Q. A thin wire of length having a linear density is bent into a circular loop with as its centre as shown in the figure. The moment of inertia of the loop about the line is
image

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Solution:

Given, length of a thin wire
Linear density of the wire
Let the mass of the wire
Now, according to the question
image
Here, circular loop of the wire with centre, .
is the radius of circular loop of wire.
Moment of inertia axis passing through the centre and perpendicular to the plane . .......(i)
Moment of inertia axis passing through the centre of circular loop about its diameter,

Now, the total moment of inertia for circular loop of wire. If is the distance between the axis, and are the respective moments of inertia about these axies,
Then,
From Eqs. (i) and (ii), we get

We know that,
Mass length density
Mass of the wire, ...(iv)
Thus, the length of the wire, circumference of the circular loop of wire
....(v)
Now, from Eq. (iv) and (v) putting the value of and in Eq. (iii), we get

So, the moment of inertia of a circular loop of wire
is, .