{s}^2\frac{\theta }{2}}$

$=2Acos\frac{\theta }{2}$      

In air $F=\frac{1}{4\pi {\in }_0}\frac{q_1{q}_2}{r^2}$  

In medium $F\text{'}=\frac{1}{4\pi \left(k{\in }_0\right)}\frac{q_1{q}_2}{r^2}$  

$F\text{'}=\frac{F_0}{K}$

$a=\frac{\left(m_1-m_2\right)g}{\left(m_1+m_2\right)}=\frac{g}{8}$

8m₁ – 8m₂ = m₁ + m₂

7m₁ = 9m₂

$\frac{m_1}{m_2}=\frac{9}{7}$          

U = nC_vT

-> $U=n_1{C}_{v_1}T+n_2{C}_{v_2}T$  

-> $8*\frac{5R}{2}*T+4*\frac{5R}{2}*T$

= 30RT

$\frac{d?}{dt}=10t-3$            

at t = 2, V = 17

$i=\frac{V}{R}=\frac{17}{8}A$              

  $X=\overline{\left(A+\overline{B}\right)+\left(\overline{A}+B\right)}$

$\left(\overline{A+\overline{B}}\right)\cdot \left(\overline{\overline{A}+B}\right)$

$\left(\overline{A}\cdot \overline{B}\right)\cdot \left(\overline{A}\cdot \overline{B}\right)$

$\left(\overline{A}\cdot B\right)\cdot \left(A\cdot \overline{B}\right)=\overline{A}\cdot B\cdot A\cdot \overline{B}=0$  

  $\frac{1}{f}=\left(\mu -1\right)\left(\frac{2}{R}\right)$                    f = 20 cm

$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$

$\frac{1}{v}+\frac{1}{10}=\frac{1}{20}$