1. http://www.physics.usyd.edu.au/~gfl/Lecture Physics 1901 (Advanced) A/Prof Geraint F. Lewis Rm 557, A29 gfl@physics.usyd.edu.au www.physics.usyd.edu.au/~gfl/Lecture

2. http://www.physics.usyd.edu.au/~gfl/Lecture Rotational Motion So far we have examined linear motion; Newton’s laws Energy conservation Momentum Rotational motion seems quite different, but is actually familiar. Remember: We are looking at rotation in fixed coordinates, not rotating coordinate systems.

3. http://www.physics.usyd.edu.au/~gfl/Lecture Rotational Variables Rotation is naturally described in polar coordinates, where we can talk about an angular displacement with respect to a particular axis. For a circle of radius r, an angular displacement of  corresponds to an arc length of Remember: use radians!

4. http://www.physics.usyd.edu.au/~gfl/Lecture Angular Variables Angular velocity is the change of angle with time There is a simple relation between angular velocity and velocity

5. http://www.physics.usyd.edu.au/~gfl/Lecture Angular Variables Angular acceleration is the change of  with time Tangential acceleration is given by

6. http://www.physics.usyd.edu.au/~gfl/Lecture Rotational Kinematics Notice that the form of rotational relations is the same as the linear variables. Hence, we can derive identical kinematic equations: LinearRotational a=constant constant v=u+at  o + t s=s o +ut+½at 2  o + o t+½t 2

7. http://www.physics.usyd.edu.au/~gfl/Lecture Net Acceleration Remember, for circular motion, there is always centripetal acceleration The total acceleration is the vector sum of a rad and a tan. What is the source of a rad ?

8. http://www.physics.usyd.edu.au/~gfl/Lecture Rotational Dynamics As with rotational kinematics, we will see that the framework is familiar, but we need some new concepts; LinearRotational MassMoment of Inertia ForceTorque

9. http://www.physics.usyd.edu.au/~gfl/Lecture Moment of Inertia This quantity depends upon the distribution of the mass and the location of the axis of rotation.

10. http://www.physics.usyd.edu.au/~gfl/Lecture Moment of Inertia Luckily, the moment of inertia is typically; where c is a constant and is <1. ObjectI Solid sphere2/5 M R 2 Hollow sphere2/3 M R 2 Rod (centre)1/12 M L 2 Rod (end)1/3 M L 2

11. http://www.physics.usyd.edu.au/~gfl/Lecture Energy in Rotation To get something moving, you do work on it, the result being kinetic energy. To get objects spinning also takes work, but what is the rotational equivalent of kinetic energy? Problem: in a rotating object, each bit of mass has the same angular speed , but different linear speed v.

12. http://www.physics.usyd.edu.au/~gfl/Lecture Energy in Rotation For a mass at point P Total kinetic energy

13. http://www.physics.usyd.edu.au/~gfl/Lecture Parallel Axis Theorem The moment of inertia depends upon the mass distribution of an object and the axis of rotation. For an object, there are an infinite number of moments of inertia! Surely you don’t have to do an infinite number of integrations when dealing with objects?

14. http://www.physics.usyd.edu.au/~gfl/Lecture Parallel Axis Theorem If we know the moment of inertia through the centre of mass, the moment of inertia along a parallel axis d is; The axis does not have to be through the body!

15. http://www.physics.usyd.edu.au/~gfl/Lecture Torque Opening a door requires not only an application of a force, but also how the force is applied;  It is ‘easier’ pushing a door further away from the hinge.  Pulling or pushing away from the hinge does not work! From this we get the concept of torque.

16. http://www.physics.usyd.edu.au/~gfl/Lecture Torque Torque causes angular acceleration Only the component of force tangential to the direction of motion has an effect Torque is

17. http://www.physics.usyd.edu.au/~gfl/Lecture Torque Like force, torque is a vector quantity (in fact, the other angular quantities are also vectors). The formal definition of torque is where the £ is the vector cross product. In which direction does this vector point?

18. http://www.physics.usyd.edu.au/~gfl/Lecture Vector Cross Product The magnitude of the resultant vector is and is perpendicular to the plane containing vectors A and B. Right hand grip rule defines the direction

19. http://www.physics.usyd.edu.au/~gfl/Lecture Torque and Acceleration At point P, the tangential force gives a tangential acceleration of This becomes

20. http://www.physics.usyd.edu.au/~gfl/Lecture Torque and Acceleration For an arbitrarily shaped object We have the rotational equivalent of Newton’s second law! Torque produces an angular acceleration. Notice the vector quantities. All rotational variables point along the axis of rotation. (Read torques & equilibrium 11.0-11.3 in textbook)