Gravitational potential energy pg. 43 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).

Objectives Physics terms Investigate examples of gravitational potential energy. Calculate the potential energy, mass, or height of an object using the gravitational potential energy equation. Choose the reference frame and coordinate system best suited to a particular problem. potential energy gravitational potential energy Mass Gravity Height The second and third objective require the student to understand and apply the formula for gravitational potential energy. Once we can calculate an object’s gravitational potential energy, we will be able to analyze transformations between types of energy. Example: a waterfall “powering” a turbine that generates electrical current.

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? (Hint: what would the container do if its released)? If the container is released, the stored energy turns into kinetic energy as the container starts moving.

Gravitational potential energy If the mass of the container increases, what do you think happens to its potential energy? If the height of the container increases, what do you think happens to its potential energy?

Gravitational potential energy Gravitational potential energy comes from work done AGAINST gravity. Think about it, gravity is always pulling things down towards Earth so whenever you lift anything up, you’re opposing the work gravity is doing to pull the item down. For example the work you do when you lift a bottle of water up. Whoever or whatever raised it had to “fight” gravity and work against a force, namely the bottle’s weight.

W = Fd = mgh = PE F = mg d = h Gravitational potential energy The gravitational potential energy stored in an object equals the work done to lift it. W = Fd = mgh = PE F = mg d = h To lift an object, you must exert an upward force equal to the object’s weight. The distance you lift it is the height h. 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.

Equations mass Potential Energy height gravity The change in gravitational potential energy of an object is its mass multiplied by “g” and by the change in height. You can think of PE as the energy of position. 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). Remember at Earth’s surface, g = 9.81 m/s2

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?”

An example How much PE does this water bottle gain when it’s lifted 1m? G U E S 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.

Athletics and energy How much energy does it take to raise a 70 kg (154 lb) person one meter off the ground? G U E S 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? 686 J

Math Practice 1. What is the potential energy of a 0.15kg ball when it is 1.37 meters above the floor? 2. What is the energy of the same ball when it is 13.7 m above the floor? 3. How does the potential energy of a 1.5 kg ball raised 10 m off the floor, compare to the 0.15 kg ball? 4. Suppose a battery contains 500 J of energy. What is the heaviest object the battery can raise to a height of 30 meters? 5. 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. Multiplying the height by 10 (while keeping the mass constant) has multiplied the energy tenfold.

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.

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.)

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.”

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.”

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.

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

How do you choose? The height you use depends on the problem you are trying to solve. Only the change in height actually matters when solving potential energy problems. So how do you know where h = 0? YOU get to set h = 0 wherever it makes the problem easiest to solve.

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). In this case potential energy is equal to 14.7J relative to the floor. 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). In this case, the potential energy to 58.8J relative to the bottom of the hole.

Does the path matter? Thing 1 & Thing 2 want to get to the top of a mountain. Thing 1 hikes up a winding trail. Thing 2 takes the secret elevator straight to the top. Which one has the greatest potential energy at the top? They 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. Thing 1 Thing 2

Homework What do each of the symbols mean in this equation: PE = mgh? Translate the equation PE = mgh into a sentence with the same meaning. How much PE does a 1 kg mass gain when raised by a height of 10 meters? For question 3, ask “Would it be less or more potential energy than that on the Moon?”

Homework How high would a 2 kg mass have to be raised to have a gravitational potential energy of 1,000 J? 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? This is the most convenient reference point: the altitude of base camp.