One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the $P-V$ diagrams below. In cycle I, processes $a, b, c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes $a^{\prime}, b^{\prime}, c^{\prime}$ and $d^{\prime}$ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle $\mathrm{I}$ is $W_I$ and that during cycle II is $W_{I I}$. The ratio $W_I / W_{I I}$ is ________.

$S_1$ and $S_2$ are two identical sound sources of frequency $656 \mathrm{~Hz}$. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4 \sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$ on a circular path around $O$, as shown in the figure. There are three points $P, Q$ and $R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$. The source $S_1$ can move along direction $O P$.

[Given: The speed of sound in air is $324 \mathrm{~m} \mathrm{~s}^{-1}$ ]

When only $S_2$ is emitting sound and it is at $Q$, the frequency of sound measured by the detector in $\mathrm{Hz}$ is _________.

$S_1$ and $S_2$ are two identical sound sources of frequency $656 \mathrm{~Hz}$. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4 \sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$ on a circular path around $O$, as shown in the figure. There are three points $P, Q$ and $R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$. The source $S_1$ can move along direction $O P$.

[Given: The speed of sound in air is $324 \mathrm{~m} \mathrm{~s}^{-1}$ ]

Consider both sources emitting sound. When $S_2$ is at $R$ and $S_1$ approaches the detector with a speed $4 \mathrm{~m} \mathrm{~s}^{-1}$, the beat frequency measured by the detector is _________ $\mathrm{Hz}$.

A cylindrical furnace has height $(H)$ and diameter $(D)$ both $1 \mathrm{~m}$. It is maintained at temperature $360 \mathrm{~K}$. The air gets heated inside the furnace at constant pressure $P_a$ and its temperature becomes $T=360 \mathrm{~K}$. The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \mathrm{~m}$ and height $h=9 \mathrm{~m}$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density $\rho_a=$ $1.2 \mathrm{~kg} \mathrm{~m}^{-3}$, pressure $P_a$ and temperature $T_a=300 \mathrm{~K}$ enters the furnace. Assume air as an ideal gas, neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.

[Given: The acceleration due to gravity $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ and $\pi=3.14$ ]

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _________ $\mathrm{gm} \mathrm{s}^{-1}$.