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C. Energy and Momentum · Work, Energy and Simple Harmonic Motion

Gravitational Potential Energy and Conservative Forces

C2.1 C2.2About 65 minutes
Learning goals
  • I can calculate gravitational potential energy and its change using a reference level I choose.
  • I can relate the work done by gravity to the change in gravitational potential energy (W_g = −ΔEg).
  • I can explain why only the change in height matters, whatever the path.
  • I can distinguish conservative forces (gravity, springs) from non-conservative forces (friction, air resistance).

Words to know

gravitational potential energy
Energy stored because of an object's height in a gravitational field: Eg = mgh near Earth's surface.
reference level
The height chosen as h = 0. Any level can be chosen; only changes in Eg matter.
conservative force
A force whose work between two points does not depend on the path, so its work around a closed path is zero (for example, gravity).
non-conservative force
A force whose work depends on the path, such as kinetic friction or air resistance; it changes mechanical energy into thermal energy.
path independence
The property that the work done by a force depends only on the starting and ending positions.

Energy waiting to happen

A toboggan sitting at the top of a snowy hill is not moving, but it clearly has the potential to speed up. A sap bucket hanging from a maple tree, water behind a dam, a skier on a chairlift: each has energy stored because of its position in Earth's gravitational field. In this lesson you will connect that stored energy to the work done by gravity, and learn why gravity is a special kind of force.

Remember from Grade 11: gravitational potential energy near Earth's surface is Eg = mgh, where h is the height above a chosen reference level, and g = 9.8 N/kg.

Gravitational potential energy and the work done by gravity

When an object moves from height h₁ to height h₂, gravity does work

W_g = −mg(h₂ − h₁) = −ΔEg

So the work done by gravity equals the negative of the change in gravitational potential energy.

  • Going up: Δh > 0, gravity does negative work and Eg increases. Energy is stored.
  • Going down: Δh < 0, gravity does positive work and Eg decreases. Stored energy is released, often as kinetic energy.

Only height matters

In Work Done by a Constant Force in One and Two Dimensions you saw that gravity's work on a ramp, mgΔd cos θ, is the same as mgΔh. That is true for any path, straight or curved: break the path into tiny pieces, and gravity only does work on the vertical part of each piece. So ΔEg depends only on the starting and ending heights, not on the route.

loading dockA: liftB: ramp 4.0 mC: up and down a hill pathΔh = 1.2 m
Three routes from the floor to a loading dock 1.2 m higher. Gravity does the same work, −mgΔh, on a crate along all three, so its change in Eg is the same.

Conservative and non-conservative forces

A force is conservative if the work it does on an object moving between two points does not depend on the path. Equivalently, the work it does around any closed path (one that returns to its start) is zero. Gravity is conservative. So is the force of an ideal spring (you will see this in Elastic Potential Energy).

Only conservative forces have a potential energy. Energy put in by working against them can be completely recovered.

Kinetic friction and air resistance are non-conservative. They always oppose the motion, so they do negative work on every piece of the path, and a longer path means more negative work. Around a closed path their work is not zero. The energy they remove becomes thermal energy (and some sound), and it cannot be fully recovered as mechanical energy.

ceiling (2.5 m)floor (0 m)booktabletop (0.80 m)Reference levelEg of bookΔEg if it falls to floorfloor+15.7 J−15.7 Jtabletop0 J−15.7 Jceiling−33.3 J−15.7 J
The same book on the same table has different values of Eg depending on the reference level chosen, but the change in Eg when it falls to the floor is −15.7 J in every case.

Choosing the reference level

You may choose any convenient reference level (h = 0): the floor, the ground, the lowest point of the motion, the top of a hill. Below the reference level, Eg is negative, which is fine. Only changes in Eg have physical meaning, and changes are the same whatever reference you choose. Pick one level, say which it is, and use it for the whole problem.

A limit: Eg = mgh assumes g is constant, which is true close to Earth's surface. For satellites and spacecraft far from Earth, g gets weaker, and you need Eg = −GMm/r from the sibling unit Gravitational, Electric and Magnetic Fields.

Worked example

Example 1: A ski hill

A 65 kg skier rides a chairlift to the top of a ski hill with a vertical drop of 150 m, then skis to the bottom. Find (a) the change in her Eg on the way up, (b) the work done by gravity on the way down.

  1. (a) ΔEg = mgΔh = (65)(9.8)(+150) = +9.6 × 10⁴ J (95 550 J).
  2. (b) On the way down Δh = −150 m. W_g = −ΔEg = −(65)(9.8)(−150) = +9.6 × 10⁴ J.
  3. The length and shape of the run do not matter: only the 150 m vertical drop.

(a) +9.6 × 10⁴ J; (b) +9.6 × 10⁴ J.

Worked example

Example 2: Reference levels

A 2.0 kg book rests on a table 0.80 m above the floor. The ceiling is 2.5 m above the floor. Find the book's Eg using the floor, the tabletop and the ceiling as reference levels. Then find ΔEg if it falls to the floor, for each choice.

  1. Floor: h = +0.80 m, Eg = (2.0)(9.8)(0.80) = +15.7 J.
  2. Tabletop: h = 0, Eg = 0 J.
  3. Ceiling: h = 0.80 − 2.5 = −1.7 m, Eg = (2.0)(9.8)(−1.7) = −33.3 J.
  4. Falling to the floor, Δh = −0.80 m in every case, so ΔEg = (2.0)(9.8)(−0.80) = −15.7 J every time.

Eg is +15.7 J, 0 J or −33.3 J, depending on the reference; ΔEg is always −15.7 J.

Worked example

Example 3: Path does not matter for gravity, but does for friction

A 30 kg crate is moved at constant speed from the floor onto a loading dock 1.2 m high: (a) by lifting it straight up, (b) by pushing it up a smooth 4.0 m ramp, (c) by pushing it up the same ramp when μ_k = 0.25. Find the work you do in each case.

  1. (a) W = mgΔh = (30)(9.8)(1.2) = 353 J (352.8 J).
  2. (b) Ramp angle: sin θ = 1.2 ÷ 4.0, θ = 17.5°. Push = mg sin θ = 88.2 N; W = (88.2)(4.0) = 353 J. Same as lifting: a ramp needs less force over a longer distance.
  3. (c) Friction: F_f = μ_k mg cos θ = 0.25(294)(0.954) = 70.1 N. Extra work over 4.0 m: (70.1)(4.0) = 280.5 J. Total W = 352.8 + 280.5 = 633 J.
  4. In all three, gravity does −353 J and ΔEg = +353 J. Only (c) turns some of your work into thermal energy.

(a) 353 J, (b) 353 J, (c) 633 J. Gravity is conservative; friction is not.

Worked example

Example 4: A long hiking trail

A 55 kg hiker follows a winding 6.0 km trail from a parking lot at an elevation of 120 m to a lookout at 480 m. What is her change in Eg?

  1. Only the change in height matters: Δh = 480 − 120 = 360 m.
  2. ΔEg = (55)(9.8)(360) = 1.9 × 10⁵ J (194 040 J).
  3. The 6.0 km length of the trail does not appear in the calculation.

About 1.9 × 10⁵ J. Her body uses much more energy than this, because muscles are not very efficient (see Power and Efficiency in Mechanical Systems).

Just for fun

Choosing a reference level is the one time in physics where you get to decide where zero is. Choose the ceiling and everything in the room has negative potential energy, which sounds gloomy but changes no answers.

Hands-on activity

Path independence with a marble and a ruler

You need: a marble, a flexible plastic ruler or a strip of cardboard about 30 cm long, two short pieces of a cardboard tube (or a folded index card) to make a curved track, books, a tape measure, and a towel to catch the marble.

  1. Make a straight ramp: prop one end of the ruler on a stack of books so the top is 10 cm above the table, with the bottom ending at the table edge.
  2. Release the marble from the top and see how far from the table it lands on the towel on the floor. Do 3 trials.
  3. Make a curved ramp from the cardboard with the same 10 cm starting height and the same flat exit at the table edge. Repeat.
  4. Compare the landing distances.

What to look for: the landing distances should be similar, because gravity did about the same work (mgΔh) on both paths. Any difference comes from friction and rolling, which depend on the path. Discuss: which route was longer, and did it lose a bit more energy?

Your turn

1

Find Eg of a 0.20 kg apple 3.0 m above the ground (ground = reference).

2

A 70 kg climber goes up a 25 m climbing wall. Find ΔEg.

3

A 500 kg elevator car descends 12 m. Find ΔEg and the work done by gravity.

4

Can Eg be negative? Explain.

5

A crate is moved up 2.0 m by a long, gentle ramp and an identical crate by a short, steep ramp, both without friction. Compare the changes in Eg.

6

A 3.0 kg box sits on a shelf 1.8 m above the floor. Find its Eg relative to a desk 0.60 m above the floor.

7

A ball is thrown up and caught at the same height it was thrown from. What total work did gravity do on it?

More practice: warm-up

1

Write the formula for gravitational potential energy near Earth's surface.

2

What is the value of g used in this course?

3

Find Eg for a 1.0 kg object 10 m above the reference level.

4

Name a conservative force and a non-conservative force.

5

What is the relationship between the work done by gravity and ΔEg?

More practice: on your own

1

A 1.5 kg bag of flour falls from a 2.2 m shelf to a 0.90 m counter. Find ΔEg.

2

A cyclist (total mass 80 kg) rides up a hill that rises 45 m over 1.2 km of road. Find ΔEg.

3

Explain, using work, why kinetic friction is non-conservative.

4

A 4.0 kg rock has Eg = −78.4 J relative to a bridge deck. Where is it?

5

You carry a 10 kg box up a flight of stairs (rise 3.0 m) and back down. Find the total work done by gravity on the box.

Challenge

1

A 0.50 kg ball moves along a curved track from point A (4.0 m high) up over a hump 6.0 m high and down to point B (1.0 m high). Find the work done by gravity from A to the hump, from the hump to B, and from A to B. Show that the parts add to the whole.

2

At an altitude of 1000 km, g is about 7.3 N/kg instead of 9.8 N/kg. Would Eg = mgh with g = 9.8 N/kg overestimate or underestimate the energy needed to lift a satellite that high? Explain.

3

A 20 kg crate is pushed 5.0 m across a level floor and then 5.0 m back to where it started. μ_k = 0.30. Find the total work done by friction and by gravity, and explain what this shows.

Check your understanding

Quiz

1.

A 5.0 kg box is raised from 1.0 m to 3.0 m above the floor. What is ΔEg?

2.

Gravity is called a conservative force because:

3.

A crate is pushed up a rough ramp at constant speed. Compared with lifting it straight up to the same height, the work you do is:

4.

Changing the reference level for Eg changes:

5.

A ball is thrown up and returns to the thrower's hand. The total work done by gravity is:

6.

Which statement about friction is correct?

For parents and teachers

Big idea (C2.1, C2.2): gravitational potential energy is stored by working against gravity, and the work gravity does depends only on the change in height. That makes gravity a conservative force. Friction is non-conservative because its work depends on the path.

How to help: on a walk or drive, compare two routes up the same hill: the change in potential energy is the same, but which route needs more force at each moment? Ask your teen to choose a reference level and then change it, and show that the answer to a 'how much energy' question does not change.

Common difficulties:

  • Using the path length instead of the vertical height.
  • Mixing reference levels within one problem.
  • Being uneasy with negative Eg (it is fine).
  • Forgetting the minus sign in W_g = −ΔEg.