# What is center of inertia?

• Chemist@
In summary: Where is center of inertia located?In summary, the center of inertia is the point at which the body's mass can be assumed to be concentrated when considering its rotational motion.

#### Chemist@

What is the center of inertia? Is it the same as the center of mass?

What other component of inertia can have space-like - center - dimensions?

I don't understand.

"A https://www.physicsforums.com/wiki/point [Broken], near or https://www.physicsforums.com/wiki/within [Broken] a https://www.physicsforums.com/wiki/body [Broken], at which the body's https://www.physicsforums.com/wiki/mass [Broken] can be https://www.physicsforums.com/wiki/assumed [Broken] to be https://www.physicsforums.com/wiki/concentrated [Broken] when considering its rotational motion and https://www.physicsforums.com/wiki/moment_of_inertia [Broken]. This may be different from its https://www.physicsforums.com/wiki/centre_of_mass [Broken] which is the equivalent for linear motion."

Above from wikidictionary.

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Mathman, all of the links in the body of your text point to nonexistent pages. PF doesn't have a wiki.

How to calculate it?

Can anyone give an example where it is easy to see that the center of mass is not the center of inertia?

As I understand the concept from the rather terse dictionary definition...

Suppose that you have a turntable. On this turntable you have an object. The "center of inertia" of the object is where you could place its entire mass and wind up with the same moment of inertia as the original object.

Suppose, for instance that the object is a thin hoop of mass m, radius r placed flat on the turntable with its center R units from the center of the turntable. The moment of inertia of this hoop with respect to the center of the turntable is ##mr^2 + mR^2##.

Its "center of inertia", C, is at distance ##\sqrt{r^2+R^2}## from the center of the turntable so that the moment of inertia works out to ##mC^2 = mr^2 + mR^2##

Shouldn't moment of inertia be used instead of the masses?

Have a look at radius of gyration (mechanics), it assumes all the mass is concentrated at a single point and radius.
For instance a cylinder with a mass of 10 kg and a radius of 0.1 metres, rotating about its longitudinal axis has a moment of inertia of
0.04 kg - m², the radius of gyration = 0.0632 metres

Is center of inertia a valid concept? I think inertia is quantitative concept.

## 1. What is center of inertia?

The center of inertia, also known as the center of mass, is a point in an object or system of objects where the mass is evenly distributed in all directions. This point is the balance point of the object, meaning that if the object were placed on a pivot at this point, it would remain balanced.

## 2. How is the center of inertia different from the center of gravity?

The center of inertia and the center of gravity are often used interchangeably, but they are slightly different. The center of gravity is the point where the force of gravity acts on an object, while the center of inertia is the point where the mass is evenly distributed. They can be the same point in a uniform gravitational field, but in more complex situations, they may differ.

## 3. How is the center of inertia calculated?

The center of inertia can be calculated by finding the weighted average of the positions of all the particles that make up the object or system. This can be done using the equation:

xc = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn)
where xc is the center of inertia, mn is the mass of the nth particle, and xn is the position of the nth particle.

## 4. Why is the center of inertia important?

The center of inertia is an important concept in physics and engineering because it helps us understand the behavior of objects and systems. It is used in calculations involving rotational motion, stability, and balance. Knowing the position of the center of inertia can also help in designing structures and machines that are stable and efficient.

## 5. Can the center of inertia change?

Yes, the center of inertia can change depending on the distribution of mass in an object or system. For example, if a person moves their arms while standing, the center of inertia of their body will shift. In more complex systems, such as a rotating object, the center of inertia can also change as the object moves and its mass distribution shifts.