# Effect of density and volume on inertia

I recently came across this question :

What will be the effect on inertia if -

  - density is halved

- volume is one third


I searched about it but everywhere found the answer that there will be no effect on inertia. But i fell this is wrong. The reason given was that inertia depends on mass, which is not changed. But

       Mass = density × volume


It means that mass is directly proportional to density and volume. So if density or volume is decreased, then mass should also decrease and hence inertia SHOULD decrease.

Am i right ? Why ?

Edit : one more question arised that should we consider the mass to be conserved (even if its not mentioned in the question) why ?

• Are you asking about inertia? Or moment of inertia? When the density and volume are changed, what stays the same? eg When density is halved, does the mass change? – sammy gerbil Sep 20 '16 at 15:47

The inertia of a body depends on mass, but not on density.

If you increase the volume of the bulk material (like by stretching), then what you are actually doing is just extending the amount of matter over a larger volume, which will decrease the amount of matter present per unit volume and hence thereby decreasing the density. Hence mass, which is density times volume suffers no net change (and it has to be as required by the conservation of matter) as the increase in volume is cancelled by the decrease in density.

So the mass remain constant and hence inertia suffers no change.

Update:

It's a fundamental requirement that the mass is conserved. Mass is amount of matter. You cannot create or destroy matter by changing the volume. By increasing the volume, you spread the matter to more space and hence the amount of matter per volume decreased and so by definition, the density decreases. Hence, in effect there is no change in mass.

I don't understand why you feel that there is a change in mass in this problem. What you have to say about where the mass has gone, by changing the volume? I think you have misunderstood about the volume and density. Density and volume is a property of the body itself. If nothing else is specified in the problem, it assumes the basic requirement that the matter is conserved.

• But the question didn't mentioned that mass is conserved. Why should we consider the mass to be conserved ? There are other ways also to decrease volume and density. – Nilaksh Singh Sep 20 '16 at 7:42
• Please See update – UKH Sep 20 '16 at 13:21

I presume that these 2 changes happen together to the same object.

Suppose we have a solid cylinder of mass $M$ and radius $R$. Its moment of inertia is $\frac12MR^2$.

If we halve the density while keeping the volume (and radius) the same, we are halving the mass. If we multiply the volume by a factor of $\frac13$ (while keeping the mass the same) then we are reducing the radius by a factor of $(\frac13)^{\frac13}$ because $V \propto R^3$. So overall the moment of inertia is multiplied by a factor of $(\frac12)(\frac13)^{\frac23} \approx 0.2404$.