Question

Water flows steadily through a horizontal pipe of a variable cross-section. If the pressure of the water is $p$ at a point, where the speed of the flow is $v$ , what is the pressure at another point, where the speed of the flow is $2v$ ? Let the density of water be $1 \, g \, cm^{- 3}$ .

• A. $p+\left(\frac{3}{2}\right)\rho \, v^{2}$
• B. $p-2\rho v^{2}$
• C. $p-3\rho v^{2}$
• D. $p-\left(\frac{3}{2}\right)\rho v^{2}$
• E. $p-\left(\frac{5}{2}\right)\rho v^{2}$

Answer 1

Answer: (D)

Two point are situated in a pipe and their height from ground is same
$\because $ According Bernoulli's theorem,
$p+\rho gh+\frac{1}{2}\rho v^{2}=$ constant
At a point pressure of water is $p$ and speed of flow is $v$
therefore, $p+\frac{1}{2}\rho v^{2}=c$ .....(i)
At another point speed of flow is $2v$ therefore,
$p_{1}+\frac{1}{2}\rho \left(2 v\right)^{2}=c$ ......(ii)
To find pressure $p_{1}$ at another point, equating the equation (i) and (ii)
$\Rightarrow \, \, \, p+\frac{1}{2}\rho v^{2}=p_{1}+\frac{1}{2}\rho \left(4 v^{2}\right)$
$\Rightarrow \, p_{1}=p-\frac{3}{2}\rho v^{2}$