College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA Chapter 11ROTATION 11.1 The Motion of Rigid Bodies Rigid bodies A rigid body is defined as a body of which the distance between any two particles is not changed. This is another ideal model for dealing with physical problems.

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA The classification of the motion of rigid bodies (i) Translational motion: The motion in which any straight line in a rigid body is always parallel; (ii) Rotation about a fixed axis: The motion in which any particle in a body is in a circular motion of a radius about an identical straight line keeping constant position; (iii) Rotation about a fixed point; (iv) A planar motion of a rigid body: The motion in which any particle in the body is always in its plane; (v) A general motion of a rigid body.

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA 11.2Angular Kinematics Rotation about a fixed axis In a rotation about a fixed axis, any particle in the body turns the same angle as others though the distance to the axis may be different. Therefore, we can chose an angular coordinate to represent the position of the rigid body. Angular displacement: Average angular velocity: Instantaneous angular velocity:

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA Average angular acceleration: Instantaneous angular acceleration: On the other hand, with some initial conditions, angular equation can be solved:

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA Relation of linear and angular quantities The relationship between linear and angular quantities: Further definition of angular velocity vector: And angular acceleration vector:

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA Thus,

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA 11.3 Kinetic Energy of Rotation Kinetic energy of rotation The kinetic energy of a rigid body is Rotational inertia:

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA Calculating the rotational inertia For continuously distributed mass: For a collection of point masses : It is related to the mass distribution about an axis.

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA Parallel-axis theorem: Normal-axis theorem:

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA 11.4 Newton’s Second Law in Angular Form Generally, torque is defined as: Torque

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA The torque of a force about a axis is the component of the torque about a point on the axis.

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA Newton’s second law for rotation We consider the simple situation: The torque: Generally, we have

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA Newton’s second law for rotation We consider the simple situation: The torque: Generally, we have

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA From the rotation law: We get the work-kinetic energy theorem for rotation

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA If the force acting on the body is conservative, then we have The conservation of mechanical energy in rotational motion:

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA Problems: (on page 239), , , , ,