Q: A U - shaped tube contains a liquid of density rho and it is rotated about the left dotted line as shown in the figure. Find the difference in the levels of liquid column.

Weight of liquid of height $ H = d ^ 2 4 H g $ (i) Let us consider a mass dm situated at a distance $x$ from $A$ as shown in the figure. The centripetal force required for the mass to rotate $=( dm ) ^ 2 $  The total centripetal force required for the mass of length $L$ to rotate $= _ 0 ^ 1 ( x d ^ 2 4 ) xdx ( x ^ 2 )= d ^ 2 4 ^ 2 1^ 2 2 $ (ii) This centripetal force required is provided by the weight of liquid of height 'H' From (i) and (ii) \( aligned & d ^ 2 4 H g = d ^ 2 4 ^

Question 1

Pressure inside two soap bubbles are 1.01 and 1.02 atmosphere, respectively. The ratio of their volumes is :

Accepted Answer

According to question, pressure inside, 1st soap bubble, P_1=P_1-P_0=0.01= 4 T R_1 .....(i) And P_2=P_2-P_0=0.02= 4 T R_2 .....(ii) Dividing, equation (ii) by (i), 1 2 = R_2 R_1 R_1=2 R_2 Volume V= 4 3 R^3 V_1 V_2 = R_1^3 R_2^3 = 8 R_2^3 R_2^3 = 8 1

Question 2

A cube of ice floats partly in water and partly in kerosene oil. The radio of volume of ice immersed in water to that in kerosene oil (specific gravity of Kerosene oil =0.8, specific gravity of ice =0

Accepted Answer

Let V_1 be, the volume immersed in water. V_2 be the volume immersed in oil. In equilibrium condition, v_1 _w g+v_2 _0 g= (v_1+v_2 ) _e g v_1+ v_2 _0 _w = (v_1+v_2 ) _c _w v_1+0.8 v_2=0.9 v_1+0.9 v_2 0.1 v_1=0.1 v_2 v_1: v_2=1: 1

Question 3

Eight equal drops of water are falling through air with a steady speed of 10 ~cm / s . If the drops coalesce, the new velocity is:

Accepted Answer

Let r be the radius of small drops of water. R = radius of big drop formed as volume remain same. 8 . 4 3 r ^3- 4 3 R ^3 R =2 r Terminal velocity, aligned & v _ T = 2 9 ( - ) r ^2 ~g \ & v _ T r ^2 \ & v _1 v _2 = ( r R )^2 \ & 10 v _2 = ( 1 2 )^2 ( v _1=10 ~cm / s given ) \ & v _2=40 ~cm / s aligned

Question 4

If 1000 droplets of water of surface tension 0.07 N / m . having same radius 1 ~mm each, combine to from a single drop. In the process the released surface energy is( Take . = 22 7 )

Accepted Answer

We have V_f=V_i 4 3 r_ f ^3=1000 4 3 r _ i ^3 r _ f ^3=1000 r _ i ^3 r_f=10 r_i So, released energy = Initial surface energy - final surface energy =1000 T 4 r _ i ^2- T 4 r _ f ^2 =4 T (1000 r _ i ^2- r _ f ^2 ) =4 0.07 (1000 r _ i ^2-100 r _ i ^2 ) =4 0.07 900 r_i^2 =4 63 10^ -6 =7.92 10^ -4 ~J

Question 5

A capillary tube is attached horizontally to a constant heat arrangement. If the radius of the capillary tube is increased by 25 \%, then the rate of flow of liquid will change nearly by

Accepted Answer

Liquid volume coming out of the tube per second aligned & V= p r^4 8 l \ & \ & V_2 V_1 = [ r_2 r_1 ]^4 V_2=V_1 [ 125 100 ]^4=V_1 ( 5 4 )^4 \ & \ & So, now V V = V_2-V_1 V_1 = (244) V_1-V_1 V_1 =144 \% aligned

Question 6

A U - shaped tube contains a liquid of density rho and it is rotated about the left dotted line as shown in the figure. Find the difference in the levels of liquid column.

Accepted Answer

Weight of liquid of height $ H = d ^ 2 4 H g $ (i) Let us consider a mass dm situated at a distance $x$ from $A$ as shown in the figure. The centripetal force required for the mass to rotate $=( dm ) ^ 2 $  The total centripetal force required for the mass of length $L$ to rotate $= _ 0 ^ 1 ( x d ^ 2 4 ) xdx ( x ^ 2 )= d ^ 2 4 ^ 2 1^ 2 2 $ (ii) This centripetal force required is provided by the weight of liquid of height 'H' From (i) and (ii) \( aligned & d ^ 2 4 H g = d ^ 2 4 ^

Question 7

A spherical drop of radius R is divided into eight equal droplets. If surface tension is T , then the work done in this process is

Accepted Answer

Volume of 8 small droplets = Volume of 1 big drop 4 3 r^ 3 8= 4 3 R ^ 3 r= R (8)^ 1 / 3 Work done =( Change in area ) surface tension = (4 r^ 2 n-4 R ^ 2 ) T \ 4 ( R ^ 2 (8)^ 2 / 3 ) 8-4 R ^ 2 \ T =4 R ^ 2 ~T

Mechanical Properties of Fluids — Previous Year Questions & Solutions