Physics

$n_1=1\to monotomic$

$n_2=3\to diatomic$ $C_v=\frac{{\alpha }^{2}R}{4}J/mo{l}^{-k}$

As, Cv _(mix) = $\frac{1*\frac{3}{2}R+3*\frac{5}{2}R}{4}=\frac{9R}{4}=\frac{{\alpha }^{2}R}{4}$

$\therefore \alpha =3$

For adiabatic process – PV^Y = const

$T_1{V_1}^{Y-1}=T_2{V}_2^{Y-1}$

$\frac{T_2}{T_1}={\left(\frac{v_1}{v_2}\right)}^{Y-1}={\left(\frac{d_2}{d_1}\right)}^{Y-1}={\left(32\right)}^{\left(\frac{7}{5}-1\right)}={\left(32\right)}^{2/5}$

$={\left(2\right)}^2=4$

$g_{\left(ath\right)}=g_{\left(atdepth\alpha h\right)}$ h << R

$\Rightarrow g\left(1-\frac{2h}{R}\right)=g\left(1-\frac{\alpha h}{R}\right)$

$\begin{array}{l}1-\frac{2h}{R}=1-\frac{\alpha h}{R}\Rightarrow \frac{2h}{R}=\alpha \frac{h}{R}\\ \alpha =2\end{array}$

$R=\frac{u^{2}sin\left(2*45°\right)}{g}=\frac{u^2}{g}$

$\frac{R}{2}=\frac{u^2}{2g}=\frac{u^{2}sin20}{g}$

$sin2\theta =\frac{1}{2}$

$\Rightarrow 2\theta =30°\Rightarrow \theta =15°$

$I_{OO\text{'}}=\left[\frac{1}{4}M{\left(\frac{a}{2}\right)}^2\right]*2+\left[\frac{5}{4}m{\left(\frac{a}{2}\right)}^2\right]*2$

$=\frac{6M{a}^2}{8}=\frac{3}{4}M{a}^2$

x=3

loss in k = Gain in U spring

$\Rightarrow \frac{3E}{4}=\frac{1}{2}k{\left(\frac{25}{100}\right)}^2$

$\Rightarrow \frac{3E}{4}=\frac{k}{32}\Rightarrow k=\frac{3F}{4}*32=24E$                

d₁ = 2mm = 2r

$\rho =1750kg/m^3$ coeff of viscosity H

  $\frac{d}{dt}\left(2r\right)=0.35cm/s$              

here,   $\left(\frac{4}{3}\pi r^3\right)\rho g=6\pi nrv$

$\Rightarrow \eta =\frac{4r^2\rho g}{3*6v}=\frac{2r^2\rho g}{9v}$

$\frac{350}{9*35}=\frac{10}{9}?1$  

An excellent example of pure rotational motion is the motion of the blades of a ceiling fan when it is switched on. In this type of motion, the object rotates about a fixed axis, and all particles of the object move in circles centred on that axis. Windmills and turbines also follow the same rotational mechanics. 

${\upsilon }_0=320Hz.v_{train}=36km/h=\frac{36*1000}{60*60}=10m/s$

${\upsilon }_{\left(reflectedSound\right)}={\upsilon }_0\left(\frac{v}{v-v_{train}}\right)$

$=320\left(\frac{330}{330-10}\right)=330Hz$

$\Rightarrow {\upsilon }_{echo}={\upsilon }_{reflected}\left(\frac{v+v_{train}}{V}\right)$

$=330\left(\frac{330+10}{330}\right)=340Hz.$