A conducting solid sphere of radius $R$ and mass $M$ carries a charge $Q$. The sphere is rotating about an axis passing through its center with a uniform angular speed $\omega$. The ratio of the magnitudes of the magnetic dipole moment to the angular momentum about the same axis is given as $\alpha \frac{Q}{2 M}$. The value of $\alpha$ is ___________.

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency $v_1$ and ejects the electron with a kinetic energy of 10 eV . The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency $v_2$. The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV . It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV) is ____________.

An ideal monatomic gas of $n$ moles is taken through a cycle $W X Y Z W$ consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic $V-T$ diagram. The volume of the gas at $W, X$ and $Y$ points are, $64 \mathrm{~cm}^3, 125 \mathrm{~cm}^3$ and $250 \mathrm{~cm}^3$, respectively. If the absolute temperature of the gas $T_W$ at the point $W$ is such that $n R T_W=1 \mathrm{~J}$ ( $R$ is the universal gas constant), then the amount of heat absorbed (in J ) by the gas along the path $X Y$ is ___________.

A geostationary satellite above the equator is orbiting around the earth at a fixed distance $r_1$ from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance $r_2$ from the center of the earth, such that $r_1=1.21 r_2$. The time period of the second satellite as measured from the geostationary satellite is $\frac{24}{p}$ hours. The value of $p$ is _________.