Question 7: Properties of Matter

Question

A liquid of density $600 \mathrm{~kg} / \mathrm{m}^3$ flowing steadily in a tube of varying cross-section. The cross-section at a point $A$ is $1.0 \mathrm{~cm}^2$ and that at $B$ is $20 \mathrm{~mm}^2$. Both the points $A$ and $B$ are in same horizontal plane, the speed of the liquid at $A$ is $10 \mathrm{~cm} / \mathrm{s}$. The difference in pressures at $A$ and $B$ points is $\_\_\_\_$ Pa.

Accepted Answer

The density of liquid is $ho=600 \mathrm{~kg} / \mathrm{m}^3$.
Area of cross-section at point A is $\mathrm{A}_{\mathrm{A}}=1.0 \mathrm{~cm}^2=1.0 	imes 10^{-4} \mathrm{~m}^2$.
Area of cross-section at point $B$ is $A_B=20.0 \mathrm{~mm}^2=20.0 	imes 10^{-6} \mathrm{~m}^2=0.2 	imes 10^{-4} \mathrm{~m}^2$.
The velocity at A is $\mathrm{v}_{\mathrm{A}}=10 \mathrm{~cm} / \mathrm{s}=0.1 \mathrm{~m} / \mathrm{s}$.
For an incompressible liquid in steady flow, the mass flow rate must remain constant.
Using equation of continuity,
$$ A_A v_A=A_B v_B $$
Substituting the values :
$$ \left(1.0 	imes 10^{-4}ight) 	imes(0.1)=\left(0.2 	imes 10^{-4}ight) 	imes v_B $$
$$\Rightarrow $$ $$ \mathrm{v}_{\mathrm{B}}=\frac{1.0 	imes 0.1}{0.2}=\frac{0.1}{0.2}=0.5 \mathrm{~m} / \mathrm{s} $$
According to Bernoulli's equation for a horizontal flow (where height $h$ is constant), the sum of pressure and kinetic energy per unit volume is constant :
$P_A+\frac{1}{2} ho v_A^2=P_B+\frac{1}{2} ho v_B^2$
$$\Rightarrow $$ $$ P_A-P_B=\Delta P_{A B}=\frac{1}{2} ho\left(v_B^2-v_A^2ight) $$
Substituting the given values $ho=600 \mathrm{~kg} / \mathrm{m}^3, \mathrm{v}_{\mathrm{B}}=0.5 \mathrm{~m} / \mathrm{s}$ and $\mathrm{v}_{\mathrm{A}}=0.1 \mathrm{~m} / \mathrm{s}$
$$ P_A-P_B=\frac{1}{2} 	imes 600 	imes\left(0.5^2-0.1^2ight) $$
$$\Rightarrow $$ $P_A-P_B=300 	imes(0.25-0.01)$
$$\Rightarrow $$ $\mathrm{P}_{\mathrm{A}}-\mathrm{P}_{\mathrm{B}}=300 	imes 0.24$
$$\Rightarrow $$ $$ \mathrm{P}_{\mathrm{A}}-\mathrm{P}_{\mathrm{B}}=72 \mathrm{~Pa} $$
Therefore, the difference in pressures at points A and B is 72 Pa .
Hence, the correct option is (D).

Answer

18

Answer

144

Answer

36

Answer

72

Properties of Matter

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