Two large circular discs separated by a distance of 0.01 m are connected to a battery via a switch as shown in the figure. Charged oil drops of density 900 kg m⁻³ are released through a tiny hole at the center of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of 200 V across the discs. As a result, an oil drop of radius 8 $$ \times $$ 10⁻⁷ m stops moving vertically and floats between the discs. The number of electrons present in this oil drop is ________.
(neglect the buoyancy force, take acceleration due to gravity = 10 ms⁻² and charge on an electron (e) = 1.6 $$ \times $$ 10^–19 C)

A point charge q of mass m is suspended vertically by a string of length l. A point dipole of dipole moment $$\overrightarrow p $$ is now brought towards q from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is N $$ \times $$ (mgh), where g is the acceleration due to gravity, then the value of N is _________ .
(Note that for three coplanar forces keeping a point mass in equilibrium, $${F \over {\sin \theta }}$$ is the same for all forces, where F is any one of the forces and $$\theta $$ is the angle between the other two forces)

A beaker of radius r is filled with water (refractive index $${4 \over 3}$$) up to a height H as shown in the figure on the left. The beaker is kept on a horizontal table rotating with angular speed $$\omega$$. This makes the water surface curved so that the difference in the height of water level at the center and at the circumference of the beaker is h (h << H, h << r), as shown in the figure on the right. Take this surface to be approximately spherical with a radius of curvature R. Which of the following is/are correct? (g is the acceleration due to gravity)

A student skates up a ramp that makes an angle 30$$^\circ$$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed v₀ and wants to turn around over a semicircular path xyz of radius R during which he/she reaches a maximum height h (at point y) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then (g is the acceleration due to gravity)