Q. Water flows steadily through a horizontal pipe of variable cross-section. If the pressure of water is $p$ at a point where flow speed is $v$, the pressure at another point where the flow of speed is $2v$, is (take density of water as$\ \rho$)
A) $p - \frac{3{\rho v}^{2}}{2}$
B) $p - \frac{{\rho v}^{2}}{2}$
C) $p - \frac{3{\rho v}^{2}}{4}$
D) $p - \rho v^{2}$
Explanation
(a) From Bernoulli’s equation, the sum of all forms of energy in a fluid flowing along an enclosed path (a streamline) is the same at any two points in the path. Therefore,
$p + \frac{1}{2}\rho v_{1}^{2} = p' + \frac{1}{2}\rho v_{2}^{2}$
$Given,\ \ v_{2} = 2v,\ \ \ \ {v\ }_{1} = v$
$\therefore\ \ \ \ p + \frac{1}{2}\rho v^{2} = p' + \frac{1}{2}\rho(2v)^{2}$
$\Longrightarrow \ \ \ \ \ p^{'} = \ p - \frac{3}{2}\rho v^{2}\ \ $
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.