describe radius of gyration?
Answers
Radius of Gyration
Radius of gyration is defined as the distance from the axis of rotation to a point where the total mass of the body is supposed to be concentrated, so that the moment of inertia about the axis may remain the same. Simply, gyration is the distribution of the components of an object. It is denoted by K. In terms of radius of gyration, the moment of inertia of the body of mass M is given as,
Inertia (I) = MK2
Suppose a body consists of n particles each of mass m. Let r1, r2, r3, ... , rn be their perpendicular distances from the axis of rotation. Then, the moment of inertia I of the body about the axis of rotation is
Formula of moment of inertia
If all the particles are of same mass m, then
Moment of inertia if all mass is same
Since mn = M, total mass of the body,
Formula of inertia in terms of total mass or body and radius
From the above equations, we have
Radius of gyration is the root mean square distance of particles from axis formula
Therefore, the radius of gyration of a body about a given axis may also be defined as the root mean square distance of the various particles of the body from the axis of rotation.
Radius of gyration
To understand radius of gyration, you need to know what moment of inertia, which is the quantity that relates how much torque is needed to change the angular velocity of a particle or a collection of particles.
Now, for the interesting part. you can treat the collection of particles / continuous object as if it were a single point where the all the mass was concentrated at certain distance from the center. This distance is call Radius of Gyration.
Couple of other examples are given below:
1) It gives you an idea of the load under which a pillar or a column will buckle.
2) One example I am familiar with is the measure of comfort level on a ship; the radius of gyration has an effect on it.
Ships at sea as they encounter waves experience different motions like rolling or pitching and each of these motions have radius of gyration associated with them as the axis of rotation is different for each motion. So for pitch and roll we have pitch radius of gyration and roll radius of gyration. Now these radii depends upon the distribution of mass of the vessel.
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