D. Energy and Society · Work, Energy and PowerGravitational Potential Energy
D3.3 D2.3 D2.1 D3.1About 65 minutes
Learning goals- I can calculate gravitational potential energy with Eg = mgh, measured from a chosen reference level.
- I can explain why only changes in gravitational potential energy (ΔEg = mgΔh) have physical meaning.
- I can relate the work done lifting an object to its gain in gravitational potential energy.
- I can explain why ΔEg does not depend on the path taken.
Words to know
gravitational potential energy (Eg)
Energy stored because of an object's height above a reference level: Eg = mgh.
reference level
The height chosen as h = 0 (for example, the floor or the ground). Eg is measured relative to it.
change in gravitational potential energy (ΔEg)
ΔEg = mgΔh, where Δh is the change in height. It is the same whatever reference level you choose.
gravitational field strength (g)
The force of gravity on each kilogram: about 9.8 N/kg near Earth's surface.
elastic potential energy
Energy stored in an object that is stretched, compressed or bent, such as a spring or a drawn bow.
Energy waiting at the top
A toboggan sitting at the top of a hill is not moving, yet everyone knows what will happen when it is nudged. A dam holds back a lake of water high above the turbines. A heavy icicle hangs from the eaves. Each of these has energy stored because of its height: gravitational potential energy. Lift something and you store energy in it; let it fall and that energy comes back out, usually as kinetic energy.
The formula
To lift an object of mass m at constant velocity, you push up with a force mg. Over a height Δh you do work W = mgΔh (Lesson 3). That work is stored as gravitational potential energy. Measured from a reference level where h = 0:
Eg = mgh
- m in kg, g = 9.8 N/kg, h in m (height above the reference level), Eg in J.
The change in gravitational potential energy when the height changes by Δh is
ΔEg = mgΔh
Raising an object (Δh positive) increases Eg. Lowering it (Δh negative) decreases Eg. Gravity's own work is the opposite: W_gravity = −ΔEg. When a ball falls, gravity does positive work and Eg goes down by the same amount.
You choose the reference level
The value of Eg depends on where you put h = 0. The floor, a tabletop or sea level can all be the reference. Below the reference level, h is negative and so is Eg.
This is not a problem, because only changes in Eg matter. A drop from the shelf to the table releases the same energy whichever reference level you picked. Choose whatever makes the problem easiest, usually the lowest point in the problem, and use it for the whole problem.
Worked exampleExample 1: Two reference levels
A 2.0 kg can of paint sits on a shelf 1.80 m above the floor. A table top is 0.80 m above the floor. (a) Find the can's Eg relative to the floor. (b) Find its Eg relative to the tabletop. (c) The can is moved from the shelf onto the table. Find ΔEg using each reference.
- (a) h = 1.80 m: Eg = (2.0)(9.8)(1.80) = 35.28 J ≈ 35 J.
- (b) h = 1.80 − 0.80 = 1.00 m: Eg = (2.0)(9.8)(1.00) = 19.6 J ≈ 20 J.
- (c) Floor reference: on the table Eg = (2.0)(9.8)(0.80) = 15.68 J, so ΔEg = 15.68 − 35.28 = −19.6 J.
- Tabletop reference: on the table Eg = 0, so ΔEg = 0 − 19.6 = −19.6 J. The same!
Eg is 35 J (floor) or 20 J (tabletop), but the change is −19.6 J either way.
Worked exampleExample 2: Climbing the stairs
A 65 kg student climbs three floors, a vertical height of 10.5 m. By how much does her gravitational potential energy increase?
- ΔEg = mgΔh
- ΔEg = (65 kg)(9.8 N/kg)(10.5 m) = 6688.5 J
- ΔEg ≈ 6.7 × 10³ J
Her Eg increases by about 6.7 × 10³ J (6.7 kJ).
Worked exampleExample 3: Finding a height, and going below zero
(a) A 0.50 kg ball has 24.5 J of Eg relative to the ground. How high is it? (b) A 1.2 kg bucket is lowered into a well, 6.0 m below the ground. What is its Eg relative to the ground?
- (a) h = Eg ÷ (mg) = 24.5 J ÷ (0.50 kg × 9.8 N/kg) = 24.5 ÷ 4.9 = 5.0 m.
- (b) The bucket is below the reference level, so h = −6.0 m.
- Eg = (1.2)(9.8)(−6.0) = −70.56 J ≈ −71 J.
(a) 5.0 m; (b) about −71 J. A negative Eg just means "below the reference level".
Taking the stairs or the ramp to a platform 3.0 m high gives the same increase in Eg. Gravity only cares about the vertical change in height. (Not to scale.)Path does not matter
Whether you take the stairs, a long ramp or an elevator, your Eg at the top is the same. Gravity is vertical, so only vertical motion counts. A ramp lets you push with a smaller force over a longer distance, but the work stored as Eg, mgΔh, is the same. (Friction on a long ramp can add extra work, but that becomes thermal energy, not Eg.)
Forces like gravity, whose work depends only on the start and end points, are the forces we can attach a potential energy to.
Other kinds of potential energy
A stretched elastic, a compressed spring, a drawn bow and a flexed hockey stick store elastic potential energy. Food, fuel and batteries store chemical potential energy. SPH3U calculations focus on Eg; you will meet the formula for elastic potential energy in SPH4U.
Just for funA medium apple holds about 400 kJ of chemical energy. If your muscles could turn all of it into Eg, a 50 kg student could climb about 800 m on one apple: (400 000 J) ÷ (50 kg × 9.8 N/kg) ≈ 816 m. Real muscles are only about one-quarter efficient, so the apple actually gets you about 200 m up. Still an impressive snack.
Hands-on activityFlour craters
You need: a baking pan, flour (enough for a layer about 3 cm deep), a tennis ball or similar ball, a kitchen scale, a ruler or tape measure, and a spoon. Do this on a floor you can sweep, or outside.
- Weigh the ball (in kg). Smooth the flour flat with the spoon or a ruler.
- Drop the ball from 0.25 m above the flour. Lift it out carefully and measure the diameter of the crater.
- Smooth the flour and repeat from 0.50 m, 1.00 m and 1.50 m.
- For each height, calculate the ball's Eg relative to the flour surface: Eg = mgh.
- Make a table of Eg and crater diameter. Does more Eg make a bigger crater? Is the relationship a straight line?
What to look for: a 58 g tennis ball from 1.00 m has Eg = (0.058)(9.8)(1.00) ≈ 0.57 J. Higher drops give bigger craters, because more Eg becomes kinetic energy, and then work done pushing flour aside. The crater usually does not grow in direct proportion to the height, because the work goes into moving and packing flour in a complicated way. That is a good thing to discuss in a conclusion.
Your turn
1Find the Eg of a 3.0 kg rock on a ledge 12 m above the ground (ground = reference level).
Eg = (3.0)(9.8)(12) = 352.8 J ≈ 3.5 × 10² J.
2A 45 kg student rides an escalator up 6.0 m. Find her ΔEg.
ΔEg = (45)(9.8)(6.0) = 2646 J ≈ 2.6 × 10³ J.
3A 0.25 kg icicle has 9.8 J of Eg relative to the ground. How high is it?
h = 9.8 ÷ (0.25 × 9.8) = 4.0 m.
4A 5.0 kg box is lowered 2.0 m. Find ΔEg and the work done by gravity.
ΔEg = (5.0)(9.8)(−2.0) = −98 J. W_gravity = −ΔEg = +98 J.
5A climber's Eg relative to the base of a cliff is 1.0 × 10⁴ J. What is her Eg relative to the top of the 20 m cliff if her mass is 51 kg? Explain.
Relative to the base, h = 10 000 ÷ (51 × 9.8) ≈ 20 m, so she is at the top. Relative to the top, h = 0, so Eg = 0 J.
6Why do we say only changes in Eg have physical meaning?
Because the value of Eg depends on the reference level we choose, but the energy released or stored when the height changes, mgΔh, is the same for every reference level.
More practice: warm-up
1Write the formula for gravitational potential energy.
2Find Eg for a 1.0 kg book 2.0 m above the floor.
(1.0)(9.8)(2.0) = 19.6 J ≈ 20 J.
3What value of g do we use near Earth's surface?
4What is the Eg of an object at the reference level?
5Name one object that stores elastic potential energy.
Sample: a stretched elastic band, a compressed spring, a drawn bow, a trampoline mat when stretched.
More practice: on your own
1A 1200 kg elevator rises from the 2nd floor (6.0 m above ground) to the 10th floor (30 m above ground). Find ΔEg.
Δh = 24 m. ΔEg = (1200)(9.8)(24) = 282 240 J ≈ 2.8 × 10⁵ J.
2Ana carries a 20 kg bag up a 15 m staircase. Ben pushes the same bag up a 40 m long ramp to the same landing. Compare the increase in Eg of the bag in each case.
Both raise it 15 m, so ΔEg = (20)(9.8)(15) = 2940 J in both cases. The path doesn't matter.
3On the Moon, g = 1.6 N/kg. How high would you have to lift a 10 kg rock on the Moon to give it the same Eg as lifting it 1.0 m on Earth?
Earth: (10)(9.8)(1.0) = 98 J. Moon: 98 = (10)(1.6)h, so h ≈ 6.1 m.
4Water behind a dam is 50 m above the turbines. How much Eg does each kilogram of water lose as it falls to the turbines?
ΔEg = (1)(9.8)(−50) = −490 J per kilogram.
5A student says, "Eg can't be negative, because energy can't be negative." Correct the statement.
Eg can be negative when an object is below the reference level (h < 0). This is fine because the reference level is chosen by us; only changes in Eg are physically meaningful.
Challenge
1A waterfall has 2.0 × 10³ kg of water going over its edge each second and drops 40 m. How much gravitational potential energy is released each second?
ΔEg = (2.0 × 10³)(9.8)(40) = 7.8 × 10⁵ J each second (that is 784 kW of power, a word you will meet in the Power lesson).
2A 60 kg student climbs a 4.0 m rope in gym class, then slides back down. Using the floor as the reference, find her Eg at the top, and the total change in Eg for the whole climb and slide.
Top: Eg = (60)(9.8)(4.0) = 2352 J ≈ 2.4 × 10³ J. She ends where she started, so total ΔEg = 0.
3A 2.0 kg object is lifted from 1.0 m below a reference level to 3.0 m above it. Find ΔEg two ways: from Eg values and from mgΔh.
Eg values: (2.0)(9.8)(3.0) − (2.0)(9.8)(−1.0) = 58.8 − (−19.6) = 78.4 J. With Δh = 4.0 m: (2.0)(9.8)(4.0) = 78.4 J ≈ 78 J. ✓
Quiz
1. The gravitational potential energy of a 5.0 kg object 2.0 m above the reference level is:
2. Which quantity does NOT depend on the choice of reference level?
3. A box is carried up a long ramp instead of being lifted straight up the same height. Its increase in Eg is:
4. An object below the reference level has:
5. When a ball falls 3.0 m, gravity does +15 J of work on it. The ball's ΔEg is:
6. A 0.50 kg ball has 9.8 J of Eg. Its height above the reference level is:
Big idea: Gravitational potential energy (D3.3) is stored work. Its value depends on a chosen reference level, but its change does not, and that change is what we use in conservation problems next lesson (D2.3).
How to help: Ask "Where did you choose h = 0?" for each problem. Point out that the path does not matter: taking the stairs or the ramp at a mall to the same level stores the same Eg.
Common difficulties:
- Mixing heights measured from different reference levels in one problem.
- Using the length of a ramp instead of its vertical height.
- Thinking negative Eg is impossible.
- Using mass in place of weight (mg).
Safety and mess: the flour activity is messy but safe. Work over a sweepable floor or outside, and keep flour away from open flames.