Plane Kinematics of rigid bodies
Plane Kinematics of rigid bodies: A rigid body executes plane motion when all parts of the body move in parallel planes.We generally consider the plane of motion to be the plane that contains the mass center and we treat the body as a thin slab with motion confined to the plane of the slab. The plane motion of a rigid body may be divided into several categories:
Translation: It is a motion in which every line in the body remains parallel to its original position at all times. In translation, there is no rotation of any line in the body. In rectilinear transition, all points in the body move in parallel straight line.
In curvillinear transition , all points move on congruent curves.
Notice that in translation, the motion of the body is completely specified by the motion of any point in the body, since all point have the same motion. The velocity of the translating body can vary with time in both direction and magnitude.
Rotation: Rotation about a fixed axis is the angular motion about the axis. Play the following animation for obtaining the rotation of the rod about the axis passing through point O and perpendicular to the plane.
It follows that all particles move in circular paths about the axis of rotation and all lines in the body which are perpendicular to the axis of rotation rotate through the same angle in the same time.
General plane motion of rigid body is a combination of transition and rotation.
Rotation: If a rigid body moves so that along some straight line all the particles of the body, or a hypothetical extension of the body, have zero velocity relative to some reference, the body is said to be in rotation relative to this reference. The line of stationary particles is called the axis of rotation.
The rotation of a rigid body is described by its angular motion of any straight line of the body.
Supposing there are two straight lines, 1 and 2 in the body, which are at an angle of b with respect to each other. At an instant of time, the lines make an angle of with a fixed horizontal line, which is taken as a reference.
In the figure, Upon differentiation with respect to time,
Thus, all lines on a rigid body in its planes of motion have the same angular displacement , the same angular velocity, and the same angular acceleration.