Given below are two statements: Statement I: When the speed of liquid is zero everywhere, the pressure difference at any two points depends on the equation $$ P_1 - P_2 = \rho g (h_2 - h_1). $$ Statement II: In the ventury tube shown, $$ 2gh = v_1^2 - v_2^2. $$ In the light of the above statements, choose the most appropriate answer from the options given below.
Statement I is incorrect but Statement II is correct.
Both Statement I and Statement II are incorrect.
Statement I is correct but Statement II is incorrect.
Show Solution
The Correct Option isD
Solution and Explanation
Analysis of Statement I: Statement I asserts that when the liquid speed is zero universally, the pressure differential between any two points is dictated by the equation: \[ P_1 - P_2 = \rho g(h_2 - h_1) \] This statement is accurate, deriving from the principles of hydrostatic pressure, which are applicable to fluids at rest or in uniform motion without velocity gradients.
Analysis of Statement II Using Bernoulli’s Equation: For a fluid in motion within a venturi tube, Bernoulli’s equation is applicable: \[ P_1 + \rho gh + \frac{1}{2}\rho v_1^2 = P_2 + \rho gh + \frac{1}{2}\rho v_2^2 \]
The pressure difference, derived from this equation, is: \[ P_1 - P_2 = \frac{1}{2}\rho (v_2^2 - v_1^2) \] The provided statement, \(2gh = v_2^2 - v_1^2\), does not represent a general outcome of Bernoulli’s equation and is erroneous as presented.
Conclusion: Consequently, Statement I is correct (applicable to static fluids or uniform motion without speed variations), whereas Statement II is incorrect in the context of a venturi tube.
