1. Work
Work
is the bridge between
Force
and
Energy.

2. The General Work Equation
W = F Dx cos 
F: force (N)
Dx : displacement (m)
: angle between force and displacement (often 0o)

3. Units of Work SI System: Joule (N m) British System: foot-pound
(not used in Phys B)
cgs System: erg (dyne-cm)
Atomic Level: electron-Volt (eV)

4. More on the Joule SI unit for force = N SI unit for length = m
SI unit for work is = N•m or Joule(I J = 1N•m)
1 Joule of work is done when 1 N acts on a body moving it a distance of 1 meter

5. Counterintuitive Results
There is no work done by a force if it causes no displacement.
Forces perpendicular to displacement can do no work. The normal force and gravity do no work when an object is slid on a flat floor, for instance.

6. Work and Energy
By doing positive work on an object, a force or collection of forces increases its mechanical energy in some way.
The two forms of mechanical energy are called potential and kinetic energy.

7. Kinetic Energy Energy due to motion K = ½ m v2
K: Kinetic Energy in Joules.
m: mass in kg
v: speed in m/s

8. The Work-Energy Theorem
Wnet = DKE
When net work due to all forces acting on an object is positive, the kinetic energy of the object will increase (positive acceleration).
When net work due to all forces is negative, the kinetic energy of the object will decrease (deceleration).
When there is no net work due to all forces acting on an object, the kinetic energy is unchanged (constant speed).

9. Work and graphs
F(x)
The area under the curve of a graph of force vs displacement gives the work done by the force in performing the displacement. xa xb x

10. Power The rate of which work is done.
When we run upstairs, t is small so P is big.
When we walk upstairs, t is large so P is small.

11. Power in Equation Form
P = W/t
work/time
P = F V
(force )(velocity)

12. Unit of Power SI unit for Power is the Watt. 1 Watt = 1 Joule/s
Named after the Scottish engineer James Watt ( ) who perfected the steam engine.

13. How We Buy Energy… 1 kWh = 1000J/s • 3600s = 3.6 x 106J
The kilowatt-hour is a commonly used unit by the electrical power company.
Power companies charge you by the kilowatt-hour (kWh), but this not power, it is really energy consumed.
1 kW = 1000 W
1 h = 3600 s
1 kWh = 1000J/s • 3600s = 3.6 x 106J

14. Hooke’s Law: Springs stretching
Fapp = kx
100
-100
-200
200
F(N) 1 2 3 4 5
x (m) m m x F
Wapp = ½ kx2

15. Springs:compressing Fapp = kx m m x F Wapp = ½ kx2 100 -100 -200 200
Fapp = kx
100
-100
-200
200
F(N) -4 -3 -2 -1
x (m) m m x F
Wapp = ½ kx2

16. Work and Energy
Work done on a system by a force or forces causes a change in energy of the system.
Positive work on a system increases energy in some way.
Negative work decreases energy in some way.

17. More about force types Conservation of Mechanical Energy holds
Conservative forces:
Work in moving an object is path independent.
Work in moving an object along a closed path is zero.
Work done against conservative forces increases potential energy; work done by them decreases it.
Ex: gravity, springs
Non-conservative forces:
Work is path dependent.
Work along a closed path is NOT zero.
Work may be related to a change in total energy (including thermal energy).
Ex: friction, drag
Conservation of Energy holds
only for ALL Forces
Conservation of Mechanical Energy holds
only for Conservative Forces

18. Potential energy
Energy an object possesses by virtue of its position or configuration.
Represented by the letter U.
Examples:
Gravitational Potential Energy
Spring Potential Energy

19. Potential gravitational energy
Ug = mgh

20. Another form of Potential Energy
Springs also can possess potential energy (Us).
Us = ½ kx2
Us: spring potential energy (J).
k: force constant of a spring (N/m).
x: the amount the spring has been stretched or compressed from its equilibrium position (m).

21. Where is Spring Potential Energy Zero?
Us is zero when a spring is in its preferred, or equilibrium, position.
This means the spring is neither compressed nor extended.

22. The Work-Energy Theorem again
Wnet =  U
Work is done when the potential energy of an object changes
For example, if you apply a force to a spring and store energy in the spring, you have done work on the spring. This work is against the way the spring wants to be (relaxed), so in this example, we say this work is negative.

23. Law of Conservation of Energy
In any isolated system, the total energy remains constant.
E = constant

24. Law of Conservation of Energy
Energy can neither be created nor destroyed, but can only be transformed from one type of energy to another.

25. Law of Conservation of Mechanical Energy
E = U + K = C
E = U + K = 0
for gravity
Ug = mghf - mghi
K = ½ mvf2 - ½ mvi2

26. Law of Conservation of Mechanical Energy
E = U + K = C
E = U + K = 0
for springs
Ug = ½ kxf2 - ½ kxi2
K = ½ mvf2 - ½ mvi2

27. Pendulums and Energy Conservation
Energy goes back and forth between K and U.
At highest point, all energy is U.
As it drops, U goes to K.
At the bottom , energy is all K.

28. Pendulum Energy ½mvmax2 = mgh K1 + U1 = K2 + U2
For minimum and maximum points of swing h
K1 + U1 = K2 + U2
For any points 1 and 2.

29. Springs and Energy Conservation
Transforms energy back and forth between K and U.
When fully stretched or extended, all energy is U.
When passing through equilibrium, all its energy is K.
At other points in its cycle, the energy is a mixture of U and K.

30. Spring Energy K1 + U1 = K2 + U2 = E m -x m ½kxmax2 = ½mvmax2 m x All U
All U
All K
K1 + U1 = K2 + U2 = E
For any two points 1 and 2 m -x m
½kxmax2 = ½mvmax2
For maximum and minimum displacements from equilibrium m x

31. Law of Conservation of Energy (with dissipative forces)
E = U + K + Eint= C
U + K +  Eint = 0
Eint is thermal energy.
Mechanical energy may be converted to and from heat.