Rotational Dynamics

1. Rotational Dynamics Chapter 9

2. Expectations After Chapter 9, students will: • calculate torques produced by forces • recognize the condition of complete equilibrium • calculate the location of the center of gravity of a collection of objects • use the rotational form of Newton’s second law of motion to analyze physical situations • calculate moments of inertia

3. Expectations After Chapter 9, students will: • calculate the rotational work done by a torque • calculate rotational kinetic energy • calculate angular momentum • apply the principle of the conservation of angular momentum in an isolated system

4. Preliminary Definitions • Torque • Complete Equilibrium • Center of Gravity

5. Torque Torque: the rotational analog to force Force produces changes in linear motion (linear acceleration). A force is a push or a pull. Torque produces changes in angular motion (angular acceleration). A torque is a twist.

6. Torque length of lever arm Mathematical definition: The lever arm is the line through the axis of rotation, perpendicular to the line of action of the force. SI units: N·m torque force

7. Torque Torque is a vector quantity. It magnitude is given by and its direction by the right-hand rule:

8. Torque For a given force, the torque depends on the location of the force’s application to a rigid object, relative to the location of the axis of rotation. more torque less torque

9. Torque For a given force, the torque depends on the force’s direction.

10. Complete Equilibrium A rigid object is in complete equilibrium if the sum of the forces exerted on it is zero, and the sum of the torques exerted on it is zero. An object in complete equilibrium has zero translational (linear) acceleration, and zero angular acceleration.

11. Center of Gravity In analyzing the equilibrium of an object, we see that where a force is applied to an object influences the torque produced by the force. In particular, we sometimes need to know the location at which an object’s weight force acts on it. Think of the object as a collection of smaller pieces.

12. Center of Gravity In Chapter 7, we calculated the location of the center of mass of this system of pieces: Multiply numerator and denominator by g:

13. Center of Gravity But: Substituting: It is intuitive that the weight force acts at the effective location of the mass of an object.

14. Newton’s Second Law: Rotational Consider an object, mass m, in circular motion with a radius r. We apply a tangential force F: The result is a tangential acceleration according to Newton’s second law.

15. Newton’s Second Law: Rotational The torque produced by the force is But the tangential acceleration is related to the angular acceleration: Substituting:

16. Newton’s Second Law: Rotational This is an interesting result. If we define the quantity as the moment of inertia, we have the rotational form of Newton’s second law.

17. Moment of Inertia The equation gives the moment of inertia of a “particle” (meaning an object whose dimensions are negligible compared with the distance r from the axis of rotation). Scalar quantity; SI units of kg·m2

18. Moment of Inertia Not many real objects can reasonably be approximated as “particles.” But they can be treated as systems of particles …

19. Moment of Inertia The moment of inertia of an object depends on: • the object’s total mass • the object’s shape • the location of the axis of rotation

20. Rotational Work and Energy By analogy with the corresponding translational quantities: Translational Rotational SI units: N·m = J SI units: (kg·m2) / s2 = N·m = J

21. Total Mechanical Energy We now add a term to our idea of the total mechanical energy of an object: total energy gravitational potential energy translational kinetic energy rotational kinetic energy

22. Angular Momentum By analogy with linear momentum: Angular momentum is a vector quantity. Its magnitude is given by and its direction is the same as the direction of w. w must be expressed in rad/s. SI units: kg·m2 / s

23. Angular Momentum: Conservation If a system is isolated (no external torque acts on it), its angular momentum remains constant. [If a system is isolated (no external force acts on it), its linear momentum remains constant.]