A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre ‘O’ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is :

A plane electromagnetic wave, has
frequency of 2.0 $$ \times $$ 10¹⁰ Hz and its energy density is 1.02 $$ \times $$ 10^–8 J/m³ in vacuum. The amplitude of the magnetic field of the wave is close to
( $${1 \over {4\pi {\varepsilon _0}}} = 9 \times {10^9}{{N{m^2}} \over {{C^2}}}$$ and speed of light
= 3 $$ \times $$ 10⁸ ms^–1)

A charged particle (mass m and charge q)
moves along X-axis with velocity V₀. When it
passes through the origin it enters a region having uniform electric field
$$\overrightarrow E = - E\widehat j$$ which extends upto x = d.
Equation of path of electron in the region x > d is

A cylindrical vessel containing a liquid is rotated about its axis so that the liquid rises at its sides as shown in the figure. The radius of vessel is 5 cm and the angular speed of rotation is $$\omega $$ rad s^–1. The difference in the height, h (in cm) of liquid at the centre of vessel and at the side will be :