JEE Advanced · Physics · 13. Thermodynamics

Paragraph :
$P_{14-16}$ : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius $R$ and height $L_0$. The cylinder is open at its bottom and has a small hole at its top. A piston of mass $M$ is held at a distance $L$ from the top surface as shown in the figure. The atmospheric pressure is $p_0$.

Question :
While the piston is at a distance $2 L$ from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is

A $\left(\frac{2 p_0 \pi R^2}{\pi R^2 p_0+M g}\right)(2 L)$
B $\left(\frac{p_0 \pi R^2-M g}{\pi R^2 p_0}\right)(2 L)$
C $\left(\frac{p_0 \pi R^2+M g}{\pi R^2 p_0}\right)(2 L)$
D $\left(\frac{p_0 \pi R^2}{\pi R^2 p_0-M g}\right)(2 L)$

Verified Solution

Answer & Solution

Correct Answer

(D) $\left(\frac{p_0 \pi R^2}{\pi R^2 p_0-M g}\right)(2 L)$

Step-by-step Solution

Detailed explanation

Let $p$ be the pressure in equilibrium
Then,
$
\begin{aligned}
& \therefore p=p_0-\frac{M g}{A}=p_0-\frac{M g}{\pi R^2} \\
& \text { Applying } \quad p_1 V_1=p_2 V_2 \\
& p_0(2 A L)=(p)\left(A L^{\prime}\right) \\
& L^{\prime}=\frac{2 p_0 L}{p}=\left(\frac{p_0}{p_0-\frac{M g}{\pi R^2}}\right) \text { (2L) } \\
& =\left(\frac{p_0 \pi R^2}{\pi R^2 p_0-M g}\right)(2 L) \\
&
\end{aligned}
$
$\therefore$ Option (d) is correct.

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

List I describes four systems, each with two particles

A

and

B

in relative motion as shown in figure. List II gives possible magnitudes of their relative velocities (in

m s^{- 1}

) at time

t = \frac{π}{3} s

	List-I		List-II
(I)	$A$ and $B$ are moving on a horizontal circle of radius $1 m$ with uniform angular speed $ω = 1 rad s^{- 1}$ . The initial angular positions of $A$ and $B$ at time $t = 0$ are $θ = 0$ and $θ = \frac{π}{2}$ respectively.	(P)	$\frac{\sqrt{3} + 1}{2}$
(II)	Projectiles $A$ and $B$ are fired (in the same vertical plane) at $t = 0$ and $t = 0.1 s$ respectively, with the same speed $v = \frac{5 π}{\sqrt{2}} m s^{- 1}$ and at $45 °$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide, $(g = 10 m s^{- 2})$ .	(Q)	$\frac{(\sqrt{3} - 1)}{\sqrt{2}}$
(III)	Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_{A} = x_{0} \sin \frac{t}{t_{0}}$ and $x_{B} = x_{0} \sin (\frac{t}{t_{0}} + \frac{π}{2})$ respectively, starting from $t = 0$ . Take $x_{0} = 1 m, t_{0} = 1 s$ .	(R)	$\sqrt{10}$
(IV)	Particle $A$ is rotating in a horizontal circular path of radius $1 m$ on the $x y$ plane, with constant angular speed $ω = 1 rad s^{- 1}$ . Particle $B$ is moving up at a constant speed $3 m s^{- 1}$ in the vertical direction as shown in the figure. (Ignore gravity.)	(S)	$\sqrt{2}$
		(T)	$\sqrt{25 π^{2} + 1}$

Which one of the following options is correct?

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	LIST -I		LIST-II
A)	$\frac{v_{1}}{v_{2}}$	P)	$\frac{1}{8}$
B)	$\frac{L_{1}}{L_{2}}$	Q)	1
C)	$\frac{K_{1}}{K_{2}}$	R)	2
D)	$\frac{T_{1}}{T_{2}}$	S)	8