Question

Discuss the necessity of radius of gyration. Define it. On what factors does it depend and it does not depend? Can you locate some similarity between the centre of mass and radius of gyration?​

Answer 1

☆Radius of gyration:

》If some rigid system has some moment of inertia 'I', and we somehow concentrate that object in whole mass 'm',converting it into particle-mass system,

And placing that particle at distance (k),from the axis such that it gives the same inertia as that of continuous rigid system.

》That 'k' will be defined here as Radius of gyration.

$\boxed{\bf{I=mk²}}$

》let's take for a disc,

• MR²/2 = Mk²

•$\ k= \dfrac{R}{ \sqrt{2}}$

》so, k is only dependent on the distance from the axis.

It is totally the measure of distance.

☆Uses;

》We can observe here that, more the 'k' will be for an object, more will be its moment of inertia.

》so, its generally used for estimating the extent of inertia of different random objects.

》like,for eg, let's take a wheel, if we say that it has some "k", then it will be having estimated 'I', and we get how much required torque we needed to apply for its rotation.

☆Similarities between CM and k.

》Centre of mass: It is that point where whole mass of the system is assumed to be located.

• if initially some $\ F_{net}$ are applied on the system, and those forces if we now apply on that CM point we will get the same results.

》same thing we'd done with k also, we assume the whole body as point mass and placed it at some distance ,ie, k.

• and after rotating it about the axis, the results of inertia is same, as it was during rotating the whole continuous rigid system.

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Answer 2

Radius of Gyration :

• It is the radius of a ring for which it will have same Moment of Inertia (along geometrical axis) as that of a body rotating along a specified axis.

• Let Moment of Inertia for any object along a specific axis be I , then

$\therefore \: I = m {r}^{2}$

$\implies \: {r}^{2} = \dfrac{I}{m}$

$\implies \: r = \sqrt{ \dfrac{I}{m} }$

Necessity of Radius of Gyration:

• It is a direct index representing the moment of inertia of a body , i.e. directly proportional.

• Now, whenever, the mass of an object is distributed near the axis, Radius of Gyration will be less and so will be MI.

• Similarly vice versa can be stated.

Similarity with Centre of Mass :

• Centre of mass imagines a particular point where the entire mass can be thought to be concentrated. Using this we can easily understand the translatory motion of an object.

• Similarly, radius of direction imagines a particular distance from an access via the entire mask and it ought to be concentrated with which we can easily understand the rotational motion of an object.

Hope It Helps.