A bottle has an opening of radius a and length b. A cork of length b and radius (a + $$\Delta $$a) where ($$\Delta $$a < < a) is compressed to fit into the opening completely (See figure). If the bulk modulus of cork is B and frictional coefficient between the bottle and cork is $$\mu $$ then the force needed to push the cork into the bottle is :

A conducting metal circular-wire-loop of radius r is placed perpendicular to a magnetic field which varies with time as
B = B₀e$${^{{{ - t} \over r}}}$$ , where B₀ and $$\tau $$ are constants, at time t = 0. If the resistance of the loop is R then the heat generated in the loop after a long time (t $$ \to $$ $$\infty $$) is :

Within a spherical charge distribution of charge density $$\rho $$(r), N equipotential surfaces of potential V₀, V₀ + $$\Delta $$V, V₀ + 2$$\Delta $$V, .......... V₀ + N$$\Delta $$V ($$\Delta $$ V > 0), are drawn and have increasing radii r₀, r₁, r₂,..........r_N, respectively. If the difference in the radii of the surfaces is constant for all values of V₀ and $$\Delta $$V then :

In an engine the piston undergoes vertical simple harmonic motion with amplitude 7 cm. A washer rests on top of the piston and moves with it. The motor speed is slowly increased. The frequency of the piston at which the washer no longer stays in contact with the piston, is close to :