1. Gravitational potential energy
This is the fourth lesson in our unit on work and energy. In it, we will use the fact that weight is a force to calculate the energy required to lift something.
This energy is gravitational potential energy—a form of energy that depends on an object’s position within a gravitational field.
This energy has the potential of doing future work, and we can store energy for later use by elevating massive objects or substances (like water).

2. Equations or
The change in gravitational potential energy of an object is its mass multiplied by “g” and by the change in height.
At Earth’s surface, g = 9.8 N/kg, or 9.8 kg m/s2
Advanced students might be interested to learn that the unit “N/kg” is actually an acceleration—it is the same as “meters per second squared”!
Writing out the newton as (kilogram x meter)/(second x second), and then dividing by kg, will prove as much.
But in this context, N/kg is a more useful unit because it emphasizes that the Earth pulls on each kilogram of an object’s mass with 9.8 N (about 2.2 pounds).

3. How do we know that the energy is there?
Gravitational potential energy
This heavy container has been raised up above ground level.
Due to its height, it has stored energy—gravitational potential energy.
How do we know that the energy is there?

4. Gravitational potential energy
This heavy container has been raised up above ground level.
Due to its height, it has stored energy—gravitational potential energy.
If the container is released, the stored energy turns into kinetic energy.

5. Gravitational potential energy
If the mass of the container increases, its potential energy will also increase.
If the height of the container increases, its potential energy will also increase.

6. Gravitational potential energy
The gravitational potential energy of an object is . . .
These are two commonly used ways to write “potential energy.”
Ask: Translate this formula into a complete sentence. (Answer on next few slides.)
Let the first letter of each variable on the right-hand side of the formula be a reminder:

7. Gravitational potential energym m
The gravitational potential energy of an object is the mass m in kilograms . . .x
… “m” is for “mass” (in kilograms) …

8. Gravitational potential energy
The gravitational potential energy of an object is the mass m in kilograms multiplied by the local acceleration due to gravity g (which is 9.8 m/s2 near Earth’s surface) . . .
… “g” is for “gravity” (given in newtons per kilogram) …
Note: the parenthetical phrase is not necessarily required as part of the statement.

9. Gravitational potential energyh h
The gravitational potential energy of an object is the mass m in kilograms multiplied by the local acceleration due to gravity g (which is 9.8 m/s2 near Earth’s surface), multiplied by the height h in meters.
… and “h” is for height, in meters.

10. Gravitational potential energy
How can you give an object gravitational potential energy?

11. Gravitational potential energy
How can you give an object gravitational potential energy?
Gravitational potential energy comes from work done against gravity ...

12. Gravitational potential energy
How can you give an object gravitational potential energy?
Gravitational potential energy comes from work done against gravity ...
… such as the work you do when you lift this bottle of water.
Whoever or whatever raised it had to “fight” gravity and work against a force, namely the bottle’s weight.

13. Gravitational potential energy
Where does the formula for gravitational potential energy come from?
Ep = mgh
Give students a chance to figure this out for themselves. Hint: where did the formula for kinetic energy come from?

14. Gravitational potential energy
Where does the formula for gravitational potential energy come from?
Ep = mgh
The gravitational potential energy stored in an object equals the work done to lift it.
The gravitational potential energy is the work done against gravity to lift an object. This is a big idea! The energy acquired by a system is the work done on the system to change its state from one with lower energy to one with higher energy. Now let’s define gravitational potential energy, or “potential energy” for short.
“Ep” equals mass x g x height.

15. Deriving the formula W = Fd Work is force times distance.
Recall this formula from the previous lesson.

16. Deriving the formula W = Fd F = mg Work is force times distance.
To lift an object, you must exert an upward force equal to the object’s weight.
F is the object’s weight in this case—which is sometimes called W, but that would create a notation problem with work W!

17. Deriving the formula W = Fd F = mg d = h Work is force times distance.
The distance you lift it is the height h.
When you lift an object at constant velocity, you must apply an upward force equal to the downward force of gravity on the object.
And that force of gravity has a value that is just “mg” – the product of mass and the strength of gravity.
As for the distance is just the height to which the object has been lifted.

18. Deriving the formula W = Fd = mgh = Ep F = mg d = h
Work is force times distance.
W = Fd = mgh = Ep
F = mg
d = h
force = weight distance = height

19. An example
A 1.0 kg mass lifted 1.0 meter gains 9.8 joules of gravitational potential energy.
Advanced students may like to discuss how the units of “g” emphasize what gravity does:
namely, it exerts a certain amount of force on each kilogram of an object’s mass.

20. Exploring the ideas Click this interactive calculator on page 259
The interactive calculator that accompanies this lesson will allow us to solve problems like those we saw earlier in the assessment slide. Ask the students to find this interactive calculator button in their e-Book and click it.

21. Engaging with the concepts
What is the potential energy of a 1.0 kg ball when it is 1.0 meter above the floor?
Grav. potential energy
1.0
9.81
1.0
Let’s work through a few examples.

22. Engaging with the concepts
What is the potential energy of a 1.0 kg ball when it is 1.0 meter above the floor?
Ep = 9.8 J
Grav. potential energy
9.8
1.0
9.81
1.0
What is the energy of the same ball when it is 10 m above the floor?
Watch the position of the red ball on the vertical, left-hand scale.

23. Engaging with the concepts
What is the potential energy of a 1.0 kg ball when it is 1.0 meter above the floor?
Ep = 9.8 J
Grav. potential energy 98
1.0
9.81 10
What is the energy of the same ball when it is 10 m above the floor?
Ep = 98 J
Multiplying the height by 10 (while keeping the mass constant) has multiplied the energy tenfold.

24. Engaging with the concepts
How does the potential energy of a 10 kg ball raised 10 m off the floor, compare to the 1 kg ball?
Grav. potential energy 10
9.81 10
Now let’s investigate the effect of increasing mass. (When mass changes, the size of the red ball changes.)

25. Engaging with the concepts
How does the potential energy of a 10 kg ball raised 10 m off the floor, compare to the 1 kg ball?
Grav. potential energy
It is 10 times greater, or 980 J.
980 10
9.81
1.0
Potential energy is also directly proportional to the object’s mass (represented here by the size of the red sphere alongside the left-hand scale).

26. Engaging with the concepts
Suppose a battery contains 500 J of energy.
What is the heaviest object the battery can raise to a height of 30 meters?
Mass
500
9.81 30 10
Prompt students to suggest which of the variables in the formula is the “unknown.”
(It’s “m” and the problem is, at least implicitly, asking for mass in kilograms.)

27. Engaging with the concepts
Suppose a battery contains 500 J of energy.
What is the heaviest object the battery can raise to a height of 30 meters?
1.7 kg
Mass
500
1.7
9.81 30
Students could also be encouraged to predict how much mass could be lifted 30 meters on the Moon or another planet with a different value for g.
They could then test their predictions when working with the calculator.
Ask them what new piece of information they would need. (Hint: what variable have we not yet changed in any way? )

28. Engaging with the concepts
The energy you use (or work you do) to climb a single stair is roughly equal to 100 joules.
How high up is a 280 gram owlet that has 100 J of potential energy.
Height
100
0.280
9.81
980
This problem amounts to solving for “h” when “Ep” and “m” are given.

29. Engaging with the concepts
The energy you use (or work you do) to climb a single stair is roughly equal to 100 joules.
How high up is a 280 gram owlet that has 100 J of potential energy.
36.4 meters
Height
100
0.280
9.81
36.4
980
Point out that units for mass and height must be in kilograms and meters because a joule is a kg meter squared per second squared.

30. Athletics and energy
How much energy does it take to raise a 70 kg (154 lb) person one meter off the ground?
Advanced students may point out that “70 kg” is a measure of mass, while ‘154 lb.” is a measure of weight (which is a kind of force).
Other details: pounds (lb.) are a British unit of force; a British unit of MASS is the “slug”!
Discussion question: If you took the person from Earth to the Moon, one of these quantities (70 and 154) would change. Which one? Would it get bigger or smaller?

31. Athletics and energy
How much energy does it take to raise a 70 kg (154 lb) person one meter off the ground?
This is a good reference point. It takes 500 to 1,000 joules for a very athletic jump.
The energy it takes to raise the person becomes the person’s stored potential energy.
Discussion question: If you took the person from Earth to the Moon, one of the “givens” (70 kg and 154 lb.) would change. Which one? Would it get bigger or smaller?
What does this imply for how high a Moonwalking astronaut might be able to jump?

32. Typical potential energies
Here are some other objects and heights that students can use to get a sense of typical gravitational potential energy values. Possible point to make: If the baseball was thrown, it would ALSO have kinetic energy. Ask: “The arrow has 9.8 joules of energy at a height of 1100 meters. How many joules does it have at only one meter?”

33. Reference frames and coordinate systems
When calculating kinetic energy, you need to chose a reference frame.
Typically, we choose the Earth as our reference frame.
We treat the Earth as if it is at rest.
Point out that typically we consider the Earth to be at rest, and we attach our coordinate axis frame to the Earth.
The kinetic energy of an object depends on the choice of reference frame. If we used the Earth as our reference frame, then the ball that the robot is holding has zero kinetic energy. But if we use the Sun as our reference frame then the ball has the same velocity as the Earth, and a huge kinetic energy!

34. Reference frames and coordinate systems
When calculating gravitational potential energy, we need to choose where to put the origin of our coordinate system.
In other words, where is height equal to zero?
When a student calculates gravitational potential energy, they also have to make decisions about the frame of reference and coordinate axis. Again, typically they will attach their coordinate axis system to the Earth, but they will need to decide where to put the origin. Where is h = 0 meters? This determines the number of joules of potential energy that an object has, but it won’t effect the CHANGE in joules as the object moves from one place to another.

35. Determining height Where is zero height?
Perhaps elaborate: “That is, where does h equal zero? Where is the potential energy zero?” Point out that wherever height is zero, potential energy also is zero by definition.
Elicit opinions as to what zero potential energy might mean. Write some of these ideas on a whiteboard or blackboard for comparing to the next few slides.

36. Determining height Where is zero height? the floor?
the ground outside?
the bottom of the hole?
“Is there only one right answer to this question?” (Pause for student input if time allows.)

37. Determining height
If h = 1.5 meters, then the potential energy of the ball is 14.7 joules.
“That’s enough energy to make a flashlight glow for about 15 seconds.”

38. Determining height
If h = 4 meters, then the potential energy is 39.2 J.
“Thirty-nine joules will keep the flashlight on for about 40 seconds.”

39. Determining height
If h = 6 meters, then the potential energy is 58.8 J.
“And 59 joules will keep it lit for a minute or so!”
Note: it can be valuable for students to hear you mentally round a quantity up or down to a reasonable number of significant figures.

40. Which is correct? 14.7 J? 39.2 J? 58.8 J? Which answer is correct?
“How can we answer this question? Can we?” (Take a poll by show of hands or a shout-out.)

41. Which is correct? All are correct! 14.7 J? 39.2 J? 58.8 J?
“All are correct! … If you thought so, give yourself a hand!”

42. How do you choose?
The height you use depends on the problem you are trying to solve …

43. How do you choose?
The height you use depends on the problem you are trying to solve …
… because only the change in height actually matters when solving potential energy problems.
Discussion thread: “Suppose someone is going to drop a baseball onto your head.
Would you rather have it fall 6 meters or 1 ½ before hitting your skull?” (Pause for contributions.)
“OK, I’ll only drop it 1 ½ meters, then. I can do that by standing on a chair while you kneel on the floor.”
“So my next question is … does it matter if we do this cruel little experiment in the basement, on the ground floor, or on the roof?”
(This may take some time to discuss!) “Does it hurt any more in one place than another, as long as it always goes 1 ½ meters?”

44. How do you choose? So how do you know where h = 0?
“If I’m dropping a baseball onto your head, I might want to say zero height is where the top of your head is … even if you’re six feet tall!”

45. You decide So how do you know where h = 0?
YOU get to set h = 0 wherever it makes the problem easiest to solve.
Usually, that place is the lowest point the object reaches.
“If I’m dropping a baseball onto your head, I might want to say zero height is where the top of your head is … even if you’re six feet tall!”

46. Pick the lowest point
If the ball falls only as far as the floor, then the floor is the most convenient choice for zero height (that is, for h = 0).
We say that our reference frame is defined by setting “h” equal to zero on the floor.
What if you’re far from the Earth’s surface, the way the Moon is? What quantity do you think would determine the Moon’s gravitational potential energy?
What if you’re an electron in outer space, and the only other material object for miles around is a proton (a hydrogen nucleus)? What kind of potential energy do you think is involved here?

47. Pick the lowest point relative to the floor.
In this case, the potential energy at the position shown here (at the level of the dashed line) is
relative to the floor.
We say “in any case” because the DIFFERENCE in potential energy between two places on Earth is ALWAYS 14.7 joules if the object is 1 kg and one of those places is 1.5 meters higher than the other.

48. Reference frames
If the ball falls to the bottom of the hole, then the bottom of the hole is the best choice for zero height (that is, for h = 0).
Likewise here, what really matters is the fact that the ball will fall 6 meters.

49. Reference frames
In this case, the potential energy at the position shown here (at the level of the dashed line) is
relative to the bottom of the hole.
Saying that the bottom of the hole is “zero height” just makes it easier to figure out how much energy is involved when the ball falls 6 meters.

50. Reference frames
Gravitational potential energy is always defined relative to your choice of location for zero height.
These “E sub p’s” really tell us how much work gravity will do pulling the ball down to the floor, the ground, or the bottom of the hole.

51. Reference frames
Gravitational potential energy is always defined relative to your choice of location for zero height.
And unlike kinetic energy, it can even be negative!
Possible extension: Pause here and ask the students how much energy could come if the ball rolled off the ground and into the hole.
If needed, guide them to realize that this would be the difference between 58.8 J and 39.2 J (that is, it would be 19.6 J).

52. Does the path matter?
A set of identical twins wants to get to the top of a mountain.
One twin hikes up a winding trail.
The second twin takes the secret elevator straight to the top.
Which twin has the greatest potential energy at the top?
When a student calculates gravitational potential energy, they also have to make decisions about the frame of reference and coordinate axis. Again, typically they will attach their coordinate axis system to the Earth, but they will need to decide where to put the origin. Where is h = 0 meters? This determines the number of joules of potential energy that an object has, but it won’t effect the CHANGE in joules as the object moves from one place to another.

53. Path independence The twins have the SAME potential energy at the top.
It doesn’t matter HOW they gained height. Changes in potential energy are independent of the path taken.
When a student calculates gravitational potential energy, they also have to make decisions about the frame of reference and coordinate axis. Again, typically they will attach their coordinate axis system to the Earth, but they will need to decide where to put the origin. Where is h = 0 meters? This determines the number of joules of potential energy that an object has, but it won’t effect the CHANGE in joules as the object moves from one place to another.

54. Assessment
What does each of the symbols mean in this equation: EP = mgh?
Hint: remember that each variable is the first letter of an important term!
Allow time for students to write down their answers.

55. Assessment
What does each of the symbols mean in this equation: EP = mgh? m = mass in kg g = the strength of gravity in N/kg h = the change in height in meters
Translate the equation EP = mgh into a sentence with the same meaning.
And if you use these units (kilograms, newtons PER kilogram, and meters), your energy will always be in joules.
Allow time for students to write down their answers to the second prompt.

56. Assessment
What does each of the symbols mean in this equation: EP = mgh? m = mass in kg g = the strength of gravity in N/kg h = the change in height in meters
Translate the equation EP = mgh into a sentence with the same meaning. The change in gravitational potential energy of an object is its mass multiplied by “g” and multiplied by the change in height.
How much EP does a 1 kg mass gain when raised by a height of 10 meters?
Don’t be thrown off by the fact that we wrote “E sub p” in Number 2 and “P E” in Number 1. They mean the same thing.
Different books, teachers, and web sites use different symbols for potential energy, so it’s worth learning how to go back and forth.
On a standardized test, look for the formula sheet or reference section to see how the test does it.
Allow time for students to write down their answers to the third prompt.

57. Assessment
What does each of the symbols mean in this equation: EP = mgh? m = mass in kg g = the strength of gravity in N/kg h = the change in height in meters
Translate the equation EP = mgh into a sentence with the same meaning. The change in gravitational potential energy of an object is its mass multiplied by “g” and multiplied by the change in height.
How much EP does a 1 kg mass gain when raised by a height of 10 meters? EP = mgh = 98 joules
For question 3, ask “Would it be less or more potential energy than that on the Moon?”

58. Assessment
How high would a 2 kg mass have to be raised to have a gravitational potential energy of 1,000 J?
Hint or discussion thread: Go back to the formula. What is the unknown in this question? What are the givens?
Allow time for students to write down their answers to the fourth prompt.

59. Assessment
How high would a 2 kg mass have to be raised to have a gravitational potential energy of 1,000 J? h = EP/mg = 51 m
Mountain climbers at the Everest base camp (5,634 m above sea level) want to know the energy needed reach the mountain’s summit (altitude 8,848 m). What should they choose as zero height for their energy estimate: sea level, base camp, or the summit?
Note on problem solving: some students will do better with symbolic algebra, dividing Ep by mg and THEN putting in the given values.
Others will prefer to write 1000 J = (2 kg)(9.8 N/kg)(h) and then rearrange to isolate “h.” In that case, h = (1000)/(2X9.8) meters = 51 m. Either approach is valid.

60. Assessment
How high would a 2 kg mass have to be raised to have a gravitational potential energy of 1,000 J? h = EP/mg = 51 m
Mountain climbers at the Everest base camp (5,634 m above sea level) want to know the energy needed reach the mountain’s summit (altitude 8,848 m). What should they choose as zero height for their energy estimate: sea level, base camp, or the summit?
The climbers are located at the base camp, so their change in gravitational potential will be relative to the base camp. They should therefore set the base camp’s altitude as zero height.
This is the most convenient reference point: the altitude of base camp.

61. Assessment
Which location is most convenient to choose as the zero height reference frame if the robot tosses the ball into the hole?
Allow time for students to answer.

62. Assessment
Which location is most convenient to choose as the zero height reference frame if the robot tosses the ball into the hole?
Setting h = 0 at the lowest place that the object reaches means the potential energy will always be positive. This makes the problem easier to solve.
h = 0 m
The fewer minus signs you have in a problem, the fewer the errors!
What if you’re far from the Earth’s surface, the way the Moon is? What quantity do you think would determine the Moon’s gravitational potential energy?
What if you’re an electron in outer space, and the only other material object for miles around is a proton (a hydrogen nucleus)? What kind of potential energy do you think is involved here?