Question

A circular disc is to be made by using iron and aluminium, so that it acquires maximum moment of inertia about its geometrical axis. It is possible with-
a) Iron and aluminium layers in alternate order
b) Aluminium at interior and Iron surrounding it
c) Iron at interior and aluminium surrounding it
d) Either (a) or (c)

Answer 1

Hint: Moment of Inertia of any solid depends upon the mass and the distance from the centroid axis. Mass is proportional to density, hence more the density more will be the mass and thus maximum will be the moment of inertia.

Complete step-by-step answer:
A circular disc is made up of a larger number of circular rings. Total moment of inertia of disc can be given by the sum of moment of inertia of all these small rings about the geometrical axis. Now the moment of inertia of a circular ring is given by-
$I = M{R^2}$
$\Rightarrow I \propto M$
Since mass is proportional to the density of the material. As we know the density of the Iron is much more than that of aluminium. Hence To get maximum value of moment of Inertia the less dense material should be used at interior and denser at the surroundings of it. Therefore, using Aluminium at the interior and Iron at its surrounding will maximizes the moment of inertia.
Hence option (b) is the correct answer.

If we use Iron and aluminium layers in alternate order then it will increase the mass but not maximise it. Hence option (a) is correct.
Iron at the interior and aluminium surrounding it will lead to the minimization of mass. Hence option (c) is also correct.
We can’t use both at the same time, so option (d) is not correct.
The correct option is (b)

Note: Always keep in mind that mass will maximize the moment of inertia and mass is proportional to the density of the material.