Potential energy is the energy an object possesses because of its position in a force field (gravity, elastic force, electric force). Stored, it can be converted into another form, often kinetic energy. It is only defined up to a constant: only its variations have physical meaning.
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How to measure it in class
You do not measure Ep directly: you measure an altitude and a mass, then calculate. The point of the experiment is to verify that the potential energy lost is indeed recovered as kinetic energy.
Steps:
- Weigh the object and measure the fall height h with a tape measure
- Explicitly choose the origin of altitudes and write it on the board before any calculation; measure h between the centers of gravity, not between the edges of the objects
- Film the fall with a graduated ruler in the frame, then track it in FizziQ’s Kinematics module
- Plot Ep = mgh as a function of time, and check that it decreases as Ek increases
- Redo the full calculation with a different origin of altitudes (for example the release point) and observe that ΔEp is identical
- Compare the ΔEp lost and the ΔEk gained: the difference measures the dissipation
Scientific activities on this topic
- Pendulum: the relation a = 2gh/r - convert the height lost into speed at the lowest point and verify the energy balance
- Impact vibrations and seismics - measure what becomes of the potential energy of an object that falls and crashes
Learn more
There are different forms of potential energy depending on the force field considered. The most common forms are:
Gravitational potential energy, related to the position of an object in a gravitational field.
Elastic potential energy, associated with the deformation of an elastic material, such as a spring.
Electric potential energy, related to the position of electric charges in an electric field.
The formula for calculating potential energy depends on the specific form of potential energy considered. For example, for gravitational potential energy near the ground, the formula is Ep = m g h, where m is the mass of the object, g the strength of gravity and h the altitude relative to a chosen origin.
Ep = mgh is only valid near the ground
This formula assumes that the gravitational field is uniform, that is, that g keeps the same value over the entire height considered. This is excellent as long as h remains small compared with the Earth’s radius (6,371 km): over 10 km, g varies by only 0.3%. For a satellite or a probe, the formula no longer holds and you need the general form Ep = −GMm/r, with the origin taken at infinity. This potential energy is then negative - which is surprising, but simply follows from the choice of origin.
Defined up to a constant: difficulty number 1
Potential energy is not an absolute quantity. Adding the same constant to Ep at every point changes strictly nothing in the physics: forces are derived from the variations of Ep, not from its value. You are therefore free to choose where Ep = 0. The ground, the lowest point of the trajectory, the equilibrium position, infinity: all these choices are legitimate, and lead to the same measurable result.
Concretely, this means that a question like “what is the potential energy of this marble?” has no answer until you have said relative to what. On the other hand, “by how much has its potential energy changed while falling 2 m?” has a unique answer: ΔEp = −mg × 2. The practical advice is simple: write down the chosen origin before writing any value, and do not change it during the exercise.
A notion reserved for conservative forces
A potential energy can only be associated with a force whose work does not depend on the path followed: weight, the restoring force of a spring, the electrostatic force. Friction has no potential energy, because its work depends on the distance traveled - a round trip brings the object back to its starting point but leaves a non-zero negative work. This is the deep reason why mechanical energy is not conserved in the presence of friction.
Origin of the concept
The word itself is recent: it was William Rankine who introduced the expression potential energy in 1853, as part of the construction of the principle of conservation of energy. The underlying mathematical tool - the potential function - is older: Lagrange, Laplace, then Poisson and George Green (who coined the term potential function in 1828) developed it for gravitation and electrostatics. Newton, for his part, reasoned in terms of forces and never used the notion of potential energy: attributing it to his work is a frequent but false anachronism.
Orders of magnitude and applications
Gravitational potential energy is a convenient way to store energy on a large scale: pumped-storage power stations raise water at night to run it through turbines at peak hours, with a round-trip efficiency of about 75 to 80%. At the laboratory scale, it is very modest: lifting 1 kg by one meter costs only 9.8 J, compared with 335,000 J to melt the same kilogram of ice. This is why chemical batteries, far more energy-dense, win out for portable electronics.
Formula
Gravitational potential energy, uniform field (near the ground):
Ep = m × g × h
Variation, the only quantity with physical meaning:
ΔEp = m × g × (h_final − h_initial) = −W(weight)
General gravitational potential energy, origin taken at infinity:
Ep = −G × M × m / r
Elastic potential energy of a spring (origin at the natural length):
Ep = ½ × k × x²
General relation between a conservative force and potential energy, in one dimension:
F = −dEp/dx
where:
- Ep: potential energy (J)
- m: mass of the object (kg)
- g: strength of gravity, about 9.81 m·s⁻² in France
- h: altitude relative to the chosen origin (m)
- G: gravitational constant, 6.674 × 10⁻¹¹ N·m²·kg⁻²
- M: mass of the attracting body (kg)
- r: distance to the center of the body (m)
- k: spring constant (N·m⁻¹)
- x: extension relative to the natural length (m)
Application examples
- A 500 g book resting on an 80 cm table has Ep = 0.5 × 9.81 × 0.80 ≈ 3.9 J relative to the floor, and 0 J relative to the table top. Both answers are correct: they describe different origins.
- A 70 kg diver on a 10 m platform has 6.9 kJ available. He enters the water at about 14 m·s⁻¹, that is 50 km/h.
- A dam with a 100 m drop releases 0.98 MJ per cubic meter of water. A reservoir of 10 million m³ thus stores nearly 10 TJ, about 2.7 GWh.
- The Grand’Maison lake, in Isère, operates as a pumped-storage station: the water pumped up at night returns during the day about 80% of the energy consumed to raise it.
- To escape the Earth’s attraction, you must supply GMm/R per kilogram, that is 62.5 MJ·kg⁻¹ - which corresponds to the escape velocity of 11.2 km·s⁻¹ and explains why launchers consist almost entirely of fuel.
FAQ
Q: Where should the zero of potential energy be placed? A: Wherever you want. This choice is free and has no physical consequence, since only variations matter. Choose whatever simplifies the calculations - generally the lowest point of the motion - and above all do not change the origin in the middle of a problem.
Q: Can potential energy be negative? A: Yes, and it is not a problem. An object located below the chosen origin has a negative Ep. In the gravitational case with the origin at infinity, Ep = −GMm/r is negative everywhere. It only means that energy would have to be supplied to bring the object to infinity, not that it “lacks” energy.
Q: Does Ep = mgh always work? A: No, only when g can be considered uniform, that is, for heights small compared with the Earth’s radius. Beyond a few tens of kilometers, you must switch to Ep = −GMm/r. For all high school experiments, mgh is more than adequate.
Q: Where is potential energy “stored”? A: Not in the object. It is associated with the system {object + Earth}, that is, with the interaction between the two. Saying that a marble “contains” potential energy is a convenient shorthand: without the Earth, this energy would not exist.
Q: Why is there no friction potential energy? A: Because the work of friction depends on the path traveled, and not only on the start and end positions. Yet it is precisely this path independence that makes it possible to define a potential energy. Friction is a non-conservative force: its work ends up as heat, not as recoverable stored energy.
Related concepts
Kinetic Energy - Mechanical Energy - Elastic Energy - Conservation of Energy - Law of Gravitation - Friction Force - Pendulum - Hooke’s Law