Moment Of Inertia Ring

Author: johnharmer

Introduction to moment of inertia ring:

Moment of inertia of a ring can be explained in the following 4 parts.

a) About an axis passing through its centre and perpendicular to its plane or about a transverse axis :

Consider a circular ring of mass M and radius R. Its moment of inertia about an axis passing through its centre and perpendicular to its plane is MR2 .

`:.`  I = MR2

Moment of inertia of circular ring

Moment of Inertia ring:

b) About a Tangent in the perpendicular Plane :

Consider a tangent in the perpendicular plane to the circular ring as the axis of rotation. The moment of inertia of the circular ring about a parallel axis through the centre of mass i.e., centre of the ring is  Io = MR2

Moment of inertia of circular ring

By the parallel axes theorem,moment of inertia about a tangent perpendicular to the plane of the ring is

Io     =      MR2

=    MR2  + MR2   (since r = distance between eh tangent and center = R )

Io = 2MR2

Radius of gyration , K =  `sqrt(I/M)`   =  `sqrt(2MR^2 / M)`    = `sqrt(2)` R.

Diameter:moment of Inertia ring:

c) About a Diameter :  The moment of inertia of a circular ring about any diameter is same. Consider two diameters xx' and yy' of the circular ring. These two diameters are perpendicular to each other and intersect at the center 'O' .

About a Diameter

Moment of inertia of the circular ring about the diameter xx'= Ix .

Moment of inertia of the circular ring about the diameter yy' = Iy .

But Ix = IY = I. due to symmetry

By perpendicular axes theorem, the sum of Ix  and Iy is equal to the moment of inertia of the circular ring about the axis zz' passing through the intersecting point O and perpendicular to its plane , Iz = MR2 .

Ix +  Iy  =  Iz .

I  +  I  = MR2 .

2 I = MR2

I = `(MR^2)/(2)`

d) About a Tangent in its Plane :  The tangent in the plane of the circular ring is parallel to the diameter. As we know that the moment of inertia of the circular ring about a diameter is  MR2 / 2 , we can find the moment of inertia about a tangent in the plane of the ring by parallel axes theorem.

About a Tangent in its Plane

Moment of inertia about A'B'  =  Moment of inertia about AB through it s center of mass + M r2

=  MR2 / 2  +  MR2

I    =   `(3MR^2)/(2)`

Weight of an object is defined to as the force exerted by a object on to its support. For a free falling body the weight will be equal to zero. In the case when a body I standing in a weighing scale, the n the mass of the body will be equal to the product of mass and the gravitational acceleration which is supported by the scale's supporting forces.

Net weight

What is net weight?

Net weight can be defined to as the weight of the object, along with the components that are relating to the object that causes the body to add the weight and reduce the weight of the weighing scale on which the object is place. The very good example for calculating the net weight is in the case of weighting live chicken. Here the chicken will be weighed when it is alive and after it is killed, the feathers are removed but the weight is also the same when it was weighted with live chicken.

In order to find the net weight, one has to know two things; total weight and tare weight. So the calculation is very simple. The net weight is equal to the total weight minus tare weight.

Total weight and Tare weight

Total weight: The total weight of a object is defined to as the mass of the entire object inclusive of all the packages or coverings and anything else attached to this object.

Tare weight: The tare weight of a object is defined to as mass of the covering or the packing or even anything that are attached to the object but not the part of the actual object.

Suppose that a body's net weight need to be found out, then the person need to remove all his clothing and have to step on to the weighing scale as naked. If in case the modesty does matters, it is also done in which the weight of the cloths are considered as the tare weight and the weight of the body as total weight. So by subtracting tare weight from the total weight, the net weight can be found out.

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