Two tubes of radii $r_1$ and $r_2$, and lengths $l_1$ and $l_2$, respectively, are connected in series and a liquid flows through each of them in streamline conditions. $P_1$ and $P_2$ are pressure differences across the two tubes. If $P_2$ is $4P_1$ and $l_2$ is $ \frac{l_1}{4}$ , then the radius $r_2$ will be equal to
JEE MAIN 2017, Diffcult
Download our app for free and get started
The volume of liquid flowing through both the tubes i.e., rate of flow of liquid same.
$Therefore,V = {V_1} = {V_2}$
$i.e.,\frac{{\pi {p_1}r_1^4}}{{8\eta {l_1}}} = \frac{{\pi {p_2}r_2^4}}{{8\eta {l_2}}}$
$or\,\,\,\,\,\,\frac{{{p_1}r_1^4}}{{{l_1}}} = \frac{{{p_2}r_2^4}}{{{l_2}}}$
$\,\,\,\,{P_2} = 4\,{P_1}\,\,and\,{l_2} = {l_1}/4$
$\frac{{{p_1}r_1^4}}{{{l_1}}} = \frac{{4{p_1}r_2^4}}{{{l_1}/4}} \Rightarrow r_2^4 = \frac{{r_1^4}}{{16}}$
$ \Rightarrow {r_2} = {r_1}/2$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1Water is flowing through a tube of non-uniform cross-section. Ratio of the radius at entry and exit end of the pipe is $3$ : $2$. Then the ratio of velocities at entry and exit of liquid isView Solution
- 2A liquid flows through a horizontal tube. The velocities of the liquid in the two sections, which have areas of cross-section ${A_1}$ and ${A_2}$, are ${v_1}$ and ${v_2}$ respectively. The difference in the levels of the liquid in the two vertical tubes is $ h$View Solution
- 3A small spherical ball of radius $r$, falling through a viscous medium of negligible density has terminal velocity ' $v$ '. Another ball of the same mass but of radius $2 r$, falling through the same viscous medium will have terminal velocity:View Solution
- 4A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration $a$ along a fixed inclined plane with angle $\theta=45^{\circ} . P_1$ and $P_2$ are pressures at points 1 and 2 , respectively, located at the base of the tube. Let $\beta=\left(P_1-P_2\right) /(\rho g d)$, where $\rho$ is density of water, $d$ is the inner diameter of the tube and $g$ is the acceleration due to gravity. Which of the following statement($s$) is(are) correct?View Solution
$(A)$ $\beta=0$ when $a= g / \sqrt{2}$
$(B)$ $\beta>0$ when $a= g / \sqrt{2}$
$(C)$ $\beta=\frac{\sqrt{2}-1}{\sqrt{2}}$ when $a= g / 2$
$(D)$ $\beta=\frac{1}{\sqrt{2}}$ when $a= g / 2$
- 5A balloon of mass $m$ contains water of mass $M$ . If it is completely immersed in water, find the apparent mass of the balloon with water in itView Solution
- 6A cubical block of wood of edge $10$ $cm$ and mass $0.92$ $kg$ floats on a tank of water with oil of rel. density $0.6$ to a depth of $4$ $cm$ above water. When the block attains equilibrium with four of its sides edges verticalView Solution
- 7The rate of steady volume flow of water through a capillary tube of length $ 'l' $ and radius $ 'r' $ under a pressure difference of $P$ is $V$. This tube is connected with another tube of the same length but half the radius in series. Then the rate of steady volume flow through them is (The pressure difference across the combination is $ P$)View Solution
- 8Figure shows a container filled with a liquid of density $\rho$. Four points $A, B, C$ and $D$ lie on the diametrically opposite points of a circle as shown. Points $A$ and $C$ lie on vertical line and points $B$ and $D$ lie on horizontal line. The incorrect statement is $\left(p_A, p_B, p_C, p_D\right.$ are absolute pressure at the respective points)View Solution
- 9A horizontal pipe line carries water in a streamline flow.At a point along the tube where the cross-sectional area is $10^{-2}$ $m^2$, the water velocity is $2$ $ms^{^{-1}}$ and the pressure is $8000 $ $Pa$. The pressure of water at another point where the cross-sectional area is $0.5$ $\times 10^{-2}$ $m^2$ is ........ $Pa$View Solution
- 10A metallic sphere weighing $3 \,kg$ in air is held by a string so as to be completely immersed in a liquid of relative density $0.8$. The relative density of metallic is $10$. The tension in the string is ........ $N$View Solution