1. 6. Work, Energy, and Power

2. The Dot Product

3. 3 where  is the angle between the vectors and A and B are their magnitudes. The dot product is the scalar

4. 4 The Dot Product A few properties of the dot product:

5. 5 The Dot Product The definition of the dot product is consistent with standard trigonometric relationships. For example: Law of cosines

6. 6 The Dot Product The definition where implies

7. Work and Kinetic Energy

8. 8 Energy Principles So far we have solved motion problems by 1. adding up all the forces to get the net force 2. and applying Newton’s laws, e.g., 2 nd Law Another way to solve such problems is to use an alternative form of Newton’s laws, based on energy principles.

9. 9 Energy Principles We are about to deduce an important energy principle from Newton’s laws. However, today physicists view energy principles, such as the conservation of energy, as fundamental laws of Nature that are independent of the validity of Newton’s laws. We start with the concept of kinetic energy.

10. 10 Kinetic Energy First, take the dot product of the 2 nd law with the velocity v Next, integrate both sides with respect to time along a path from point A to point B is the kinetic energy where

11. 11 Kinetic Energy When the right-hand side is integrated, we obtain the difference between the final and initial kinetic energies, K 2 and K 1, respectively:

12. 12 Work The left-hand side is called net work can be rewritten as The quantity

13. 13 The Work-Kinetic Energy Theorem The net work, W, done by the net force on an object equals the change,  K, in its kinetic energy. Energy is measured in joules (J): J = N m Work can be positive or negative. Kinetic energy is always positive.

14. Work-Kinetic Energy Theorem – Examples

15. 15 Example – Lifting a truck A truck of mass 3000 kg is to be loaded onto a ship using a crane that exerts a force of 31 kN over a displacement of 2m. Find the upward speed of truck after its displacement.

16. 16 Example(2) Two forces act on the truck: 1. Gravity w 2. Force of crane F app Apply the work-kinetic energy theorem

17. 17 Example(3) Since the forces are constant over the displacement, we can write the work as that is, as the dot product of the net force and the displacement.

18. 18 Example(4) Work done on truck by gravity Work done on truck by crane

19. 19 Example(5) From the work-kinetic energy theorem we obtain:

20. 20 Example – Compressed Spring Find work done on block for a displacement,  x = 5 cm Find speed of block at x = 0 Hook’s Law m = 4 kg k = 400 N/m

21. 21 Example(2) Compute work done m = 4 kg k = 400 N/m

22. 22 Example(3) m = 4 kg k = 400 N/m

23. 23 Example(4) Now apply work-kinetic energy theorem → v i initial speed v f final speed m = 4 kg k = 400 N/m

24. 24 Example(5) Speed at x = 0 Why did we ignore gravity and the normal force? m = 4 kg k = 400 N/m

25. Power

26. 26 Power Power is the rate at which work is done, or energy produced, or used. If the change in work is  W, in time interval  t, then the average power is given by while the instantaneous power is

27. 27 Power The SI unit of power is the watt (W) named after the Scottish inventor James Watt. W = J / s Example: A 100 watt light bulb converts electrical energy to light and heat at the rate of 100 joules/s.

28. 28 Power Given a force F and a small displacement dr the work done is therefore, the power can be written as that is, the dot product of the force and the velocity.

29. 29 Example – Bicycling A cyclist who wants to move at velocity v while overcoming a force F must produce a power output of at least P = Fv. At 5 m/s against an air resistance of F = 30 N, P = 150 W. However, even going up a gentle slope of 5 o, an 82 kg cyclist (+ bike) needs to output 500 W!

30. 30 Summary The work-energy theorem relates the net work done on an object to the change in its kinetic energy: W = ∆K Work done on an object by a force is Power is rate at which work is done