1 Rotation of a Rigid Body Readings: Chapter 13

2 How can we characterize the acceleration during rotation? - translational acceleration and - angular acceleration Newton’s second law:

3 Angular acceleration Center of rotation Both points have the same angular velocity Linear acceleration: Both points have the same angular acceleration

4 Rotation of Rigid Body: Every point undergoes circular motion with the same angular velocity and the same angular acceleration

5 The relation between angular velocity and angular acceleration is the same as the relation between linear velocity and linear acceleration

6 The Center of Mass For Rigid Body sometimes it is convenient to describe the rotation about the special point– the center of mass of the body. Definition: The coordinate of the center of mass: Rigid body consisting of two particles: If then

7 The Center of Mass Definition: The coordinate of the center of mass:

8 The Center of Mass: Example The center of mass of a disk is the center O of the disk O

9 Torque: Rotational Equivalent of Force

10 Torque The rotation of the body is determined by the torque

11 Torque Torque is maximum if Torque is 0 if

12 Torque Torque is positive if the force is trying to rotate the body counterclockwise Torque is negative if the force is trying to rotate the body clockwise axis The net torque is the sum of the torques due to all applied forces:

13 Torque: Example Find the net torque axis

14 Torque: Relation between the torque and angular acceleration: - moment of inertia

15 Moment of Inertia Thin rod, about center Thin rod, about end Cylinder (or disk), about center Cylindrical loop, about center

16 Moment of Inertia: Parallel-axis Theorem If you know the moment of Inertia about the center of mass (point O) then the Moment of Inertia about point (axis) P will be O P

17 Parallel-axis Theorem: Example Thin rod, about center of mass

18 Equilibrium: Massless rod Two forces (which can results in rotation) acting on the rod Equilibrium:

19 Rotational Energy: Conservation of energy (no friction):

20 Kinetic energy of rolling motion Cylinder: Cylindrical loop: Solid sphere: