Question

Water is flowing through a non-uniform radius tube. If ratio of the radius of entry and exit end of the pipe is 3:2 then the ratio of velocities of entering and exit liquid is:
1) 4:9
2) 9:4
3) 8:27
4) 1:1

Answer 1

Hint: Here, the tube or pipe is a cylindrical object so, we can apply the formula for the cylinder, the density of the liquid is the same, as it is the same liquid entering and exiting the tube. The only difference is in the radius of the tube which is given to us in terms of ratio and we have to find the velocity in terms of their ratio. Apply the equation of continuity for both the radius and cancel out the common factors, one will get the ratio between the two velocities.

Complete step by step solution:
Here, apply the equation of continuity:
${\rho _1}{A_1}{V_1} = {\rho _2}{A_2}{V_2}$;
Here:
${\rho _1} = {\text{Density of first object}}$;
${A_1} = {\text{Area of the tube at entrance}}$;
${V_1} = {\text{Volume of the tube at entrance}}$;
${\rho _2} = {\text{Density of second object}}$;
${A_1} = {\text{Area of the tube at exit}}$;
${V_1} = {\text{Volume of the tube at exit}}$;

 Here, the area of the tube is $A = \pi {r^2}$:
 \[ \Rightarrow {\rho _1}\pi r_1^2{V_1} = {\rho _2}\pi r_2^2{V_2}\];
Since, the fluid is same so, the density of the fluid will be the same:
\[ \Rightarrow \pi r_1^2{V_1} = \pi r_2^2{V_2}\];
Cancel out the common:
\[ \Rightarrow r_1^2{V_1} = r_2^2{V_2}\];
Take the velocity on the RHS to LHS and the Radius on LHS to RHS:
\[ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{r_2^2}}{{r_1^2}}\];
Put in the given values in the above equation:
\[ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{{{\left( 2 \right)}^2}}}{{{{\left( 3 \right)}^2}}}\];
Solve, the above equation:
\[ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{4}{9}\];

Therefore, Option “1” is correct. The ratio of velocities of entering and exit liquid is 4:9.

Note: The way to make a fluid flow in a tube or pipe can be done in two ways. The first is to make the pipe tilt so that the flow of the liquid would go downhill, the gravitational energy will be converted to Kinetic energy. The second way is to create a pressure difference, make the pressure at one end of the pipe larger than the pressure at the other end. This pressure difference will act like a net force which in turn produces acceleration of the liquid.