15.12 : Equation of Motion: General Plane motion

In the context of a rigid body's movement within a general plane, it is important to understand that this motion is typically triggered by external forces or couple moments exerted onto it. This principle can be explained through Newton's second law, which stipulates the translational motion of the body's center of mass along each axis.

Moreover, the body's center of mass experiences a rotational effect as a result of these couple moments. This rotation can be articulated as the product of the angular acceleration and the moment of inertia. These equations of motion can be extended to any point on the body. This can be achieved by representing the moment equation as the sum of two components: the moment about the center of mass and the moment due to the translational motion about a specific point on the body.

For an object that rolls without slipping, there is a specific moment equation for the point in contact with the floor, point O. This equation is the summation of the moment due to the translational motion of the center of mass and the moment due to the center of mass itself. In such instances, the moment equation can be expressed using the parallel axis theorem. This theorem allows us to represent the moment equation in terms of the moment of inertia of the object at about point O. The exploration of these principles provides valuable insights into the intricate dynamics of a rigid body's motion within a general plane.