Rate of inflation
The surface of a solid body at any time t is given by
( a x ) 2 + ( b y ) 2 + ( c t z ) 2 = 1
where a , b , c are constants. Find the rate at which the moment of inertia of the body about the z -axis changes with time.
The answer is 0.000.
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1 solution
I was think the same the earlier problem was helpful
Since it is a body, and a deformable body, it is inherent that it's mass is constant.
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At any instant of time, the volume of the body is:
V = 3 4 π a b c t
The mass is:
M = 3 4 ρ π a b c t
Every solid body is made up of some material. Every material has a density, which is usually constant. It is, therefore, reasonable to consider the mass to be variable. In this situation, where the mass is constant, the density of the material is time-varying. A time-varying density is not a usual physical manifestation. This is why I opine that the fact that mass is constant should be specified. It is for the sake of clarity.
Furthermore, deformable bodies have a constant volume, when thought of in a classical mechanics sense. Here, that is not the case.
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While the body is deformed density changes so in my opinion mass is constant
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@Dark Angel – You're right. Here the density changes, and not the mass.
I have done this problem under the assumption that the mass of the solid body is always constant. It would be useful to specify this in the problem statement. For the given solid ellipsoid, the moment of inertia about the Z-axis is:
I = 5 m ( a 2 + b 2 )
Differentiating the above with respect to time yields zero as the expression is independent of time, assuming that the mass remains constant.