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Up to this point we have studied the kinematics and

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Unformatted text preview: m it. Up to this point, we have studied the kinematics and dynamics of a particle whose position in three-dimensional space is completely specified by three coordinates. To describe a change in the position of a body of finite extent, such as a rigid body, is much more complicated. For convenience, it is regarded as a combination of two distinct types of motion: translational motion and rotational motion. Purely translational motion occurs if every particle of the body has the same instantaneous velocity as every other particle, the path traced out by any one particle is exactly the same as the path traced out by every other particle in the body. Under translational motion, the change in the position of a rigid body is specified completely by three coordinates such as x, y, and z giving the displacement of some point, such as the center of mass. Purely rotational motion occurs if every particle in the body moves in a circle about an axis. This axis is simply called the axis of rotation. Then the radius vectors from the axis to all particles undergo the same angular displacement in the same time. The axis of rotation doesn’t have to be through the body. In 1 general, any rotation can be specified completely by the three angular displacements with respect to the rectangular-coordinate axes x, y, and z. Any change in the position of the rigid body is thus completely described by three translational and three rotational coordinates. Any displacement of a rigid body may be arrived at by first displacing the body translationally, without rotation; or conversely, first a rotation and then a displacement. We already know that for any collection of particles—whether at rest with respect to one another, as in a rigid body, or in relative motion, as in the exploding fragments of a shell—the motion of the center of mass is completely determined by the resultant of the external forces acting on the system of particles is (1) where M is the total mass of the system and acm is the acceleration of the center of mass. This just leaves the matter of describing rotation of the body about the center of mass and relating it to the external forces acting on the body. We shall find that the kinematics of rotation motion have many similarities to the kinematics of translational motion; moreover, the dynamics of rotational motion involve forms of Newton’s second law of motion and of the work-energy theorem that are altogether analogous to those used in part...
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