MHT CET · Physics · Oscillations

A U-tube of a uniform bore is with its arm vertical. The length of the liquid in the two arms or the U-tube is \(L\); the time period \(T\) of the oscillation of the liquid column when it is displaced by ' \(Y\) ' is ( \(g=\) acceleration due to gravity)

A \(2 \pi \sqrt{\frac{Y}{g}}\)
B \(2 \pi \sqrt{\frac{2 L}{g}}\)
C \(2 \pi \sqrt{\frac{L}{g}}\)
D \(2 \pi \sqrt{\frac{L}{2 g}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(2 \pi \sqrt{\frac{L}{2 g}}\)

Step-by-step Solution

Detailed explanation

Due to the tiny displacement \(y\), the force balance in the tube reads:
\(\begin{aligned} & \frac{d^2 y}{d t^2}=-\frac{F}{m}=-\left(\frac{2 y \rho g}{L A \rho}\right) \\ & \Rightarrow \frac{d^2 y}{d t^2}=-\left(\frac{2 g}{L}\right) y\end{aligned}\)
or
\(\omega^2=\frac{2 g}{L} \Rightarrow T=2 \pi \sqrt{\frac{L}{2 g}}\)

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