## Wednesday, February 24, 2016

### Moment of inertia of a Ring and Disc

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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