1

### JEE Main 2017 (Online) 9th April Morning Slot

Two tubes of radii r1 and r2, and lengths l1 and l2 , respectively, are connected in series and a liquid flows through each of them in stream line conditions. P1 and P2 are pressure differences across the two tubes.

If P2 is 4P1 and l2 is ${{{1_1}} \over 4}$, then the radius r2 will be equal to :
A
r1
B
2r1
C
4r1
D
${{{r_1}} \over 2}$

## Explanation

We know,

Rate of flow of liquid through narrow tube,

${{dv} \over {dt}}$ = ${{\pi {{\Pr }^4}} \over {8\eta l}}$

Both tubes are connected in series so rate of flow of liquid is same.

$\therefore\,\,\,$ ${{\pi {P_1}r_1^4} \over {8\eta {l_1}}}$ = ${{\pi {P_2}r_2^4} \over {8\eta {l_2}}}$

$\Rightarrow $$\,\,\, {{{P_1}{r_1}^4} \over {{l_1}}} = {{{P_2}{r_2}^4} \over {{l_2}}} Given, P2 = 4P1 and {l_2} = {{{l_1}} \over 4} \Rightarrow$$\,\,\,$ ${{{P_1}{r_1}^4} \over {{l_1}}}$ = ${{4{P_1}{r_2}^4} \over {{{{l_1}} \over 4}}}$

$\Rightarrow $$\,\,\, {r_2}^4 = {{{r_1}^4} \over {16}} \Rightarrow$$\,\,\,$ r2 = ${{{r_1}} \over 2}$
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### JEE Main 2018 (Offline)

A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere, $\left( {{dr \over r}} \right)$ is:
A
${{mg} \over {Ka}}$
B
${{Ka} \over {mg}}$
C
${{Ka} \over {3mg}}$
D
${{mg} \over {3Ka}}$

## Explanation Because of m mass the extra pressure created is,

$\Delta$P = ${{mg} \over a}$

And Bulk modulus, $\beta$ = ${{\Delta P} \over {{{\Delta V} \over V}}}$

Given $\beta$ = K

$\therefore\,\,\,$ K = ${{{{mg} \over a}} \over {{{\Delta V} \over V}}}$

We know volume of sphere,

V = ${4 \over 3}\pi {r^3}$

$\therefore\,\,\,$ ${{dV} \over V}$ = 3 ${{dr} \over r}$

$\therefore\,\,\,$ K = ${{{{mg} \over a}} \over {3{{dr} \over r}}}$

$\Rightarrow$$\,\,\,$ ${{dr} \over r}$ = ${{mg} \over {3Ka}}$
3

### JEE Main 2018 (Online) 15th April Evening Slot

When an air bubble of radius r rises from the bottom to the surface of a lake its radius becomes ${{5r} \over 4}.$ Taking the atmospheric pressure to be equal to 10 m height of water column, the depth of the lake would approximately be (ignore the surface tension and the effect of temperature) :
A
11.2 m
B
8.7 m
C
9.5 m
D
10.5 m
4

### JEE Main 2018 (Online) 15th April Evening Slot

The value closest to the thermal velocity of a Helium atom at room temperature (300 K) in ms-1 is :
[kB =1.4 $\times$ 10-23 J/K; mHe = 7 $\times$ 10 -27 kg ]
A
1.3 $\times$ 104
B
1.3 $\times$ 103
C
1.3 $\times$ 105
D
1.3 $\times$ 102