Class 11th

  $C{N}^{-},N{O}^{+}and{O}_2^{2+}$ have bond order = 3

$C_{P,m}=C_{v,m}+R$

$C_{v,m}=20.785-8.314=12.471J{K}^{-1}mo{l}^{-1}$

$n=\frac{5000}{12.471*200}=\frac{25}{12.471}\approx 2$

$K{O}_2,N{O}_2,Cl{O}_2,NO$ are paramagnetic.

$\overrightarrow{F}=5\widehat{i}+3\widehat{j}-7\widehat{k}$

$\overrightarrow{r}=2\widehat{i}+2\widehat{j}+\widehat{k}$

$\overrightarrow{\tau }=\left|\overrightarrow{r}*\overrightarrow{F}\right|=\left|\begin{array}{ccc}\hfill \widehat{i}\hfill & \hfill \widehat{j}\hfill & \hfill \widehat{k}\hfill \\ \hfill 2\hfill & \hfill 1\hfill \\ \hfill 5\hfill & \hfill 3\hfill & \hfill -7\hfill \end{array}\right|\begin{array}{cc}\hfill =\widehat{i}\left(-k_1-3\right)-\widehat{j}\left(-14-5\right)+\widehat{k}\left(6-10\right)\hfill & \hfill =-17\widehat{i}+19\widehat{j}-4\widehat{k}\hfill \end{array}$

$\frac{\Delta K}{K_i}*100=\frac{K_f-K_i}{K_i}*100$

$=\frac{\frac{P_f^2}{2m}-\frac{P_i^2}{2m}}{\frac{P_i^2}{2m}}*100=\frac{P_f^2-P_i^2}{P_i^2}*100$

$=\frac{{\left(1.2P_i\right)}^2-{\left(P_i\right)}^2}{P_i^2}*100=\frac{1.44P_i^2-P_i^2}{P_i^2}*100$

= 44%

m₁ = 5kg

m₂ = 3kg

As m₁ is in rest

T = m_gg = 30

N = m₁ g cos

T = $m_{1}gsin\theta \Rightarrow 50sin\theta =30$

sin $=\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\Rightarrow \theta =37°$

$\therefore N=50cos\theta =50cos37°=50*\frac{4}{5}=40N$

$\frac{h}{2}=\frac{1}{2}g{t}_1^2$

h = $\frac{1}{2}g{\left(t_1+t_2\right)}^2$

from (i) and (ii) :-

$\frac{1}{2}g{t}_1^2=\frac{g}{4}{\left(t_1+t_2\right)}^2$

$2t_1^2={\left(t_1+t_2\right)}^2$

$\sqrt[]{2}{t}_1=t_1+t_2$

$t_1=\frac{t_2}{\sqrt[]{2}-1}=\left(\sqrt[]{2}+1\right)t_2$

$t_2=\left(\sqrt[]{2}-1\right)t_1$

m = 1kg ; $\Delta U=\frac{-G{m}_{e}m}{\left(R+h\right)}-\left(\frac{G{m}_{e}m}{R}\right)$

$\Delta U=\frac{GMm}{R}-\frac{GMm}{R+h}$

$=m{g}_{0}R-m{g}_0\frac{R}{\left(1+\frac{h}{R}\right)}=m{g}_0\left\{1-\frac{1}{\left(1+\frac{3R}{R}\right)}\right\}$

$=m{g}_{0}R\left(\frac{3}{4}\right)$

$=\frac{3}{4}*1*10*6400km$

$=48000*10^{3}J.$

$=48MJ.$

The two conditions for equilibrium in physics are

1. Translational Equilibrium: The net force acting on the object must be zero, which means all the horizontal and vertical forces cancel each other out.
2. Rotational Equilibrium: The net torque must also be zero, indicating that the clockwise and counter-clockwise torques are balanced. 

To reach equilibrium, all the forces and torques acting on an object need to balance each other out. In short, every action force must be met with an equal and opposite reaction. Static equilibrium happens when the object is at rest. Dynamic equilibrium is when it's moving at a constant velocity. Usually, we draw a free-body diagram to visualise these forces. Further, to prove that the object has reached equilibrium state, we use Newton's laws to prove that both the net force and net torque equal zero.