**Moment of inertia of a Ring**

Moment of inertia of a
ring about an axis passing through the center and perpendicular to the plane is
the product of mass of the ring with the square of the radius of the ring. Ring
is a one dimensional body and its mass is distributed over its length.

Axis of rotation is a
axis around which a body is rotating. All the particles of the body expect
particles on axis of rotation move around this axis. Moment of inertia is a
physical quantity in rotatory motion basing on which we can study that how easy
or difficult to put a body in rotational motion. It depends on size of the
body, shape of the body and axis of rotation. Whenever the axis of rotation
changes, distance of the particles from the axis of rotation changes and hence moment
of inertia also changes. To measure the moment of inertia with different axes
of rotation easily we have parallel axes theorem and perpendicular axes
theorem.

Let us assume that the
ring of a mass

**m**is having a radius**r**and is rotating about an axis passing through its center and perpendicular to the plane. Its moment of inertia is measured using integration process and its value is the product of mass of the ring with the square of the axis of rotation.
Let us consider that
the ring is now rotated about a parallel axis to that axis and the
perpendicular distance between two axes is some known value. To find the moment
of inertia of the ring about the new axis, we can use parallel axes theorem. As
per this theorem, moment of inertia about an axis is equal to the sum of moment
of inertia about a parallel axis passing through center and the product of mass
of the body with the square of the distance between the two parallel axis.

If the parallel axis is
the one passing through one end of the ring, then the distance between the axis
will be the radius of the ring itself. In that case moment of inertia of the
system will be the double the moment of inertia of the first case.

Let the ring is
rotating about an axis passing through the center but it is in the plane. The
original axis that is taken into consideration in the first case is
perpendicular to this axis as well as
the plane. As the ring is a uniform body, its moment of inertia about an axis
in the plane shall be the same in two perpendicular axes in the plane say x
axis and y axis. All this axes are having common passing point that is the
origin and center of the ring. Here to
measure the moment of inertia about the given axis, we can use perpendicular
axes theorem.

As per this theorem, the moment of inertia of a body about a given axis the sum of moment of inertia of the same body about two perpendicular axes passing through the same point and perpendicular to the plane.

Applying this theorem
lead us to the moment of inertia of the ring about axis in the plane or about
its diameter as half the moment of inertia of the reference case.

Let the axis of
rotation is a axis tangential to the ring and in the plane. This axis is
parallel axis and we can use parallel axes theorem.

**Moment of inertia of Disc**

Disc is a two
dimensional body and its mass is distributed over two dimensions. In this kind
of two dimensional body is directly proportional to the area of the body. Let
us consider the disc is rotating about an axis passing through center and
perpendicular to the plane. We can find the moment of inertia of this disc
using the concept of mathematical integration and it is proved to be the
product of mass of the body and square of the radius of the disc divided by the
two.

Now let us assume the
disc is rotating about a different axis and the new axis is parallel to the
original axis and the separation is known to us. To find the moment of inertia
of this system about new axis can be found using parallel axis theorem.

If the axis of rotation
is in the plane and about the diameter and to find the moment of inertia of
this system, we can use perpendicular axes theorem.

Even if the body is
rotating about a tangential axis and it is in the plane and to find out moment
of inertia, we can use again parallel axis theorem and it can be noticed that
the moment of inertia of the disc is different from each axis of rotation.

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