Physics

Volume of 27 identical drops = volume of a bigger drop

$\Rightarrow 27*\frac{4}{3}\pi r^3=\frac{4}{3}\pi R^3$              

R³ = 27r³

R = 3r

Given potential of a small drop = 22v

$V_{bigger}=\frac{k\left(27q\right)}{R}=\frac{27kq}{3r}=9\frac{kq}{r}=9*22=198volt$

For closed vessel

P a T     [v = constant]

->P = kT

$\frac{0.4}{100}=\frac{1}{1}$

T = 250 K

M₁ ->Mass of satellite (1)

M₂ -> Mass of satellite (2)

M_P -> Mass of planet    

Now NLM (2) on S₁

_{$\frac{G{M}_1{M}_P}{R_1^2}=\frac{m_1{v}_1^2}{R_1}$}

Similarly v₂ =   $\sqrt[]{\frac{G{M}_P}{R_2}}$

$\frac{v_1}{v_2}=\sqrt[]{\frac{R_2}{R_1}}=\sqrt[]{\frac{800}{3200}}=\frac{1}{2}=\frac{1}{x}$                

l₁ = M. l of solid sphere about its diameter

$=\frac{2}{5}MR{\text{'}}^2=\frac{2}{5}M{\left(2R\right)}^2=\frac{8}{5}M{R}^2$               

l₂ = M. I of solid cylinder about its axis

$=\frac{M{R}^{12}}{2}=\frac{M{\left(2R\right)}^2}{2}=2M{R}^2$               

I₃ = M. I of solid circular disc about its diameter

$=\frac{M{R}^{12}}{4}=\frac{M{\left(2R\right)}^2}{2}=M{R}^2$               

I₄ = M. I of this circular ring about its diameter

  $=\frac{M{R}^{12}}{2}=\frac{M{\left(2R\right)}^2}{2}=2M{R}^2$              

$\Rightarrow 6M{R}^2+2M{R}^2=\frac{x8M{R}^2}{5}$               

x = 5

For equilibrium along the incline plane is given by,

F cos 60° = mg sin 60°

  $F=\frac{0.2*10*\sqrt[]{3}}{2*1}*2$              

  $=2\sqrt[]{3}N=\sqrt[]{12}N$              

$f=\frac{m{v}^2}{R}$            

$\Rightarrow \frac{m{v}^2}{R}=\mu N$

 

$\frac{m{v}^2}{R}=\mu mg$

$\Rightarrow 30=\sqrt[]{\mu *75*10}$                

  $\mu =\frac{30*30}{750}=1.2$              

              Now, R = 48 m (new Radius of curvature)

    $\mu =1.2$

$V_{max}=\sqrt[]{\mu Rg}$

$=\sqrt[]{1.2*48*10}$            

$=24m/sec$               

Using snell's law –

sin I = $\mu sinr$

sin45° = $\mu sin30°$

$\mu$$=$$\frac{sin45}{sin30}$$=$$\frac{1}{\text{exception 25:}}$$*$$2$$=$√2

$\frac{\Delta z}{z}*100=3\frac{\Delta x}{x}*100+\frac{1}{2}\frac{\Delta y}{y}*100$

$=3*4+\frac{1}{2}*12$

$=\left(12+6\right)\mathrm{\%}$

$=18\mathrm{\%}$

$l_g=nk$

$l_{g}G=\left(l-l_g\right)s$

$\Rightarrow nkG=\left(l-nk\right)s\Rightarrow l=\frac{nkG}{s}+nk=nk\frac{\left(G+s\right)}{s}$

$l=\frac{nk}{s}\left(G+s\right)$

m = 5 * 10^-17kg ; k = 1.38 * 10^-23 Jk^-1

$V_{rms}=\sqrt[]{\frac{3RT}{M}}=\sqrt[]{\frac{3NkT}{Nm}}=\sqrt[]{\frac{3kT}{m}}=\sqrt[]{\frac{3*1.38*10^{-23}*300}{5*10^{-17}}}$

$=\sqrt[]{18*13.8*10^{-6}}$

$=15.7*10^{-3}m/s$

$=15mm/s$