![moment of inertia of a circle by integration moment of inertia of a circle by integration](https://i.ytimg.com/vi/I3Qn4aZUzg8/maxresdefault.jpg)
A simplified version of this new relationship states that the moment will be equal to the mass times the distance squared times the angular acceleration. If we take these two substitutions and put them into the original F = m a equation, we can wind up with an equation that relates the moment and the angular acceleration for our scenario. We can also relate the linear acceleration of the mass in that the linear acceleration is the angular acceleration times the length of the rod (d). In this case the moment will be related to the force in that the force exerted on the mass times the length of the stick (d) is equal to the moment. To relate the moment and the angular acceleration, we need to start with the traditional form of Newton's Second Law, stating that the force exerted on the point mass by the stick will be equal to the mass times the acceleration of the point mass (F = m a). We are attempting to rotate the mass about it's left end by exerting a moment there. A point mass on the end of a massless stick.
![moment of inertia of a circle by integration moment of inertia of a circle by integration](https://www.intmath.com/applications-integration/img/moment-inertia-b.png)
We want to relate the moment exerted to the angular acceleration of the stick about this point. Imagine we want to rotate the stick about the left end by applying a moment there. To see why this relates moments and angular accelerations, we start by examining a point mass on the end of a massless stick as shown below. The mass moment of inertia is a moment integral, specifically the second, polar, mass moment integral. The Mass Moment of Inertia and Angular Accelerations This page will only discuss the integration method, as the method of composite parts is discussed on a separate page. Just as with Area Moments of Inertia, the mass moment of inertia can be calculated via moment integrals or via the method of composite parts and the parallel axis theorem. This is represented in an equation with the rotational version of Newton's Second Law. The Mass Moment of Inertia represents a body's resistance to angular accelerations about an axis, just as mass represents a body's resistance to linear accelerations.