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What will happen to the moment of inertia if you keep the mass of the platform the same, but you set the inner radius to zero (the hoop is now a disk or cylinder)? Explain using what you have learned from Chapter 8 in:

a) Thermodynamics
b) Quantum mechanics
c) Classical mechanics
d) Electromagnetism

1 Answer

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Final answer:

Setting the inner radius of a hoop to zero to make it a disk reduces its moment of inertia, as mass distribution is closer to the axis and the inertia halved relative to a hoop. This has implications in classical mechanics, affecting how easily the object's rotation can accelerate or decelerate.

Step-by-step explanation:

When you change the inner radius of a hoop to zero, turning it into a disk or cylinder, the moment of inertia (I) decreases. This is because I for a hoop is I = MR², where M is the mass and R is the radius of the hoop. I for a solid cylinder or disk is I = (1/2)MR², showing that the moment of inertia is cut in half relative to the hoop when other factors are the same. The distribution of mass closer to the axis of rotation reduces the inertia. In the context of thermodynamics, this change doesn't directly relate as a moment of inertia is more a mechanical property than a thermal one. In quantum mechanics, a moment of inertia can affect the rotational spectra of molecules but is not generally discussed in terms of macroscopic objects like disks or hoops. In classical mechanics, the reduction of inertia allows for easier acceleration or deceleration of the object, as the resistance to change in rotational motion is lessened. In electromagnetism, the change in moment of inertia doesn't have a primary effect, unless it relates to the rotational dynamics of electrically charged objects or the generation of magnetic fields via rotating charges.

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