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Question #42084

Difficulty: 8/10 | 🟢 Free

The Question

A hollow cylinder of mass $M$ contains a smooth, airtight piston of mass $m$ . The cylinder is closed at its upper end and open at its lower end. Initially, the piston is locked, confining an ideal gas of volume $V_0$ and pressure $P_0$ between the closed upper end and the piston. The entire system is projected up a smooth inclined plane of inclination $\theta$ with an initial velocity $v_0$ . Immediately after launch, the piston is unlocked. The gas expands, undergoing a polytropic process $PV^n = \text{constant}$ ( $n > 1$ ). At the instant the center of mass of the system has traveled a distance $D$ up the incline, the volume of the gas is $rV_0$ . Assuming the piston has not yet left the cylinder, which of the following expressions gives the velocity of the cylinder up the incline at this instant?

Options

A.
$\sqrt{v_0^2 - 2gD \sin \theta} + \sqrt{\frac{2m P_0 V_0}{M(M+m)(n-1)} \left( 1 - r^{1-n} \right)}$
B.
$\sqrt{v_0^2 - 2gD \sin \theta} + \sqrt{\frac{2 P_0 V_0}{M(n-1)} \left( 1 - r^{1-n} \right)}$
C.
$v_0 + \sqrt{\frac{2m P_0 V_0}{M(M+m)(n-1)} \left( 1 - r^{1-n} \right)}$
D.
$\sqrt{v_0^2 - 2gD \sin \theta} - \sqrt{\frac{2m P_0 V_0}{M(M+m)(n-1)} \left( 1 - r^{1-n} \right)}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Projectile on Inclined Plane|Polytropic Process|Work against Conservative Force

Why This Question Is Tricky

Common Mistakes

This question tests the decoupling of center-of-mass kinematics from internal work-energy dynamics in a uniform gravitational field.

A common trap is attempting to write coupled differential equations for the cylinder and piston using Newton's Second Law ( $Ma_M = PA - Mg\sin\theta$ ), which leads to an intractable integral since pressure $P$ is a function of relative displacement, not time.

Students also frequently fail to use the reduced mass $\mu$ when converting internal thermodynamic work into relative kinetic energy, or they incorrectly assume the internal gas expansion alters the trajectory of the system's center of mass.

By forcing the use of the CM frame, the polytropic work cleanly translates to relative velocity, which is then superposed onto the standard projectile motion of the CM.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Projectile on Inclined Plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free