MHT CET · Physics · Mechanical Properties of Fluids

A streamline flow of a liquid of density ' \(\rho\) ' is passing through a horizontal pipe of cross-sectional area \(A_1\) and \(A_2\) at two ends. If the pressure of liquid is ' P ' at a point where flow speed is ' \(v\) ', then pressure at another point where the flow of speed becomes 3 v is

A \(\mathrm{P}-\frac{3}{4} \rho v^2\)
B \(P-2 \rho v^2\)
C \(\mathrm{P}-3 \rho v^2\)
D \(P-4 \rho v^2\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(P-4 \rho v^2\)

Step-by-step Solution

Detailed explanation

Using Bernoulli's equation,
\(\begin{aligned}
& P_1+\frac{1}{2} \rho v_1^2+\rho g h_1=P_2+\frac{1}{2} \rho v_2^2+\rho g h_2 \\
& P_1+\frac{1}{2} \rho v_1^2=P_2+\frac{1}{2} \rho v_2^2 \quad ext{...(given horizontal pipe)}
\end{aligned}\)
Substituting the given values,
\(\begin{aligned} & P+\frac{1}{2} \rho v_1^2=P_2+\frac{1}{2} \rho(3 v)^2 \\ & P_2=P+\left(\frac{1}{2} \rho v^2-\frac{9}{2} \rho v^2\right)=P-4 \rho v^2\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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