A light disc made of aluminium (a nonmagnetic material) is kept horizontally and is free to rotate about its axis as shown in the figure. A strong magnet is held vertically at a point above the disc away from its axis. On revolving the magnet about the axis of the disc, the disc will (figure is schematic and not drawn to scale)

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 :

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