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)

The filament of a light bulb has surface area 64 mm² . The filament can be considered as a black body at temperature 2500 K emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of 100 m. Assume the pupil of the eyes of the observer to be circular with radius 3 mm. Then

(Take Stefan-Boltzmann constant = 5.67 $$ \times $$ 10⁻⁸ Wm⁻²K⁻⁴ , Wien’s displacement constant = 2.90 $$ \times $$ 10⁻³ m-K, Planck’s constant = 6.63 $$ \times $$ 10⁻³⁴ Js, speed of light in vacuum = 3.00 $$ \times $$ 10⁸ ms⁻¹)

A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to scale)