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. Kinetic energy K = mv2/2 SI unit: kg*m2/s2 = J (Joule)
James Prescott Joule
( )
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. Power Average power Instantaneous power – the rate of doing work
SI unit: J/s = kg*m2/s3 = W (Watt)
James 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 If the spring is compressed 18 cm, how far does the block slide?

31. Conservation of mechanical energy: pendulum

32. Potential energy curve

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

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