The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2 L}{5}$. In thermodynamic equilibrium, the piston is a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P=\frac{k L}{A} \alpha$, then the value of $\alpha$ is __________.

In a Young's double slit experiment, a combination of two glass wedges $A$ and $B$, having refractive indices 1.7 and 1.5, respectively, are placed in front of the slits, as shown in the figure. The separation between the slits is $d=2 \mathrm{~mm}$ and the shortest distance between the slits and the screen is $D=2 \mathrm{~m}$. Thickness of the combination of the wedges is $t=12 \mu \mathrm{~m}$. The value of $l$ as shown in the figure is 1 mm . Neglect any refraction effect at the slanted interface of the wedges. Due to the combination of the wedges, the central maximum shifts (in mm ) with respect to O by ____________.

A projectile of mass 200 g is launched in a viscous medium at an angle $60^{\circ}$ with the horizontal, with an initial velocity of $270 \mathrm{~m} / \mathrm{s}$. It experiences a viscous drag force $\vec{F}=-c \vec{v}$ where the drag coefficient $c=0.1 \mathrm{~kg} / \mathrm{s}$ and $\vec{v}$ is the instantaneous velocity of the projectile. The projectile hits a vertical wall after 2 s . Taking $e=2.7$, the horizontal distance of the wall from the point of projection (in m ) is ___________.

An audio transmitter $(T)$ and a receiver $(R)$ are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the $X$ axis. They are pulled from the equilibrium position in opposite directions along the $X$ axis by a small angular amplitude $\theta_0=\cos ^{-1}(0.9)$ and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is $330 \mathrm{~m} / \mathrm{s}$, the maximum variation in the frequency (in Hz ) as measured by the receiver (Take the acceleration due to gravity $g=10 \mathrm{~m} / \mathrm{s}^2$ ) is _____________.