16.8:
Equation of Motion for a Rigid Body
The motion of a rigid body can be described using equations for translational motion and rotational motion about the center of mass.
Newton's Second Law gives the equation of motion for translational motion for a center of mass, G, of the body.
The summation of all the moments created about point G is equal to the rate of change of angular momentum of the body.
If an external force is applied at point A, other than point G, it creates a moment that causes the body to rotate.
The angular momentum of point A can be expressed as a vector product of its relative position and relative velocity with respect to point G. The time derivatives of angular momentum give the moment of point A.
Summing over all points within the rigid body, the total moment of the system about point G is calculated.
Using the relative acceleration definition and the distributive law for vectors gives the equation for the total moment about point G due to an external force.
16.8:
Equation of Motion for a Rigid Body
The movement of a rigid object can be understood through the equations that explain both translational and rotational motion about the center of mass of the object, point G. This center of mass is the point where the equation of motion for translational motion comes into play, as per Newton's Second Law.
The combined moments generated about the center of mass of the object are equal to the rate of change of the angular momentum of the body. An external force, when applied at a different point other than the center of mass of the object, causes the body to rotate and generates a moment.
The angular momentum of this point is articulated as a vector product, incorporating its relative position and velocity with respect to the center of mass of the object. The derivative of angular momentum with respect to time provides us with the moment generated at a point where an external force is applied. By summing the moments of all points within the rigid body, one can calculate the total moment of the system about the center of mass of the object.