Chapters 6, 7 Energy

1. Chapters 6, 7 Energy

2. Energy • What is energy? • Energy - is a fundamental, basic notion in physics • Energy is a scalar, describing state of an object or a system • Description of a system in ‘energy language’ is equivalent to a description in ‘force language’ • Energy approach is more general and more effective than the force approach • Equations of motion of an object (system) can be derived from the energy equations

3. Scalar product of two vectors • The result of the scalar (dot) multiplication of two vectors is a scalar • Scalar products of unit vectors

4. Scalar product of two vectors • The result of the scalar (dot) multiplication of two vectors is a scalar • Scalar product via unit vectors

5. Some calculus • In 1D case

6. Some calculus • In 1D case • In 3D case, similar derivations yield • K – kinetic energy

7. James Prescott Joule (1818 - 1889) • Kinetic energy • K = mv2/2 • SI unit: kg*m2/s2 = J (Joule) • Kinetic energy describes object’s ‘state of motion’ • Kinetic energy is a scalar

8. Work-kinetic energy theorem • Wnet – work (net) • Work is a scalar • Work is equal to the change in kinetic energy, i.e. work is required to produce a change in kinetic energy • Work is done on the object by a force

9. Work: graphical representation • 1D case: Graphically - work is the area under the curve Fx(x)

10. Chapter 6 Problem 52 A force with magnitude F = a√x acts in the x-direction, where a = 9.5 N/m1/2. Calculate the work this force does as it acts on an object moving from (a) x = 0 to x = 3.0 m; (b) 3.0 m to 6.0 m; and (c) 6.0 m to 9.0 m.

11. Net work vs. net force • We can consider a system, with several forces acting on it • Each force acting on the system, considered separately, produces its own work • Since

12. Work done by a constant force • If a force is constant • If the displacement and the constant force are not parallel

13. Work done by a constant force

14. Work done by a spring force • Hooke’s law in 1D • From the definition of work

15. Work done by the gravitational force • Gravity force is ~ constant near the surface of the Earth • If the displacement is vertically up • In this case the gravity force does a negative work (against the direction of motion)

16. Lifting an object • We apply a force F to lift an object • Force F does a positive work Wa • The net work done • If in the initial and final states the object is at rest, then the net work done is zero, and the work done by the force F is

17. James Watt (1736-1819) • Power • Average power • Instantaneous power – the rate of doing work • SI unit: J/s = kg*m2/s3 = W (Watt)

18. Chapter 6 Problem 36 A 75-kg long-jumper takes 3.1 s to reach a prejump speed of 10 m/s. What’s his power output?

19. Conservative forces • The net work done by a conservative force on a particle moving around any closed path is zero • The net work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle

20. Conservative forces: examples • Gravity force • Spring force

21. Potential energy • For conservative forces we introduce a definition of potential energy U • The change in potential energy of an object is being defined as being equal to the negative of the work done by conservative forces on the object • Potential energy is associated with the arrangement of the system subject to conservative forces

22. Potential energy • For 1D case • A conservative force is associated with a potential energy • There is a freedom in defining a potential energy: adding or subtracting a constant does not change the force • In 3D

23. Gravitational potential energy • For an upward direction the y axis

24. Gravitational potential energy

25. Elastic potential energy • For a spring obeying the Hooke’s law

26. Chapter 7 Problem 37 A particle moves along the x-axis under the influence of a force F = ax2 + b, where a and b are constants. Find its potential energy as a function of position, taking U = 0 at x = 0.

27. Conservation of mechanical energy • Mechanical energy of an object is • When a conservative force does work on the object • In an isolated system, where only conservative forces cause energy changes, the kinetic and potential energies can change, but the mechanical energy cannot change

28. Conservation of mechanical energy • From the work-kinetic energy theorem • When both conservative a nonconservative forces do work on the object

29. Internal energy • The energy associated with an object’s temperature is called its internal energy, Eint • In this example, the friction does work and increases the internal energy of the surface

30. Chapter 7 Problem 53 A spring of constant k = 340 N/m is used to launch a 1.5-kg block along a horizontal surface whose coefficient of sliding friction is 0.27. If the spring is compressed 18 cm, how far does the block slide?

31. Conservation of mechanical energy: pendulum

32. Potential energy curve

33. Neutral equilibrium Unstable equilibrium Stable equilibrium Potential energy curve: equilibrium points

34. Questions?

35. Answers to the even-numbered problems Chapter 6 Problem 14: 9.6 × 106 J

36. Answers to the even-numbered problems Chapter 6 Problem 40: The hair dryer consumes more energy.

37. Answers to the even-numbered problems Chapter 6 Problem 50: 360 J

38. Answers to the even-numbered problems Chapter 7 Problem 14: (a) 7.0 MJ (b) 1.0 MJ

39. Answers to the even-numbered problems Chapter 7 Problem 24: (a) ± 4.9 m/s (b) ± 7.0 m/s (c) ≈ 11 m

40. Answers to the even-numbered problems Chapter 7 Problem 38: 95 m