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Radius of Gyration

To understand the concept of;radius of gyration, suppose that a rigid body is madeup of large number of identical particles; each of mass. Let r1, r2, r3.rn be theperpendicular distances of these particles from the axisof rotation as shown in

figure .Then the M.I. of the body about the axis is given by,

Thus K2 is the mean of the squares of the distances Of the individual particles,i.e.'mean square distance of the particles from the axis.Taking root of equation

If the entire mass of the body were concentration at a distance K from the axis ofrotation as shown in figure then the concentrated point mass would have momentof inertia MK2about ithe axis, which is the same as the moment of inertia of thebody about the axis. Thus the motion of concentrated mass M about the given axisof rotation in a circle of radius K is equivalent to the rotation of the body of mass Mabout the axis. Therefore the quantity K is called radius if gyration of the bodyabout its axis of rotation. It is also called root mean square distance about the axisof rotation.

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Radius of gyration of a body about a given axis of rotation is defined as'the distancebetiveen axis of rotation and a point at which in hole mass of the body is supposedto be concentrated so as to possess the same moment of inertia as that of thebody.

ExplanationConsider a disc of mass M rotating about an axis passing through its centre andperpendicular to its plane. Let I be the moment of inertia of the disc.Suppose the disc is now replaced by a single particle of same mass as that of thedisc which is at distance K from the axis of rotation. It gives same M.I. as that ofthe disc. The moment of inertia of a single particle of mass M is,

The radius of gyration of a body depends upon,a..It is independent of mass of' the body.

b.Distribution of mass !c.Position of axis of rotation.

SI and CGS unit of radius of gyration is metre (m) and centimetre cm respectively.The dimensions of 1K are [M L1T]Physical significance of radius of gyration :

1. If distribution of mass within the body is close to the axis-of rotation, then themoment of inertia of the body is minimum, hence K is minimum.

2.If the distribution of mass within of the body is away from the axis of rotation,

then moment of inertia is maximum. Hence value of K is maximum.

3.The radius of gyration is the measure of the distribution of mass within body. Thisexplains the physical significance of radius of gyration the body.