Moment of Inertia – In this Chapter we will discuss what is moment of inertia with formula, How to get moment of inertia formula, how to calculate moment of inertia formula etc.

The moment of inertia of an object is a calculated measure for a rigid body that is undergoing rotational motion around a fixed axis: that is to say, it measures how difficult it would be to change an object’s current rotational speed. That measurement is calculated based upon the distribution of mass within the object and the position of the axis, meaning that the same object can have very different moment of inertia values depending upon the location and orientation of the axis of rotation.

The **moment of inertia**, otherwise known as the **angular mass** or **rotational inertia**, of a rigid body, is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for the desired acceleration.

It depends on the body’s mass distribution and the axis chosen, with larger moments requiring more torque to change the body’s rotation rate.

It is an extensive (additive) property: for a point mass, the moment of inertia is just the mass times the square of the perpendicular distance to the rotation axis.

The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis).

Its simplest definition is the second moment of mass with respect to distance from an axis. For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters.

For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.

Conceptually, the moment of inertia can be thought of as representing the object’s resistance to change in angular velocity, in a similar way to how mass represents a resistance to the change in velocity in non-rotational motion, under Newton’s laws of motion. The moment of inertia calculation identifies the force it would take to slow, speed up or stop an object’s rotation.

The International System of Units (SI unit) of the moment of inertia is one kilogram per meter squared (kg-m^{2}). In equations, it is usually represented by the variable *I* or *I _{P}* (as in the equation shown).

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## Simple Examples of Moment of Inertia

How difficult is it to rotate a particular object (move it in a circular pattern relative to a pivot point)? The answer depends on the shape of the object and where the object’s mass is concentrated. So, for example, the amount of inertia (resistance to change) is fairly slight in a wheel with an axis in the middle.

All the mass is evenly distributed around the pivot point, so a small amount of torque on the wheel in the right direction will get it to change its velocity.

However, it’s much harder, and the measured moment of inertia would be greater, if you tried to flip that same wheel against its axis, or rotate a telephone pole.

## Using Moment of Inertia

The moment of inertia of an object rotating around a fixed object is useful in calculating two key quantities in rotational motion:

**Rotational kinetic energy:***K*=*Iω*^{2}**Angular Momentum:**=*L**Iω*

You may notice that the above equations are extremely similar to the formulas for linear kinetic energy and momentum, with moment of inertia “*I”* taking the place of mass “*m”* and angular velocity “*ω”* taking the place of velocity “*v*,” which again demonstrates the similarities between the various concepts in rotational motion and in the more traditional linear motion cases.

## Calculating Moment of Inertia

How to Choose breadth and depth in moment of inertia problems?

The graphic on this page shows an equation of how to calculate the moment of inertia in its most general form. It basically consists of the following steps:

- Measure the distance
*r*from any particle in the object to the axis of symmetry - Square that distance
- Multiply that squared distance times the mass of the particle
- Repeat for every particle in the object
- Add all of these values up

For an extremely basic object with a clearly-defined number of particles (or components that can be *treated* as particles), it’s possible to just do a brute-force calculation of this value as described above. In reality, though, most objects are complex enough that this isn’t particularly feasible (although some clever computer coding can make the brute force method fairly straightforward).

Instead, there are a variety of methods for calculating the moment of inertia that are particularly useful. A number of common objects, such as rotating cylinders or spheres, have a very well-defined moment of inertia formulas. There are mathematical means of addressing the problem and calculating the moment of inertia for those objects which are more uncommon and irregular, and thus pose more of a challenge.

## Moment of Inertia Formulas

### What is moment of inertia formula?

The general formula represents the most basic conceptual understanding of the moment of inertia. Basically, for any rotating object, the moment of inertia can be calculated by taking the distance of each particle from the axis of rotation (*r* in the equation), squaring that value (that’s the *r*^{2} term), and multiplying it times the mass of that particle. You do this for all of the particles that make up the rotating object and then add those values together, and that gives the moment of inertia. Now we will discuss what is the formula of moment of inertia?

### what is the formula of moment of inertia?

We defined the moment of inertia I of an object to be for all the point masses that make up the object. Because rr is the * distance to the axis of rotation* from each piece of mass that makes up the object, the moment of inertia for any object depends on the chosen axis.

Using Above Formula You can solve Following types of Question -:

The moment of inertia of a uniform semicircular ring of mass M and radius R about the axis perpendicular to plane of ring is 1 ??

or

How we can find the centre of mass of Earth by using concept of moment of inertia?

To see this, let’s take a simple example of two masses at the end of a massless (negligibly small mass) rod and calculate the moment of inertia about two different axes. In this case, the summation over the masses is simply because the two masses at the end of the barbell can be approximated as point masses, and the sum, therefore, has only two terms.

The consequence of this formula is that the same object gets a different moment of inertia value, depending on how it is rotating. A new axis of rotation ends up with a different formula, even if the physical shape of the object remains the same.

This formula is the most “brute force” approach to calculating the moment of inertia. The other formulas provided are usually more useful and represent the most common situations that physicists run into.

### Write the formula for moment of inertia of a circular disc perpendicular to its plane and explain the symbols.

### Moment of Inertia Formula for Slender Rod, Axis Through One End

A slender rod rotating on an axis that goes through the end of the rod (perpendicular to its length), with mass *M* and length *L*, has a moment of inertia determined by the formula:

I = (1/3)

ML^{2}

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