Rotational Dynamics

1. Rotational Dynamics

2. Moment of Inertia The angular acceleration of a rotating rigid body is proportional to the net applied torque: a is inversely proportional to a property of the body called its moment of inertia. Newton’s second law for rotation: SI unit: kg.m2

3. There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object. The moment of inertia also depends upon the location of the axis of rotation.

4. Example: Moment of inertia of a thin ring of mass M about the central axis. The ring is divided into a number of small segments, m1 … These segments are equidistant from the axis.

5. Torque and Angular Acceleration • When a rigid object is subject to a net torque (≠ 0), it undergoes an angular acceleration. • The angular acceleration is directly proportional to the net torque. • This is Newton’s second law for rotation:

6. Other Moments of Inertia

7. Rotational Kinetic Energy An object rotating about some axis with an angular speed, ω, has rotational kinetic energy ½Iω2. Energy concepts can be useful for simplifying the analysis of rotational motion.

8. Total Energy of a System Conservation of Mechanical Energy -- translational, rotational, gravitational -- only conservative forces present (KEt + KEr + PEg)i = (KEt + KEr + PEg)f Potential energies of other conservative forces could be added.

9. If dissipative forces such as friction are present, use the generalized Work-Energy Theorem: Wnc = DKEt + DKEr + DPE

10. Angular Momentum Recall linear momentum: Angular momentum is defined by: For a system with no external torques, total system angular momentum is constant. Conservation of angular momentum:

11. Conservation Rules • In an isolated system, the following quantities are conserved: • Mechanical energy • Linear momentum • Angular momentum