As shown in figure, when a spherical cavity (centered at O) of radius 1 is cut out of a uniform sphere of radius R (centered at C), the centre of mass of remaining (shaded) part of sphere is at G, i.e, on the surface of the cavity. R can be detemined by the equation :

The series combination of two batteries, both of the same emf 10 V, but different internal resistance of 20$$\Omega $$ and 5$$\Omega $$, is connected to the parallel combination of two resistors 30$$\Omega $$ and R $$\Omega $$. The voltage difference across the battery of internal resistance 20$$\Omega $$ is zero, the value of R (in $$\Omega $$) is : _______

An asteroid is moving directly towards the centre of the earth. When at a distance of 10R (R is the radius of the earth) from the earths centre, it has a speed of 12 km/s. Neglecting the effect of earths atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is 11.2 km/s) ? Give your answer to the nearest integer in kilometer/s _____.

The first member of the Balmer series of hydrogen atom has a wavelength of 6561 Å. The wavelength of the second member of the Balmer series (in nm) is: