# What is the purpose of radius of gyration

• I
• TheBlueDot
In summary, the conversation is about finding the Lagrangian for a mass rolling down an incline with given values for mass, radius, and radius of gyration. The confusion lies in whether to use r or Rg for the moment of inertia, and it is clarified that Rg is given to be equal to r/\sqrt{2}. The object in question is not explicitly stated to be a solid disk or cylinder, hence the uncertainty in using the \frac{m}{2}r^2 formula.
TheBlueDot
Hello,

I'm given a problem of a mass rolling down an incline with mass 'm', radius 'r', and radius of gyration Rg, and I need to write the Lagrangian for the motion. I'm confused on why both r and Rg are given. Don't I just need one of the two for the moment of inertia?
Thanks!

The radius of gyration gives the moment of inertia via the equivalence of the given object and an object with all its mass concentrated at a distance of the radius of gyration from the center of mass. So $I = m R_g^2$.

You can treat the object as a massless cylinder of the given physical radius with a hollow cylinder of mass m and radius $R_g$ embedded within it.

Hi jambaugh,

Thank you for your response. The part I'm confused on is if I use Rg the moment of inertia (assume a cylinder) would be I= mRg^2, but if I use 'r', then I=1/2 *mr^2. I guess I don't understand where the question is leading...

Why would $I = \frac{m}{2} r^2$? Is $R_g$ given to be equal to $r/\sqrt{2}$? Did the problem specifically say the object was a solid disk or cylinder? Maybe the object is not what you are assuming with the $\frac{m}{2}r^2$ formula.

## What is the purpose of radius of gyration?

The radius of gyration is a measure of how spread out or concentrated the mass of an object is in relation to its axis of rotation. It is commonly used in the study of rigid body dynamics and can help determine the stability, strength, and performance of structures and systems.

## How is radius of gyration calculated?

The radius of gyration is calculated by taking the square root of the ratio of the object's moment of inertia to its mass. It is represented by the symbol k and can be calculated for any object, regardless of its shape or size.

## What does a larger radius of gyration indicate?

A larger radius of gyration indicates that the mass of an object is more spread out from its axis of rotation. This means that the object is less compact and may have a lower moment of inertia, making it easier to rotate.

## Why is the radius of gyration important in sports and athletics?

In sports and athletics, the radius of gyration can help determine the performance and stability of equipment, such as bats, clubs, and racquets. It can also be used to analyze the movements and techniques of athletes and improve their performance.

## What is the difference between radius of gyration and center of mass?

The radius of gyration and center of mass are both measures of an object's distribution of mass, but they are not the same. The center of mass is the point at which an object's mass is balanced, while the radius of gyration is a measure of how spread out the mass is from the axis of rotation. The two concepts are related, but serve different purposes in the study of physics and mechanics.

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