Consider one mole of helium gas enclosed in a container at initial pressure P₁ and volume V₁. It expands isothermally to volume 4V₁. After this, the gas expands adiabatically and its volume becomes 32V₁. The work done by the gas during isothermal and adiabatic expansion processes are W_iso and W_adia, respectively. If the ratio $${{{W_{iso}}} \over {{W_{adia}}}}$$ = f ln 2, then f is ______.

Shown in the figure is a semicircular metallic strip that has thickness t and resistivity $$\rho $$. Its inner radius is R₁ and outer radius is R₂. If a voltage V₀ is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage $$\Delta $$V develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale)

A particle of mass m moves in circular orbits with potential energy V(r) = Fr, where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle’s orbit is denoted by R and its speed and energy are denoted by v and E, respectively, then for the n^th orbit (here h is the Planck’s constant)

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity X as follows: [position] = [X^{$$\alpha $$}]; [speed] = [X^{$$\beta $$} ]; [acceleration] = [X^p]; [linear momentum] = [X^q]; [force] = [X^r]. Then