Dynamics of Rotational Motion
In classical mechanics, rotational motion of a rigid body can be defined as a motion of an object that keeps the same relative orientation to its initial position. In other words, it is a motion of a rigid body about one or more of its fixed axes, with the fixed axis remaining either fixed or rotating relative to the initial position. Rotational motion can occur about any of three fixed axes, called the principal axes of rotation. These axes lie in a plane which is fixed in space, perpendicular to the initial position, and perpendicular to the body, and the orientation of the object remains the same. The motions of bodies in a rotating reference frame can be divided into rotational motion about a fixed axis, and rotational motion about any of the other axes. Rotational motion about a fixed axis, or rotational motion about any axis of a rotating reference frame, can be described in either a "right-hand" or "left-hand" sense. In the right-hand frame, the motion is counterclockwise when viewed from the position of the axis. In the left-hand frame, the motion is clockwise when viewed from the axis. The rotational motion of a rigid body is produced by a couple of forces: the gravity, and the Coriolis force. The gravity is a force that attracts the object to the center of the earth. By Newton's third law, the object's acceleration is the vector sum of the gravitational force and the frictional force. The frictional force is the sum of the forces that each part of the object exerts on every other part of the object. The frictional force is the sum of the forces of static friction and kinetic friction. The Coriolis force is a force that acts on the object due to the rotation of the earth. It acts on the object in a direction perpendicular to both the rotation axis and the velocity vector of the object. The rotational motion of an object, viewed from a rotating reference frame, can be divided into three motions: translational motion, rotational motion about a fixed axis, and rotational motion about any axis of the rotating reference frame. For example, the motion of a rigid body (body) in a rotating reference frame (frame) about the body's center of mass can be described in terms of the following motions: In a similar way, the motion of a rigid body (body) in the rotating reference frame (frame) can be described in terms of the following motions: The direction of the body's velocity is along the body's axis of rotation. In a rotating reference frame, the velocity of a rigid body is a vector whose direction is perpendicular to the rigid body's rotation axis. The magnitude of the velocity in the rotating reference frame is the object's linear velocity. The magnitude of the velocity in the rotating reference frame is also the object's speed in the rotating reference frame. The rotational motion of a rigid body (object) about a fixed axis can be described in terms of a rotation matrix which relates the linear velocity in a body's local frame of reference to the linear velocity in the rotating reference frame. The rotation matrix for a rigid body about a fixed axis is given by the following formula: where "?" is the rotation angle, "?" is the rotation axis vector in the local frame of reference, and "v" is the speed of the rigid body.
