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

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)

An open-ended U-tube of uniform cross-sectional area contains water (density 10³ kg m⁻³ ). Initially the water level stands at 0.29 m from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density 800 kg m⁻³ is added to the left arm until its length is 0.1 m, as shown in the schematic figure below. The ratio $$\left( {{{{h_1}} \over {{h_2}}}} \right)$$ of the heights of the liquid in the two arms is :