Figure shows a counterweight of mass m suspended by a cord wound around a spool of radius r, forming part of a turntable supporting the object. The turntable can rotate without friction. When the counterweight is released from rest, it descends through a distance h, acquiring a speed v. The moment of inertia I of the rotating apparatus is

Figure shows a counterweight of mass m suspended by a cord wound around a spool of radius r, forming part of a turntable supporting the object. The turntable can rotate without friction. When the counterweight is released from rest, it descends through a distance h, acquiring a speed v. The moment of inertia I of the rotating apparatus is

  1. A

    mr2(2ghv2+1)

  2. B

    mr2(ghv2-1)

  3. C

    mr2(2ghv2-1)

  4. D

    mr2(ghv2+1)

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    Solution:

    Each point on the cord moves at a linear speed of v = ωr where r is the radius of the spool. The energy conservation equation for the counter weight turn table-Earth system is:
    (K1+K2+Ug)i+Wother = (K1+K2+Ug)f

    Specializing, we have
    0+0+mgh+0 = 12mv2+122+0

    mgh = 12mv2+12Iv2r2

    2mgh -mv2 = Iv2r2 and finally, I = mr2(2ghv2-1)

     

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