JEE (Advanced) 2009 Paper-2

A cylindrical vessel of height 500 mm has an orifice (small hole) at its bottom. The orifice is initially closed and wateris filled in it up to height H. Now the top is completely sealed with a cap and theorifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being 200 mm. Find the fall in height (in mm) of water level due to opening ofthe orifice.
[Take atmospheric pressure $$= 1.0 \times 10^5 N/m^2$$, density of water $$= 1000 kg/m^3$$ and $$g = 10 m/s^2$$. Neglect any affect of surface tension.]

Two soap bubbles A and B are kept in a closed chamber where the air is maintained at pressure 8 N/m$$^2$$. The radii of bubbles A and B are 2 cm and 4 cm, respectively. Surface tension of the soap-water used to make bubbles is 0.04 N/m. Find the ratio $$\frac{n_B}{n_A}$$. Where $$n_A$$ and $$n_B$$ are the number of moles of air in bubbles A and B, respectively. [Neglect the effect of gravity.]

Three objects A. B and C are kept in a straight line on a frictionless horizontal surface. These have:masses m, 2m and m, respectively. The object A moves towards B with a speed 9 m/s and makes an elastic collision with it. Thereafter, B makes completely inelastic collision with C. All motions occur on the same straight line. Find the final speed (in m/s) of the object C.

A steady current I goes through a wire loop PQR having shape of a right angle triangle with PQ = 3x, PR = 4x and QR = 5x. If the magnitude of the magnetic field at P due to this loop is $$k\left(\frac{\mu_0 I}{48 \pi x}\right)$$, find the value of k.

A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10 m/s$$^2$$, find the work
done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest.

A solid sphere of radius R has a charge Q distributed in its volume with a charge densityy $$\rho = kr^a$$, where k and a are constants and r is the distance from its center. If the electric field at $$r = \frac{R}{2}$$ is $$\frac{1}{8}$$ times that at r = R, find the value of a,

A 20 cm long string, having a mass of 1.0 g, is fixed at both the ends. The tension in tne string is 0.5 N. The string is set into vibrations using an external vibrator of
frequency 100 Hz. Find the separation (in em) between the successive nodes on the string.