The moment of inertia of a body about an axis is sometimes represented using the radius of gyration. Radius of gyration or gyradius is an important concept and finds its applications in structural engineering, mechanics.
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Radius of gyration is the distance from the axis of rotation, where the whole total mass of the system is assumed to be concentrated so that the moment of inertia remains the same.
So, the relation between the moment of inertia and the radius of gyration is given by,
Where is the effective distance from the axis rotation and the whole total mass of the system is assumed to be concentrated.
Consider the body of mass is made up of particles each having mass . Let the perpendicular distance of all the particles from the axis about which they are rotating as . Then the moment of inertia of the body can be written as:
Since all the particles have the same mass, therefore
Thus moment of inertia can be written as,
Since the moment of inertia in terms of radius of gyration can be written as,
Comparing this equation with equation,
Or
Thus, radius of gyration can also be defined as the root-mean-square distance of various particles of the body from the axis of rotation
Q1. Find the radius of gyration of a solid uniform sphere of radius R about its tangent.
Ans. By using the parallel axis theorem, we get the following:
On equating the moment of inertia of the sphere with the radius gyration formula, we get,
Q2. A circular disc of radius has a hole of radius at its centre (see figure). If the mass per unit area of the disc varies as , then the radius of gyration of the disc about its axis passing through the centre is :
A circular disc has a hole of radius at its centre. The radius of the disc is . The mass per unit area of the disc is proportional to . What is the radius of gyration of the disc about an axis perpendicular to its plane passing through its centre?
Ans. The moment of inertia about an axis passing through the centre is given by,
Considering a small element of mass and thickness at a distance from the centre,
Mass of the disc,
Radius of gyration,
Q3. Consider a uniform wire of mass and length . It is bent into a semicircle. Find the radius of gyration about a line perpendicular to the plane of the wire passing through the centre.
Ans.
When the wire is bent into the shape of a semicircle, the length of the wire will become the circumference of the semi-circle.
Now, the moment of inertia of the semicircular wire will be,
Q4. Consider a badminton racket with length scales as shown in the figure.Consider the badminton racket with the dimension as shown in the figure. If the mass of the linear and circular portions of the badminton racket are the same and the mass of the threads are negligible,The mass of the racket without the threads is M. The mass of the threads are negligible. find the radius of gyration of the racket about an axis perpendicular to the handle and in the plane of the ring circular portion, at distance from the handle end, of the handle.
Ans.
Total moment of inertia about the shown axis,
Moment of inertia of linear part + Moment of inertia of circular part
Q1. What factors influence the radius of gyration?
Ans. The size and state of the body are two factors that influence the estimation of the radius of gyration. It is the positioning and arrangement of the rotational axis. It also relies on mass appropriation in relation to the body’s rotational axis.
Q2. Is the gyration radius constant?
Ans. No, the radius of gyration is not fixed or constant. Its value is determined by the rotating axis and the distribution of the body's mass around the axis.
Q3. What is the dimensional formula for the radius of gyration?
Ans.
Q4. Why do we calculate the radius of gyration?
Ans. The radius of gyration is used to compare how various structural shapes will behave under compression along an axis. It is used to predict buckling in a compression beam or member.
The radius of gyration is very useful in the analysis of buckling of various structural shapes. The radius of gyration is used to predict the buckling under compressive stress along the axis of a structure. Lesser the radius of gyration the higher the chances of buckling under compression along the axis of the structure. This is why circular or square cross section is the most preferred shape for columns as they have an equal radius of gyration in every radial direction.
Q5. Radius of gyration does not depend on
Ans. (d)
Radius of gyration is independent of the mass of the body but depends on the shape and size of the body, position and configuration of axis of rotation and distribution of mass of the body with respect to axis of rotation.