MHT CET · Physics · Mechanical Properties of Fluids

A hemispherical portion of radius ' \(R\) ' is removed from the bottom of a cylinder of radius ' R '. The volume of the remaining cylinder is ' \(V\) ' and its mass is ' \(M\) '. It is suspended by a string in a liquid of density ' \(\rho\) ' where it stays vertical. The upper surface of the cylinder is at a depth ' \(h\) ' below the liquid surface. The force on the bottom of the liquid is

A Mg
B \(\mathrm{Mg}-\mathrm{V} \rho \mathrm{g}\)
C \(\mathrm{Mg}+\pi \mathrm{R}^2 \mathrm{~h} \rho g\)
D \(\quad \rho g\left(V+\pi r^2 h\right)\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\quad \rho g\left(V+\pi r^2 h\right)\)

Step-by-step Solution

Detailed explanation

Net upward zone force on the bottom of the liquid \(=\) weight of the liquid displaced by cylinder + thrust force on upper surface of the cylinder due to h column of liquid
\(\begin{aligned}
& F_{\text {net }}=\rho V g+\rho g h \times \pi r^2 \\
&=\rho g\left(V+\pi r^2 h\right)
\end{aligned}\)

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.

Same subject

Explore more questions on app

From MHT CET

Explore more questions on app