what do you mean by radius of gyration derive expression for it
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Answer:
The moment of inertia of a body about an axis is sometimes represented using the radius of gyration. Now, what do you mean by the radius of gyration? We can define the radius of gyration as the imaginary distance from the centroid at which the area of cross-section is imagined to be focused at a point in order to obtain the same moment of inertia. It is denoted by k.
Radius of Gyration Formula
The formula of moment inertia in terms of the radius of gyration is given as follows:
I = mk2 (1)
where I is the moment of inertia and m is the mass of the body
Accordingly, the radius of gyration is given as follows
k=Im−−√ (2)
The unit of the radius of gyration is mm. By knowing the radius of gyration, one can find the moment of inertia of any complex body equation (1) without any hassle.
Consider a body having n number of particles each having a mass of m. Let the perpendicular distance from the axis of rotation be given by r1, r2, r3,…,rn. We know that the moment of inertia in terms of radius of gyration is given by the equation (1). Substituting the values in the equation, we get the moment of inertia of the body as follows
Radius of Gyration
………… (3)
If all the particles have the same mass then equation (3) becomes
Radius of Gyrationbecomes
Radius of Gyration
We can write mn as M which signifies the total mass of the body. Now the equation becomes
Radius of Gyration
………… (4)
From equation (4), we get
Radius of Gyration
From the above equation, we can infer that the radius of gyration can also be defined as the root-mean-square distance of various particles of the body from the axis of rotation.