1. Work and Energy

2. Scalar (Dot) Product When two vectors are multiplied together a scalar is the result:

3. Dot Product Unit vector has a magnitude of 1 and is in the x direction

4. Dot Product Unit vector has a magnitude of 1 and is in the y direction

5. Dot Product Unit vector has a magnitude of 1 and is in the y direction

6. Dot Product

9. 9 Work: The _______done by a constant force acting on an object is equal to the product of the magnitudes of the displacement and the component of the force __________to that displacement 5.1 Work Done by a Constant Force Unit of work: Newton meter (N m) 1 N m = 1 ________(J)

10. 10 5.1 Work Done by a Constant Force If there is a force but no displacement: no work is done.

11. 11 5.1 Work Done by a Constant Force If the force is _______________to the displacement, work is done

12. 12 5.1 Work Done by a Constant Force If the force is at an angle to the displacement, the ________________component of the force does the work

13. 13 5.1 Work Done by a Constant Force If the force (or a component) is in the direction of motion, the work done is _____________. If the force (or a component) is opposite to the direction of motion, the work done is ______________.

14. 14 5.1 Work Done by a Constant Force If there is more than one force acting on an object, it is useful to define the _________work: The total, or net, work is defined as the work done by all the forces acting on the object, or the scalar sum of all those quantities of _________________.

15. 15 5.3 The Work–Energy Theorem: Kinetic Energy

16. 16 5.3 The Work–Energy Theorem: Kinetic Energy We can use this relation to calculate the work done:

17. 17 5.3 The Work–Energy Theorem: Kinetic Energy Kinetic energy is therefore defined: (Kinetic Energy) Unit: Joule (J) The net work on an object changes its kinetic energy.

18. Work Done by Varying Force

19. Work by a Varying Force We allow the size of the Δx’s to approach zero The work done by the varying force is: and

20. Work Done by a Varying Force In vector form:

21. Work Done by a Spring Force exerted by a spring: Where k is the spring constant in N/m and x is the displacement The neg sign indicates that the force is always opposite the displacement from equilibrium

22. Work done by a spring

23. Work Done by a Spring The work done by a spring:

24. Work done by an applied force on a spring

25. Potential Energy

26. From a previous slide

27. Conservative Forces Gravity and the Spring Force are conservative The work done by conservative forces:

28. Conservative Forces Two properties for a force to be conservative 1.The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle 2.The work done by a conservative force on a particle moving through any closed path is zero. (A closed path is one for which the beginning point and the endpoints are identical) Do not change the mechanical energy of a system

29. Conservative & Nonconservative Forces Some conservative forces: – – – Nonconservative Forces – – –

30. Conservative Forces We can define the potential energy function, U

31. Potential Energy Function Suppose we had this equation: The integral would be:

32. Potential Energy Function If we have this equation: We can do the reverse of what we did on the previous slide: The x component of a conservative force acting on an object within a system equals the negative derivative of the PE of the system w/ respect to x

33. Check of Previous Equation

34. Energy Diagrams & Equilibrium of a System   