Radius of Gyration

 

Radius of Gyration 

Definition: The radius of gyration of a body about an axis may be defined as the distance from the axis of a mass point whose mass is equal to the mass of the whole body and whose moment of inertia is equal to the moment of inertia of the body about the axis. 

Explanation: The radius of gyration gives the distance from a body to an imaginary point. This imaginary point is such that the moment of inertia at that point is equal to the moment of inertia of the entire body. It essentially means that if the entire mass of the body were concentrated at that single point, we would observe the same amount of moment of inertia about that rotational axis. 

 In terms of the radius of gyration, the formula for moment inertia is: 

The moment of inertia is represented by I, while the mass of the body is represented by m. 

 As a result, the gyration radius is as follows:

The moment of inertia of any rigid body may be calculated using the radius of gyration (1). 

Consider a body made up of n particles, each has weight of m. Let us consider for the perpendicular distance from the pivot of rotation to particles. In terms of the radius of gyration, we can see that the condition produces the moment of inertia (1). 

We can calculate the body’s moment of inertia by substituting the qualities in the condition. 

If all particles have the same mass, equation (3) may be rewritten as

As a result, we may express mn as M, which denotes the body’s overall mass. 

Therefore, the equation will be:

We may deduce the following from equation (4) and (1):

From final equation, we can say that the radius of gyration is the root-mean-square distance of various parts of the body. It is derived from the rotational axis. 

Physical significance of radius of gyration : 

(a) Radius of gyration depends upon shape and size of the body. 

(b) It measures the distribution of mass about the axis of rotation. a thin rod has a smaller radius of gyration than a thick rod of the same length because its mass is concentrated closer to the axis of rotation. 

(c ) If the particles of the body are distributed close to the axis of rotation, the radius of gyration is less. (d) If the particles are distributed away from the axis of rotation the radius of gyration is more. 

(e) Radius of gyration is independent of mass of the body 

The radius of gyration changes with respect to change in the orientation of the axis of rotation. One of the most prominent uses of the radius of gyration is to understand how an object would behave if it were compressed along a given axis of rotation. Through the radius of gyration, it becomes easier to find the moment of inertia of a body and subsequently solve other problems with respect to the rotational motion of a body.

The center of mass (CM) coordinates locates a point where if the entire mass M of a system of particles or that of a rigid body can be thought to be concentrated such that the acceleration of this point mass obeys Newton's second law of motion, viz., Fnet=MaCM , where Fnet is the sum of all the external forces acting on the body or on the individual particles of the system of a particles. Similarly, radius of gyration locates a point from the axis of rotation where the entire mass M can be thought to be concentrated such that the angular acceleration of that point mass about the axis of rotation obeys the relation, τnet=Mα , where τnet is the sum of all the external torques acting on the body or on the individual particles of the system of particles. 

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