### RADIUS OF GYRATION:

- Radius of Gyration is the distance from the axis of rotation to a point where all of the mass or area can be concentrated to yield the same resistance to angular motion.

- An averaging out of the radii (r) of all the mass particles. This allows all the mass or area to be represented by a single radius (k).

- Mathematically the radius of gyration is the root mean square distance of the object’s parts from either its center of mass or a given axis, depending on the relevant application.

I = mk² or I = Ak²

- The distribution of an object’s mass has a much The distribution of an object’s mass or area has a much greater affect on the moment of inertia than mass.

### Principal Axes for any Object

**Maximum Moment of Inertia Axis (Imax):** It is the Axis along which the moment of inertia is largest i.e. maximum moment of inertia.

** Minimum Moment of Inertia Axis (Imin):** It is the Axis along which the moment of inertia is smallest i.e. minimum moment of inertia.

** Intermediate Moment of Inertia Axis (Iint):** It has an intermediate moment of inertia. Determined not by its moment of inertia value, but rather because it is perpendicular to the both Imax and Imin. Note: All three axes are perpendicular to each other

**Note:** All three axes are perpendicular to each other.

### Use of Radius of Gyration

Calculating the radius of gyration is relatively easy, so it is often used to study various dynamic systems. For example,

**In the molecular system**, we can use the radius of gyration (**K**) to measure the effective size of a polymer. If the value of**K**is small, then we can say the polymer is relatively compact.

- Similarly,
**in structural engineering**, we can use**K**to measure the stiffness of the column.

- We can also use the radius of gyration to determine the distribution of the mass in the axis of rotation. If the value of
**K**is small, then the mass is close to the axis of rotation. However, if**K**is large, the mass is far from the axis of rotation.

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