The mechanical energy of a system is the sum of its kinetic energy and its potential energy: Em = Ek + Ep. It is expressed in joules (J). In the absence of friction, it is conserved during motion.
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How to measure it in class
Mechanical energy cannot be read on any sensor: it has to be reconstructed. You need a speed and an altitude at the same instant, hence video analysis, then FizziQ’s spreadsheet to compute the sum.
Steps:
- Weigh the object and note its mass m
- Film the motion with a graduated ruler in the frame, camera fixed and perpendicular to the plane of motion
- Mark the position frame by frame in FizziQ’s Kinematics module
- Choose an origin for altitudes (the lowest point of the trajectory is the most convenient choice) and record h(t)
- Calculate in the spreadsheet Ek = ½ m v², Ep = m g h, then Em = Ek + Ep
- Plot the three curves together: Ek and Ep should be in phase opposition, Em nearly horizontal
Scientific activities on this topic
Various hands-on activities with a smartphone or tablet help students better grasp the concept of mechanical energy:
- Mechanical energy of a pendulum - verify by kinematic study that Ek + Ep remains constant
- Pendulum: the relation a = 2gh/r - recover the speed at the lowest point from centripetal acceleration
- Pole vault: energy balance - follow the conversion run → pole → height for a pole vaulter
- G-forces on a roller coaster - relate height lost to speed gained
- Elastic collision: billiards - a collision in which mechanical energy is almost conserved
- Coefficient of restitution - quantify the fraction of mechanical energy lost at each bounce
Learn more
Mechanical energy can be calculated using the formula: E = Ek + Ep, where E is the total mechanical energy, Ek is the kinetic energy and Ep is the potential energy.
For a moving object, Ek = 1/2 * m * v², where m is the mass of the object and v its speed. And for an object close to the ground: Ep = m * g * h, where g is the gravitational acceleration and h the altitude relative to a freely chosen origin. The mechanical energy E is therefore: E = 1/2 * m * v² + m * g * h.
When Em is conserved - and when it is not
This is the only really important point of this entry. Mechanical energy is conserved if, and only if, the only forces doing work are conservative: weight, the restoring force of a spring, the electrostatic force. A conservative force is a force whose work does not depend on the path followed, but only on the starting and ending positions - this is precisely what makes it possible to associate a potential energy with it.
As soon as a non-conservative force does work - solid friction, air friction, soft impact, plastic deformation - the mechanical energy decreases:
ΔEm = W(non-conservative forces)
This quantity is negative for friction, which always opposes the displacement. Caution: this is not a violation of the conservation of energy. The total energy is intact; it has simply passed from a mechanical form to a thermal form. See the Conservation of Energy entry for this distinction.
Two useful special cases
A force perpendicular to the displacement does no work, and therefore does not change Em: this is the case for the normal reaction of a frictionless support, or the tension of a pendulum’s string. This is why an ideal pendulum conserves its mechanical energy even though two forces act on it. Conversely, a driving force (motor, muscle) increases Em: it too is non-conservative, simply with positive work.
A tool that bypasses time
The practical value of Em is that it relates two instants without needing to know what happened in between. To find the speed of a skier at the bottom of a slope, there is no need to describe the trajectory: if friction is neglected, ½mv² = mgh gives v = √(2gh), regardless of the profile of the slope and of the mass. The work-energy theorem, covered in the Kinetic Energy entry, is the more general form of this reasoning.
Technical applications
Simple machines (levers, pulleys) transfer work without creating it: they trade force for distance, at constant energy apart from friction. Dams convert the potential energy of water into kinetic energy and then into electricity; wind turbines draw on the kinetic energy of the wind. In both cases, the efficiency measures what fraction of the available mechanical energy can be converted without degrading it into heat.
Formula
Mechanical energy of a system:
Em = Ek + Ep
Near the Earth’s surface, for an object in translation:
Em = ½ m v² + m g h
Conservation, only if no non-conservative force does work:
Em(initial) = Em(final)
Otherwise:
ΔEm = W(non-conservative forces)
Speed acquired after a descent of height h without friction:
v = √(2gh)
where:
- Em: mechanical energy (J)
- Ek: kinetic energy (J)
- Ep: potential energy (J)
- m: mass (kg)
- v: speed in the reference frame of study (m·s⁻¹)
- g: gravitational acceleration, about 9.81 m·s⁻² in France
- h: altitude relative to the chosen origin (m)
- W: work of a force (J)
Application examples
- A 70 kg skier descending 100 m of vertical drop has mgh ≈ 68.7 kJ available. Without friction he would arrive at v = √(2 × 9.81 × 100) ≈ 44 m·s⁻¹, that is 159 km/h. An actual speed of 90 km/h shows that more than half of the mechanical energy goes into the snow and the air.
- A pendulum of length 1 m released at 30° from the vertical descends by h = L(1 − cos 30°) ≈ 0.134 m and passes through the lowest point at v = √(2gh) ≈ 1.6 m·s⁻¹ - independently of its mass.
- A 60 kg pole vaulter taking off at 9 m·s⁻¹ has 2.4 kJ of kinetic energy, enough to raise her center of gravity by 4.1 m. Records above 5 m are explained by the push of the arms and by the initial position of the center of gravity, already 1 m above the ground.
- An elevator descending with regenerative braking returns part of the potential energy to the grid instead of dissipating it in braking resistors.
- A satellite in circular orbit has constant mechanical energy: no friction and no engine. This is what allows it to orbit indefinitely without consuming fuel.
FAQ
Q: Is mechanical energy always conserved? A: No, and this is the most common mistake. It is conserved only if the only forces doing work are conservative. With friction, it always decreases. What is always conserved is the total energy, not the mechanical energy.
Q: How should I choose the origin of altitudes to calculate Ep? A: However you like, it is a free choice. Only energy changes have physical meaning, so the choice of origin does not change any measurable result. Take the lowest point of the trajectory: Ep stays positive and the calculations are easier to read.
Q: My graph shows that Em decreases by 4%. Did I miss something? A: Probably not. In a video analysis, the uncertainty on Ek easily reaches 10%. A fluctuation of a few percent is within the measurement noise. Conclude that there is real dissipation only if the decrease is monotonic and continues over several periods.
Q: Does mechanical energy depend on the reference frame? A: Yes, through both of its terms. Ek depends on the reference frame chosen to measure v, and Ep on the chosen reference level. Conservation, however, remains true in any inertial reference frame, independently of these choices.
Q: What is the difference between mechanical energy and work? A: Mechanical energy is a state quantity: it characterizes the system at a given instant. Work is a transfer quantity: it describes what is exchanged between two instants. Both are measured in joules, but one does not say that a system “possesses” work.
Related concepts
Kinetic Energy - Potential Energy - Elastic Energy - Conservation of Energy - Friction Force - Pendulum - Inertial Reference Frame - Inelastic Collision