1. Elastic potential energy
This lesson introduces the concept and mathematical relationship for elastic potential energy. An interactive simulation is used to give students a feel for the concept of elastic potential energy and typical values, rather than immediately bogging them down with the mathematics. The elastic potential energy equation itself is then derived using an important concept: that work done is the area under a force-distance graph.

2. Equations or
The elastic potential energy of a spring is one half the product of its spring constant multiplied by the square of its extension or compression.
Ask: What other energy equation look similar to this one?
Point out that we are using two different notations for elastic potential energy.

3. Work and energy
Energy may be stored in a system when work is done on the system.
Use a demonstration spring or rubber band. Ask students: what can you do to store energy in the spring?
Answer: expand it or compress it. In both cases, you do work on the spring, and your work is stored in the spring as elastic potential energy.
Some springs only compress and some will only stretch, and some types of spring can do both.

4. Springs
free length
This initial portion on springs should be considered a review from the Hooke’s law lesson in an earlier chapter on forces and Newton’s laws. Students should have prior exposure to the “equilibrium length” or “free length” for a spring from an earlier chapter on force and Newton’s laws.

5. Force and deformation When you apply a force to a spring, it deforms.
Depending on the type of spring and the force applied, a spring may either compress or extend from its initial free length.
What direction is the deformation force for a Slinky(R) spring? (Answer: you extend it.)
How about for the spring in a car’s suspension? (Answer: compression.)

6. Work The applied force does work on the spring.
The change in the spring’s length is called the deformation, x.
Ask: What is the equation for work done? (Answer: W = Fd.)
Ask: What is the distance d in this example? (Answer: it is the deformation distance, NOT the new length of the spring.)
The distance d that the force acts equals the deformation of the spring, x.

7. Elastic potential energy
The work done to stretch or compress the spring is stored in the spring as elastic potential energy.
What other kind(s) of potential energy do we know about? (Answer: gravitational potential energy.)
Remind them that any form of stored energy is called potential energy.

8. Equations
The elastic potential energy of a spring is one half the product of its spring constant multiplied by the square of its deformation.
The sentence is equivalent to the equation. At this point, students will probably assume that the equation was pulled out of the blue.
Later in the lesson, this equation will be derived—from this force-distance graph!
Ask: Translate this equation into words using a complete sentence. (Answer on next slide.)

9. What is the spring constant k ?
The spring constant tells you the stiffness of the spring.
The spring constant k is a property of the spring itself. It does not change when the spring is deformed.
Stiff springs have high spring constants.
Weak springs have low spring constants. k
Students need to understand that the spring constant is a property of the spring itself. It does not change even when the length of the spring changes, and the amount of energy stored changes.

10. Units of the spring constant
The spring constant has units of N/m, or newtons per meter.
Example: A 300 N/m spring requires 300 N of force to stretch 1 meter.
A stiff spring needs a large force to stretch it a meter, so it has a large spring constant.
A stiff spring stores more potential energy per meter of stretch.
Ask the students which spring would store more elastic potential energy if you compressed them the same distance, d: a weak spring with a low spring constant, or a stiff spring with a high spring constant.

11. What is x ?
The deformation x is the change in the length of the spring.
It can be positive or negative.
It points in the opposite direction of the spring force.
It has units of meters. x

12. Exploring the ideas Click this interactive calculator on page 261
Students can now explore elastic potential energy using an interactive calculator. The interactive calculator makes it quick and easy for students to see how the equation behaves when variables are changed.

13. Engaging with the concepts
How much elastic potential energy is stored in a spring with a spring constant of 100 N/m if its displacement is 0 meters?
Elastic potential energy
100
Ask the students how much energy they will store if the displacement (the deformation distance) equals zero.

14. Engaging with the concepts
How much elastic potential energy is stored in a spring with a spring constant of 100 N/m if its displacement is 0 meters?
Elastic potential energy
0 joules
100
The elastic potential energy in a spring is zero at its “free length”.
At this equilibrium position, the spring’s restoring force is zero and the elastic potential energy is zero.

15. Engaging with the concepts
If the spring constant is 200 N/m and the spring is deflected by 1.0 cm, how much energy is stored?
Elastic potential energy
200
0.01
Have students perform this calculation using the interactive calculator. Remind them that they should always check their units. Displacement x must be in meters.

16. Engaging with the concepts
If the spring constant is 200 N/m and the spring is deflected by 1.0 cm, how much energy is stored?
Elastic potential energy
only 0.01 J!
0.01
200
0.01

17. Engaging with the concepts
How strong a spring is needed to get 1.0 joule of energy from a 1.0 cm deflection?
Spring constant
1.0
0.01
In this case the words “how strong” are asking for the value of the spring constant, not the force to stretch the spring.

18. Engaging with the concepts
How strong a spring is needed to get 1.0 joule of energy from a 1.0 cm deflection?
Spring constant
k = 20,000 N/m
1.0
20000
0.01
This spring constant is much higher than before.

19. Engaging with the concepts
How strong a spring is needed to get 1.0 joule of energy from a 1.0 cm deflection?
Spring constant
k = 20,000 N/m
1.0
20000
0.01
This is a pretty stiff spring! What might it be used for?

20. Perfect for a mountain bike!
Inside the fork tube is a spring with a spring constant of roughly 20,000 N/m.
If half your weight is on each wheel, then your weight will depress the front fork by a centimeter or two.
(If you weigh 90 pounds, and you put half your weight of 45 pounds on the front wheel, it will depress one centimeter.)

21. Calculating force k = 20,000 N/m
How much force is needed to compress this spring one centimeter?
If half your weight is on each wheel, then your weight will depress the front fork by a centimeter or two.
(If you weigh 90 pounds, and you put half your weight of 45 pounds on the front wheel, it will depress one centimeter.)

22. Calculating force k = 20,000 N/m
How much force is needed to compress this spring one centimeter?

23. Hooke’s law
The spring pushes back in the opposite direction with a force of -200 N.

24. Engaging with the concepts
How much work must be done to stretch a spring with k = 1.0 N/m by 25 cm?
Elastic potential energy
1.0
0.25
Allow time for students to answer using the interactive calculator.
Notice that the word “work” is slipping in again: Remind the student that the work done to to compress the spring equals the potential energy stored in the spring.

25. Engaging with the concepts
How much work must be done to stretch a spring with k = 1.0 N/m by 25 cm?
Elastic potential energy
Only 0.03 J!
This is a very weak spring– looser than a Slinky®.
0.031
1.0
0.25
Note that 25 cm is a significant displacement (Spread your hands apart to demonstrate how big 25 cm is ( about 10 inches .)
Even though the compression is large, the stored energy is small. Ask the students why this is.
( Answer: This spring is very loose—even more so than a Slinky(R). The low spring constant tells us the spring is weak, not stiff.)

26. Engaging with the concepts
How about a k = 100 N/m spring?
How much work must be done to stretch a spring with k = 100 N/m by 25 cm?
Elastic potential energy
100
0.25

27. Engaging with the concepts
How about a k = 100 N/m spring?
How much work must be done to stretch a spring with k = 100 N/m by 25 cm?
Elastic potential energy
3.1
100
0.25
3.1 joules
100 times more energy
Note that when the spring is 100 times stiffer, the elastic potential energy they stored is 100 times larger.

28. Engaging with the concepts
How does the elastic potential energy change if a 100 N/m spring is compressed by 25 cm versus being extended by 25 cm?
Elastic potential energy
100
-0.25
What sign is the displacement for a compressed spring? For an extended one?
How will their stored energies differ? (They won’t if they are extended or compressed by the same displacement.)

29. Engaging with the concepts
How does the elastic potential energy change if a 100 N/m spring is compressed by 25 cm versus being extended by 25 cm?
Elastic potential energy
The potential energy is the same—try other positive and negative values!
3.1
100
-0.25
Why are the answers the same? Point out that in the equation for elastic potential energy, the deflection, x, gets squared.
Allow students time to try out different values.

30. Engaging with the concepts
How does the stored energy change if the spring constant is doubled?
Elastic potential energy
100 1
This is a proportional relationship question. Allow time for students to answer.

31. Engaging with the concepts
How does the stored energy change if the spring constant is doubled?
The energy doubles.
This is true no matter what displacement is used.
Elastic potential energy
200 1
Since energy is directly proportional to the spring constant, doubling one causes the other to double.
This can be checked by entering a value of k that is twice as big again—i.e., 400 N/m—and seeing how the energy doubles.

32. Engaging with the concepts
How does the stored energy change if the displacement is doubled?
Elastic potential energy
100 1
This question involves a power relationship. Allow students time to answer.

33. Engaging with the concepts
How does the stored energy change if the displacement is doubled?
The energy increases by a factor of four (22).
What happens if the displacement is tripled?
Elastic potential energy
100 2
This can be checked by changing the displacement by a factor of two and seeing the energy quadruple. How does the stored energy change if the spring is compressed by three times as much?
Ask: “Where have we seen this kind of relationship before?” (Answer: doubling the velocity of an object causes the kinetic energy to increase by a factor of four).

34. Elastic potential energy
Where does this formula come from?
The next series of slides derives the elastic potential energy equation using a force-distance graph.

35. Elastic potential energy
Hypothesis: The elastic potential energy is derived from the work done to deform the spring from its free length . . .
Ask: What is a hypothesis? How can we test this hypothesis?

36. Work W = Fd Work is force times distance.
By now, students should be comfortable with this equation for work done.

37. Hooke’s law
W = Fd
F = -kx
Lead students through a seemingly logical set of steps. Substitute Hooke’s law, which represents the restoring force when the spring is displaced by x.

38. Hooke’s law W = Fd F = -kx where k is the spring constant in N/m . . .
Lead students through a seemingly logical set of steps. Substitute Hooke’s law, which represents the restoring force when the spring is displaced by x.

39. Hooke’s law W = Fd F = -kx where k is the spring constant in N/m . . .
and x is the change in length of the spring in meters.
F = -kx
Take time to challenge the students: Is the stored work (or energy) for a spring equal to kx2 or ½ kx2?
From the derivation above, if you replace d with x then it would appear to be kx2!!!
What’s going on? (Advanced answer: The force varies with displacement x, so we must calculate the stored work by integrating under the force-distance curve.)
This is worked out graphically on the next series of slides.

40. Force vs. distance BUT the force F from a spring is not constant.
The next series of graphs calculate the stored work as the area under the force distance curve.
The key point is that the force is not constant: by Hooke’s law it varies with distance. So work stored is not W=Fd!

41. Force vs. distance BUT the force F from a spring is not constant.
It starts at zero and increases as the deformation x increases.
On a graph of force vs. distance it is a line of constant slope.
Ask: Why is this an upwardly sloped line? (Answer: Because of Hooke’s law, where force is proportional to distance or displacement.)

42. Force vs. distance The area on this graph . . .
What are the dimensions of the area of a square on this graph? (Answer: force times distance, or N m. Those are the same units of energy, the joule!)

43. Force vs. distance
The area on this graph is force times distance . . .
What is force times distance? (Answer: work done.)

44. Force vs. distance
The area on this graph is force times distance which is the work done!
Ask: What is the equation for the area under that curve?
(It is a triangle, so the area is ½ width times height.)

45. Deriving the equation
The area of this triangle equals the work done to stretch or compress the spring, so it equals the elastic potential energy.
If students don’t remember this easily, have them count squares (or fractions of squares).

46. Deriving the equation
Remind students that this is the formula for the area of a triangle. If students don’t remember this easily, have them count squares (or fractions of squares).

47. Deriving the equation F x
What’s the equation for the force (height) at this position? x
What is the equation for the applied force at this position?

48. Deriving the equation kx x
The applied force at that position is kx. (The spring exerts a force of –kx.)

49. Deriving the equation kx x
If the distance/displacement is x, and the force is kx, then the area of the triangle under the curve is ½ kx2.
The work done in compressing the spring by a displacement x is therefore given by this equation for elastic potential energy!

50. Elastic potential energy
Ep is equal to the work done to deform the spring by an amount x.
Work done on the spring changes the spring’s stored energy.

51. Elastic potential energy
This expression is true for more than just springs!
What other objects might have elastic potential energy?

52. Elastic potential energy
Elastic potential energy is stored in all objects that can deform and spring back to their original shape.
Can you think of examples?

53. Elastic potential energy
such as a rubber band . . .
...or maybe a bungee cord or a trampoline? In fact onlyideal springs obey Hooke’s law perfectly, but most springs can be roughly modeled by Hooke’s law for the elastic portion of their stretch.

54. Typical elastic potential energies

55. Assessment
What do each of the symbols mean in this equation: Ep = ½ kx2 ?
Allow students time to write down their answer.

56. Assessment
What do each of the symbols mean in this equation: Ep = ½ kx2 ? Ep = the elastic potential energy k = the spring constant in N/m x = the displacement of the end of the spring in meters
Translate the equation EP = ½ kx2 into a sentence with the same meaning.
Ensure that students know the units of each quantity, particular for the spring constant.
Allow students time to write down their answer to the second prompt.

57. Assessment
What do each of the symbols mean in this equation: Ep = ½ kx2 ? Ep = the elastic potential energy k = the spring constant in N/m x = the displacement of the end of the spring in meters
Translate the equation EP = ½ kx2 into a sentence with the same meaning. The elastic potential energy of a spring is one half the product of its spring constant multiplied by the square of its extension or compression distance.
How much elastic potential energy is stored in a 100 N/m spring that is compressed 0.10 meters?
Converting an equation into a complete sentence is a useful skill that is particularly good for English language learners.
Allow students time to write down their answer for the third prompt.
The students can use the interactive calculator to solve it.

58. Assessment
What do each of the symbols mean in this equation: Ep = ½ kx2 ? Ep = the elastic potential energy k = the spring constant in N/m x = the displacement of the end of the spring in meters
Translate the equation EP = ½ kx2 into a sentence with the same meaning. The elastic potential energy of a spring is one half the product of its spring constant multiplied by the square of its extension or compression distance.
How much elastic potential energy is stored in a 100 N/m spring that is compressed 0.10 meters? J
Ensure that the units are correct.

59. Assessment
A spring has an elastic potential energy of 100 J when compressed 0.10 m. What is its spring constant?
Allow time for students to answer the question.

60. Assessment
A spring has an elastic potential energy of 100 J when compressed 0.10 m. What is its spring constant?
How far is a spring extended if it has 1.0 J of elastic potential energy and its spring constant is 1,000 N/m?
k = 20,000 N/m
Students may use the interactive calculator to solve these problems. They may need to use the iterative technique (varying the values of the other quantities) in order to answer question #4.
Allow time for students to answer the fifth prompt.

61. Assessment
A spring has an elastic potential energy of 100 J when compressed 0.10 m. What is its spring constant?
How far is a spring extended if it has 1.0 J of elastic potential energy and its spring constant is 1,000 N/m?
k = 20,000 N/m
0.045 m or 4.5 cm
Students may use the interactive calculator to answer the question.

62. Assessment
Are these statements about the spring constant true or false?
___ The spring constant is a measure of the stiffness of the spring.
___ The spring constant tells you how many newtons of force it ……takes to stretch the spring one meter.
___ If a spring stretches easily, it has a high spring constant.
___ The spring constant of a spring varies with x, the amount of …….stretch or compression of the spring.
Hint: the units of the spring constant can help them remember what it means.

63. Assessment
Are these statements about the spring constant true or false?
___ The spring constant is a measure of the stiffness of the spring.
___ The spring constant tells you how many newtons of force it ……takes to stretch the spring one meter.
___ If a spring stretches easily, it has a high spring constant.
___ The spring constant of a spring varies with x, the amount of …….stretch or compression of the spring. T F
Important to reinforce: the spring constant of the spring is a property of the spring itself. It does not change. It does not depend on whether the spring is stretched or compressed, or by how much.