1. Chapter 6
Work & Energy

2. Work & Energy Unit
So far, we’ve been obsessed with motion and how it relates to Newton’s Laws and forces.
Now we will be concerned with two scalar quantities, work & energy and how they are always conserved or remain constant.

3. Work & Energy Wait, what?! the ability to do work a transfer of energy

4. Topics in Chapter 6 Work Done by a Constant Force
Work Done by a Varying Force
Kinetic Energy, and the Work-Energy Principle
Potential Energy
Conservative and Nonconservative Forces
Mechanical Energy and Its Conservation
Problem Solving Using Conservation of Mechanical Energy
Other Forms of Energy; Energy Transformations and the Law of Conservation of Energy
Energy Conservation with Dissipative Forces: Solving Problems
Power

5. Work Done by a Constant Force
Definition
Important Info
a force acting through a distance
the product of the magnitudes of the displacement times the component of force parallel to the displacement.
Scalar Quantity
Magnitude only
Symbol: W
Units: Joule (J)
1 J = 1 Nm
*W = Fd
W – Work (J)
F – Force (N)
d – displacement (m)

6. Work Done by a Constant Force
Recap - For work to be done:
A force must be applied.
The object must move.
The object must move in the direction of the force.
Is the force in the direction of motion?
No – So, is work being done?
What about a force acting at an angle to the displacement?
As long as this person does not lift or lower the bag of groceries, he is doing no work on it.
The force he exerts has no component in the direction of motion – it is perpendicular to the direction of motion.

7. Work Done by a Constant Force
The work done by a constant force is defined as the distance the object moved multiplied by the component of the force in the direction of displacement:
θ is the angle between the direction of the F and d

8. Solving Work Problems Draw a free-body diagram.
Choose a coordinate system.
Apply Newton’s laws to determine any unknown forces.
Find the work done by a specific force.
To find the net work, either:
find the net force and then find the work it does, or
find the work done by each force and add.

9. Example 1
A 50 kg crate is pulled 40 m along a horizontal floor by a constant force exerted by a person, Fp = 100 N, which acts at a 37° angle as shown. The floor is rough and exerts a friction force Ff = 50 N. Determine the work done by each force acting on the crate, and the net work done on the crate. mg

10. Example 1 - Solution Draw a FBD. Choose a coordinate system. FN Fp FfFg
Draw a FBD.
Choose a coordinate system.
Let’s choose +x to be in the direction of motion.

11. Example 1 - Solution Find any unknown forces using Newton’s 2nd.Ff FN Fp Fg
Find any unknown forces using Newton’s 2nd.
All forces are known (or can be determined easily).
Determine the work done by each force.
d = 40 m
Fpx

12. Example 1 - Solution Determine the work done by each force.
Ff = 50 N FN
Fp = 100 N Fg
d = 40 m
Determine the work done by each force.
WN = perpendicular to direction of motion = zero
Wg = perpendicular to direction of motion = zero
Wf = Fdcosθ = (50 N) (40 m) (cos 180) = J
WpII=Fdcosθ = (100 N) (40 m) (cos 37) = 3200 J
Determine the net work done on the crate.
Wnet = (-2000 J) + (3200 J) = 1200 J
θ = 37°

13. Example 2
Determine the work a hiker must do on a 15.0 kg backpack to carry it up a hill of height h = 10.0 m.
Determine the work done by gravity on the backpack.
Determine the net work done on the backpack. For simplicity, assume the motion is smooth and the velocity is constant.

14. Homework Assignment Concept Questions Practice Problems p. 160 #3-6
xkcd.com

15. WORK DONE BY A VARYING FORCE
What if the applied force is not constant - it keeps changing?
How can we determine work done by a varying force?
We could do my favorite thing!
Let’s graph it! 
Plot Fcosθ vs. d.
What represents the work done on this type of graph?
The work done between any two points on the curve is equal to the area under the curve between those two points.
Try #12 (a & b) on p.162.

16. Kinetic Energy
Kinetic Energy – the energy of motion.
A moving object can do work on another moving object that it strikes - it can exert a force on an object and move it a distance. (Therefore, W is done.)
KE is measured in Joules.
Translational Kinetic Energy:
KE is a scalar quantity that is always either zero or greater than zero. Why?

17. Work-Energy Principle
The net work done on an object is equal to the change in the kinetic energy of the object.
If the net work is positive, the kinetic energy increases.
If the net work is negative, the kinetic energy decreases.

18. Potential Energy Potential Energy is the energy of position.
Examples of potential energy:
An object at some height above the ground
A stretched or compressed spring
A stretched elastic band or string

19. Gravitational Potential Energy
In raising a mass m to a height h, the work done by the external force is
Therefore, we define the gravitational potential energy:
Ug = mgy

20. Gravitational Potential Energy
The energy an object has because of its position at some height above the ground.
Ug = mgh
Notice, the higher an object is, the more gravitational PE it has.

21. Gravitational Potential Energy
This potential energy can become kinetic energy if the object is dropped.
Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces).
If Ug = mgy, where do we measure y from?
It doesn’t matter, as long as we are consistent about where we choose y = 0.
Only changes in potential energy can be measured.

22. Elastic Potential Energy
Potential energy can also be stored in a spring when it is compressed; the figure below shows potential energy yielding kinetic energy.

23. Elastic Potential Energy
The force required to compress or stretch a spring is:
Where:
k is called the spring constant, and needs to be measured for each spring.
x is the displacement from the unstretched position.
Also known as the spring equation or Hooke’s Law.

24. Elastic Potential Energy
The force increases as the spring is stretched or compressed.
We find that the potential energy of the compressed or stretched spring, measured from its equilibrium position, can be written:
Elastic PE, or Us = ½ kx2
Note: For a spring, we choose the reference point for zero PE at the spring’s natural position.

25. Assignment Concept Question p.161 #9 Practice Problems
Gravitational PE
Elastic PE
Concept Question
p.161 #9
Practice Problems
Review Example 6-7 on p.146
Do #27, 28, 30, 31 on p.163
Concept Question
p.161 #7
Practice Problems
Do #26, on p.163

26. Conservative and Nonconservative Forces
Forces for which the work done does not depend on the path taken but only on the initial and final positions.
Example: Gravity
Forces for which the work done depends not only on the starting and ending points but also on the path taken.
Example: Friction

27. Conservative and Nonconservative Forces
Although potential energy is always associated with a force with a force, not all forces have potential energy.
Potential energy can only be defined for conservative forces.

28. Assignment Review Examples 6-4, 6-5, and 6-6 on page 143-144.
Complete the Following Problems on p.162:
#15-20

29. Conservative and Nonconservative Forces
Therefore, we must distinguish between the work done by conservative forces and the work done by nonconservative forces.
We find that the work done by nonconservative forces is equal to the total change in kinetic and potential energies:
WNC = ΔKE + ΔPE

30. Mechanical Energy and Its Conservation
If there are no nonconservative forces, the sum of the changes in the kinetic energy and in the potential energy is zero – the kinetic and potential energy changes are equal but opposite in sign.
This allows us to define the total mechanical energy (E):
E = KE + PE
And its conservation:
E1 = E2 = constant

31. Problem Solving Using Conservation of Mechanical Energy
In the image on the left, the total mechanical energy is:

32. Problem Solving Using Conservation of Mechanical Energy
If there is no friction, the speed of a roller coaster will depend only on its height compared to its starting height.

33. Problem Solving Using Conservation of Mechanical Energy
For an elastic force, conservation of energy tells us:

34. Energy
Law of Conservation of Energy: Energy cannot be created nor destroyed. (It can change form.)
Changes are NOT perfect – When energy changes form, some energy is lost as heat (thermal energy).
Friction also usually accompanies these changes – still lost as heat.
Though there are several types of energy, the total energy of a closed system remains constant.
Energy is a scalar quantity expressing the capacity to do work.
There are many types of energy. Most fit into two broad categories:
Potential Energy
Kinetic Energy

35. Other Forms of Energy, Energy Transformations and the Conservation of Energy
Some other forms of energy:
electric energy
nuclear energy
thermal energy
chemical energy
Work is done when energy is transferred from one object to another.
Accounting for all forms of energy, we find that the total energy neither increases nor decreases.
Energy as a whole is conserved.

36. Energy Conservation with Dissipative Processes; Solving Problems
Problem Solving: See p. 157
Draw a picture.
Determine the system for which energy will be conserved.
Figure out what you are looking for, and decide on the initial and final positions.
Choose a logical reference frame.
Apply conservation of energy.
Solve.
If there is a nonconservative force such as friction, where do the kinetic and potential energies go?
They become heat; the actual temperature rise of the materials involved can be calculated.

37. Power Power is the rate at which work is done.
In the SI system, the units of power are watts.
1 W = 1 J/s = 1 N∙m/s
1 horsepower = 746 W
So, the difference between walking and running up stairs is power – the work done and change in gravitational potential energy is the same.

38. Power
Power is also needed for acceleration and for moving against the force of gravity.
The average power can be written in terms of the force and the average velocity:

39. Summary of Chapter 6 Work: Kinetic energy is energy of motion:
Potential energy is energy associated with forces that depend on the position or configuration of objects.
The net work done on an object equals the change in its kinetic energy – the work-energy principle.
If only conservative forces are acting, mechanical energy is conserved.
Power is the rate at which work is done.

40. Practice Problems Conservation of Mechanical Energy
Law of Conservation of Energy
p.164 #
Power
p.165 # 58, 59, 60