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A man with mass \( m \) is standing on a rotating platform in a science museum. The platform can be approximated as a uniform disk of radius \( R \) that rotates without friction at a constant angular velocity \( \omega \). The surface of the platform is frictionless, so the only forces between the man and the platform arise from the man’s feet as he runs. Two students are discussing what the man should do if he wishes to remain directly above a single point on the platform’s surface (so that, as viewed from the ground, he does not drift relative to that point).Student A claims that the man should run clockwise, in the same direction that the platform is rotating, because doing so will decrease the system’s moment of inertia and therefore increase \( \omega \), allowing him to stay above the desired point.Student B claims that, because no external torque acts on the man–platform system, the man must instead run counter-clockwise (opposite the platform’s rotation) so that the total angular momentum of the system about its central axis is conserved.Briefly explain which student’s reasoning is correct, explicitly referring to conservation of angular momentum and the absence (or presence) of external torques.

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A system consists of two small disks, of masses \( m \) and \( 2m \), attached to a rod of negligible mass of length \( 3l \) as shown above. The rod is free to turn about a vertical axis through point \( P \). The two disks rest on a rough horizontal surface; the coefficient of friction between the disks and the surface is \( \mu \). At time \( t = 0 \), the rod has an initial counterclockwise angular velocity \( \omega_0 \) about \( P \). The system is gradually brought to rest by friction. Develop expressions for the following quantities in terms of \( \mu \), \( m \), \( l \), \( g \), and \( \omega_0 \).

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A system consists of two small disks, of masses \( m \) and \( 2m \), attached to ends of a rod of negligible mass of length \( 3x \). The rod is free to turn about a vertical axis through point \( P \). The first mass, \( m \), is located \( x \) away from point \( P \), and therefore the other mass, of \( 2m \), is \( 2x \) from point \( P \). The two disks rest on a rough horizontal surface; the coefficient of friction between the disks and the surface is \( \mu \). At time \( t = 0 \), the rod has an initial counterclockwise angular velocity \( \omega_i \) about \( P \). The system is gradually brought to rest by friction. Derive expressions for the following quantities in terms of \( \mu \), \( m \), \( x \), \( g \), and \( \omega_i \).

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