A boy of mass 4 kg is standing on a piece of wood having mass 5 kg. If the coefficient of friction between the wood and the floor is 0.5, the maximum force that the boy can exert on the rope so that the piece of wood does not move from its place is __________ N. (Round off to the Nearest Integer) [Take g = 10 ms^-2 ]

A body of mass 1 kg rests on a horizontal floor with which it has a coefficient of static friction $${1 \over {\sqrt 3 }}$$. It is desired to make the body move by applying the minimum possible force F N. The value of F will be ____________. (Round off to the Nearest Integer) [Take g = 10 ms^$$-$$2 ]

A particle of mass m moves in a circular orbit in a
central potential field U(r) = U₀r⁴. If Bohr's quantization conditions
are applied, radii of possible
orbitals r_n vary with $${n^{{1 \over \alpha }}}$$, where $$\alpha$$ is ____________.

Seawater at a frequency f = 9 $$\times$$ 10² Hz, has permittivity $$\varepsilon $$ = 80$$\varepsilon $$₀ and resistivity $$\rho$$ = 0.25 $$\Omega$$m. Imagine a parallel plate capacitor is immersed in seawater and is driven by an alternating voltage source V(t) = V₀ sin(2$$\pi$$ft). Then the conduction current density becomes 10^x times the displacement current density after time t = $${1 \over {800}}$$s. The value of x is _____________. (Given : $${1 \over {4\pi {\varepsilon _0}}} = 9 \times {10^9}$$ Nm²C^$$-$$2)