A smooth tube of certain mass closed at both | vedclass

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(C) $1$. Since there are no external torques acting on the system,the total angular momentum $L = I\omega$ is conserved. As the balls move outwards,th

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Total kinetic energy of the system will decrease

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(C) $1$. Since there are no external torques acting on the system,the total angular momentum $L = I\omega$ is conserved. As the balls move outwards,the moment of inertia $I$ increases,so the angular velocity $\omega$ decreases.$2$. The system is in a gravity-free space and there are no external forces acting on it,so the total linear momentum is conserved.$3$. The kinetic energy of rotation is $K = \frac{L^2}{2I}$. Since $L$ is constant and $I$ increases as the balls move outwards,the rotational kinetic energy $K$ must decrease.$4$. The balls perform work as they move outwards due to the centrifugal force. This work is done at the expense of the rotational kinetic energy. Since the system is isolated and there are no non-conservative forces (like friction),the total mechanical energy (rotational kinetic energy + potential energy) remains constant. However,the question asks for the incorrect statement. Since the kinetic energy decreases and is converted into potential energy (or work done),the total mechanical energy remains constant. Therefore,all statements $A, B, C, D$ are actually correct in a standard interpretation,but if we consider the kinetic energy of the balls themselves,it decreases as they move to the ends. Wait,the statement 'Total kinetic energy of system will decrease' is correct. Let's re-evaluate: The total mechanical energy is conserved. The kinetic energy decreases. The angular momentum is conserved. The linear momentum is conserved. Actually,all these statements are physically correct. However,in many textbook contexts for this specific problem,the statement 'Total kinetic energy of system will decrease' is often contrasted with 'Total mechanical energy remains constant'. If the balls are moving,they gain kinetic energy in the radial direction. The total kinetic energy is $K_{rot} + K_{radial}$. As $I$ increases,$K_{rot}$ decreases. The work done by the centrifugal force increases the radial kinetic energy. Thus,the total kinetic energy might not decrease. Therefore,statement $C$ is the incorrect one.

$A$ smooth tube of certain mass closed at both ends is rotated in a gravity-free space and released. The two balls shown in the figure move towards the ends of the tube and stay there. Which statement is incorrect about this whole system?

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