# Mechanical energy **Mechanical energy** is the sum of a system's [[Kinetic_energy|kinetic energy]] and its [[Potential_energy|potential energy]] — the energy a body carries because it is moving, plus the energy it carries because of where it sits in a field of force. The quantity matters because, for a restricted but very common class of problems, the sum stays fixed while each part swings wildly: a falling apple converts [[Gravitational_energy|gravitational potential energy]] into motion joule for joule, so that a 7 J apple passes through (5 J, 2 J), (3 J, 4 J) and (1 J, 6 J) on its way down.[^murphy-apple] In the microsim below the reader releases a pendulum from an angle θ₀ and sets a damping rate Γ. Three bars — kinetic, potential and dissipated — rise and fall, and two equations answer what they do: `E = 1/2 m L^2 theta'^2 + m g L (1 - cos theta)` fixes how the first two trade against each other, and `<E> = E0 e^(-Gamma t)` fixes how quickly the third eats them both.[^cline-pend][^cline-damp] On the Energy flagship's spine this page is the main article for Part IV — Scientific use, section *Work and mechanical energy*, and it is the root of the mechanical branch: [[Kinetic_energy]] and [[Potential_energy]] are its two halves, [[Conservation_of_energy]] is the law it is a special case of, and [[Work_(physics)|work]] is the channel through which energy enters or leaves. Its pendulum sim is the shared embed that [[Flywheel_energy_storage|flywheel storage]] and [[Pendulum_(mechanics)|pendulum mechanics]] reuse. The restriction is the whole subject. Mechanical energy is conserved only when every force that does [[Work_(physics)|work]] on the system is a [[Conservative_force|conservative force]] — one whose work depends on the endpoints and not the path. Let [[Friction|friction]], air drag or an inelastic impact in, and mechanical energy leaks into [[Internal_energy|internal energy]], which is a form the mechanical books do not track. Nothing is lost from the universe; something is lost from the ledger. ## General Work is force acting through distance, `W = F d`, "when the motion is aligned with the direction of force", which makes the [[Joule|joule]] equal to one newton-metre: 2 N through 0.5 m, 0.1 N through 10 m and 150 N through 5 m are all 1 J, 1 J and 750 J respectively.[^murphy-work] Doing work on a body either speeds it up, storing the result as kinetic energy `KE = 1/2 m v^2`, or lifts it against a field, storing the result as potential energy — near Earth's surface `PE = m g h`, in a [[Hooke's_law|linear spring]] `PE = 1/2 k x^2`, and between two masses `PE = -G m1 m2 / r`.[^downey-orbit] Mechanical energy is the sum of whichever of these apply. Two features distinguish potential energy from the other terms in an energy budget. First, it only exists for conservative forces: the work done in moving between two points must be independent of the route, otherwise no single-valued function of position can be assigned.[^cline-nonconserv] Second, its zero is arbitrary. Only differences are measurable, which is why the gravitational form carries a minus sign when the zero is placed at infinite separation and no sign at all when it is placed at the floor. The Portal Book's accounting table is a useful check on scope. Murphy lists the recurring energy expressions and where each one reappears in the energy system: `m g h` for hydro and tidal, `1/2 m v^2` for wind and ocean current, `h nu` for solar, `H - T S` for chemical, `cp m dT` for thermal, `q V` for electric and `m c^2` for nuclear.[^murphy-forms] Only the first two are mechanical. Everything else on that list is a form into which mechanical energy can be converted, or out of which it can be produced, but not a form it contains — a boundary the last section of this article draws more carefully. Power is the rate, not the amount: 1 W = 1 J/s. Lifting a 10 kg box (about 100 N) through 2 m takes roughly 200 J whether it is done quickly or slowly, but doing it in one second is 200 W and doing it in four seconds is 50 W.[^murphy-power] Some of Murphy's worked examples use g ≈ 10 m/s² rather than 9.81 m/s², so a reader checking arithmetic against them should use the same g.[^murphy-power] ## Conservation of mechanical energy When only conservative forces do work, the total `E = KE + PE` is a constant of the motion. The statement is not an extra law but a consequence of [[Newton's_laws_of_motion|Newton's second law]] integrated once along the path, and in the [[Lagrangian_mechanics|Lagrangian]] and [[Hamiltonian_mechanics|Hamiltonian]] formulations it appears as the conserved quantity that follows from the equations having no explicit time dependence. The practical use of the conservation statement is that it answers speed-at-a-place questions without integrating the equation of motion at all. A body released from rest at a height h arrives at the bottom with `v = sqrt(2 g h)` regardless of the shape of the track, provided the track is frictionless; the same logic gives [[Escape_velocity|escape velocity]] from `1/2 m v^2 = G M m / r`. It also fixes the *shape* of the orbit in [[Phase_space|phase space]]: because `E` is constant, the trajectory in the (position, velocity) plane is confined to a contour of `E`, and the family of contours is the complete qualitative picture of the dynamics before a single number is computed. The pendulum is the standard laboratory for this, and it is where the article's microsim lives. ### Swinging pendulum The undamped pendulum obeys `theta'' + (g/L) sin(theta) = 0`, written as the first-order pair `theta' = w` and `w' = -(g/L) sin(theta)`. Its critical points sit at θ = nπ: the even multiples, with the bob hanging down, are centres, and the odd multiples, with the bob balanced on top, are saddles. The quantity `(1/2) w^2 - (g/L) cos(theta)` is conserved along every trajectory.[^lebl-phase] Multiplied through by `m L^2` and shifted so that the hanging position is zero, this is the mechanical energy in the form the sim displays, `E = 1/2 m L^2 theta'^2 + m g L (1 - cos theta)`.[^cline-pend] That form splits the phase plane into three regimes at a single threshold. Below `E = 2 m g L` the bob cannot reach the top, so it swings back and forth — libration, and a closed orbit in phase space. Above `E = 2 m g L` it goes over the top and keeps going, which is rotation, an orbit that runs off one side of the plane and returns on the other. Exactly at `E = 2 m g L` lies the separatrix, the trajectory that creeps toward the inverted position and takes infinite time to arrive.[^cline-pend] Note that [[Angular_momentum|angular momentum]] is not conserved here — gravity exerts a torque about the pivot — so energy is the only constant available.[^cline-pend] Released from rest at θ₀, the speed at any later angle follows directly from constancy of E: `w = +- sqrt((2 g/L)(cos(theta) - cos(theta0)))`.[^lebl-period] Integrating `dtheta/w` over a quarter swing gives the period, `T = 4 sqrt(L/(2 g)) * integral_0^theta0 dtheta / sqrt(cos(theta) - cos(theta0))`, which has no closed form and diverges as θ₀ → π.[^lebl-period] The familiar `T_lin = 2 pi sqrt(L/g)` is the small-angle limit, recovered by replacing `1 - cos(theta)` with `theta^2/2`, which turns the energy contour into an ellipse and the motion into [[Simple_harmonic_motion|simple harmonic motion]].[^cline-pend] In the sim the reader's controls are θ₀ from 1° to 179° and Γ from zero upward, with g/L fixed at 1. Three bars show kinetic, potential and dissipated energy; with Γ = 0 the first two trade and the third stays flat, and the phase-plane inset shows the orbit change from an ellipse to an "eye" hugging the drawn separatrix as θ₀ approaches 180°. The stepper is `integrators.verlet` on (θ, ω), for the reason the next-but-one section gives. A period readout carries the [[Ordinary_differential_equation|ODE]] textbook's own numbers: at g/L = 1 the fractional period error is 15 % at θ₀ = 90°, 3.8 % at 45° and 0.048 % at 5°.[^lebl-period] Those printed values are `(T - T_lin)/T` rather than `(T - T_lin)/T_lin`; taking the linear period as the denominator instead turns the same three periods into 18.0 %, 4.00 % and 0.048 %, so the sim states which definition it is showing.[^lebl-period] The lesson is that the "constant" period of a pendulum is a small-angle illusion. ### Irreversibilities Add damping and the ledger acquires a third column. With a resistive force proportional to speed, the small-angle equation becomes `theta'' + Gamma theta' + w0^2 theta = 0`, where `Gamma = b/m` and the damped frequency is `w1 = sqrt(w0^2 - (Gamma/2)^2)`. The amplitude envelope decays with time constant 2/Γ, but energy goes as the square of amplitude, so it decays twice as fast: `<E> = E0 e^(-Gamma t)`, with time constant 1/Γ.[^cline-damp] That factor of two is the single most common slip in reading a decaying oscilloscope trace. The quality factor `Q = w1/Gamma` counts how many radians of oscillation fit into one energy decay time, and its range in nature is enormous. Cline's table gives Q ≈ 250–1400 for the whole Earth ringing after a large earthquake, about 3,000 for a piano string, 10⁴ for a quartz watch crystal and for a microwave cavity, 10⁷ for an excited atom, 10¹² for a neutron star, 10¹³ for the LIGO detector and 10¹⁴ for a Mössbauer resonance.[^cline-q] Two conventions for Q are in circulation — ω₁/Γ for the free oscillator, ω₀/Γ for the driven one — and quoting one against the other is a real source of disagreement in the literature.[^cline-q] Where does the missing energy go? Into [[Internal_energy|internal energy]]: the pivot warms, the air is stirred, the wire is worked. In the microsim the dissipated bar is not a loss but an account, and the three bars are constrained to sum to E₀ at every frame; when they fail to, the integrator is at fault, not the physics. This is the point at which mechanical energy stops being conserved and [[Conservation_of_energy|total energy]] carries on being conserved, and it is the boundary at which [[Thermodynamics|thermodynamics]] takes over the bookkeeping. Critical damping, `Gamma = 2 w0`, is the fastest return to rest without overshoot, which is why door closers and galvanometer movements are built to sit near it.[^cline-damp] ### Satellite A satellite in [[Orbit|orbit]] is the clean case: over one pass the only force doing work is gravity, so `KE + PE` is fixed while both halves swing by large factors around an eccentric ellipse. With the primary treated as fixed, `KE = m v^2/2` and `PE = -G m1 m2 / r`, and the specific mechanical energy is `E = v^2/2 - GM/r`, negative for a bound orbit and zero at escape.[^downey-orbit] Orbits are also where numerical energy accounting fails most visibly, which is why the pendulum sim uses the stepper it does. An adaptive general-purpose solver run on a circular orbit does not conserve energy; tightening the relative tolerance from the default 10⁻³ to 10⁻⁵ slows the drift without stopping it.[^downey-orbit] The reason is structural rather than a matter of accuracy: applied to an undamped oscillator, the forward Euler step multiplies the energy by `1 + (w dt)^2` every step, so it grows without bound, while backward Euler damps the motion that should not be damped and the trapezoidal rule keeps the amplification factor at exactly 1.[^vuik-stability] Over long runs a second-order symplectic step such as [[Verlet_integration|Verlet]] beats fourth-order [[Runge–Kutta_methods|Runge–Kutta]] on energy, because it conserves a nearby "shadow" energy exactly: E oscillates with an amplitude of order Δt² rather than staying constant, but it never walks away. For a page whose subject is a conserved sum, choosing the integrator is choosing whether the subject survives on screen. ## Conversion Mechanical energy is the most convertible form in the energy system, which is why it sits at the hinge of so many machines. An [[Electric_generator|electric generator]] converts shaft work into [[Electrical_energy|electrical energy]]; an [[Electric_motor|electric motor]] runs the exchange backwards; a [[Turbine|turbine]] takes it out of a moving fluid, and a pump puts it back in. [[Hydroelectricity|Hydroelectric]] plants work on `m g h` and wind machines on `1/2 m v^2`, exactly the two mechanical entries in Murphy's table of forms.[^murphy-forms] [[Pumped-storage_hydroelectricity|Pumped storage]] and [[Flywheel_energy_storage|flywheels]] are mechanical stores, holding potential and kinetic energy respectively, and are among the few stores that can be charged and discharged without a chemical change. Conversions in the direction of heat are cheap and conversions away from it are expensive. Friction converts mechanical energy to [[Thermal_energy|thermal energy]] at 100 % and needs no machine; converting thermal energy back into shaft work requires a [[Heat_engine|heat engine]] and is capped by the Carnot limit. Because of that asymmetry, the same joule is worth different amounts depending on which column of the ledger it is sitting in, and [[Entropy|entropy]] is the quantity that keeps score of the difference. The unit conversions that cross these boundaries are worth keeping to hand, because most real energy arguments are lost in them rather than in the physics: 1 kWh = 1,000 J/s × 3,600 s = 3.6 MJ, 1 horsepower = 745.7 W, 1 cal = 4.184 J and 1 kcal = 4,184 J.[^murphy-power] A 100 hp engine is about 75 kW.[^murphy-power] The unit "kW per hour" does not exist and should never be written; a kilowatt-hour is already a quantity of energy, not a rate.[^murphy-power] ## Distinction from other types Mechanical energy is one column in a wider ledger, and the boundary is drawn by which variables the model tracks. [[Chemical_energy|Chemical energy]] is accounted as `H - T S`, [[Thermal_energy|thermal energy]] as `cp m dT`, radiant energy as `h nu` per [[Photon|photon]], electrical energy as `q V` and nuclear energy through [[Mass–energy_equivalence|mass–energy equivalence]] as `m c^2`.[^murphy-forms] None of these is kinetic or potential energy in the mechanical sense, even though the microscopic picture of thermal energy is precisely the kinetic and potential energy of molecules — which is the point: what counts as "mechanical" depends on where the observer draws the system boundary and which degrees of freedom are resolved. Engineering [[Thermodynamics|thermodynamics]] makes this explicit. In the first law for a closed system, the energy change is split into internal energy plus a bulk kinetic term plus a bulk potential term, `dE = dU + dKE + dPE`, and the mechanical terms are usually dropped as negligible in a stationary system — but they are dropped by assumption, not by law, and a textbook's worked problems keep them wherever a mass is raised or accelerated.[^yan-first] [[Internal_energy|Internal energy]] is the catch-all for the energy the mechanical description no longer resolves, and the first law is the statement that the whole sum is conserved even when the mechanical part alone is not. Two further distinctions are worth stating plainly. Mechanical energy is a scalar and adds arithmetically, unlike [[Momentum|momentum]], which is a vector; a head-on [[Inelastic_collision|inelastic collision]] can leave a system with zero mechanical energy while momentum is conserved exactly. And mechanical energy is frame-dependent through its kinetic term, so its numerical value is not a property of the system alone: a ball at rest on a moving train has kinetic energy in the platform frame and none in the carriage frame, while conservation holds in both. *See also:* [[Work_(physics)]] · [[Conservative_force]] · [[Kinetic_energy]] · [[Potential_energy]] · [[Conservation_of_energy]] · [[Pendulum_(mechanics)]] ## References [^cline-pend]: Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, Revised 2nd ed. Portal Book 073, Chapter 3 Linear oscillators, p. 79 (the plane-pendulum energy `E = (1/2) m L² θ'² + m g L (1 − cos θ)`; libration below E = 2mgL, rotation above it, separatrix at E = 2mgL; angular momentum not conserved; the ellipse approximation requires 1 − cos θ ≈ θ²/2). https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics [^cline-damp]: Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, Revised 2nd ed. Portal Book 073, Chapter 3 Linear oscillators, pp. 80–83 (`x'' + Γx' + ω₀²x = 0` with Γ = b/m and ω₁ = √(ω₀² − (Γ/2)²); amplitude decays with time constant 2/Γ and energy with 1/Γ, `⟨E⟩ = E₀e^(−Γt)`; critical damping returns to zero fastest). [^cline-q]: Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, Revised 2nd ed. Portal Book 073, Chapter 3 Linear oscillators, p. 83, Table 3.1 (Q = ω₁/Γ for the free oscillator; Earth 250–1400, piano string 3,000, watch crystal 10⁴, microwave cavity 10⁴, excited atom 10⁷, neutron star 10¹², LIGO 10¹³, Mössbauer 10¹⁴; the two Q conventions, ω₁/Γ free and ω₀/Γ driven, at pp. 83, 87). [^cline-nonconserv]: Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, Revised 2nd ed. Portal Book 073, Chapter 10 Nonconservative systems, pp. 259–268 (page to pin) (systems for which no single-valued potential exists, and the treatment of dissipative forces). [^lebl-phase]: Lebl, Jiří (2014). *Notes on Diffy Qs: Differential Equations for Engineers*. Portal Book 031, Chapter 9 Nonlinear systems, pp. 363–365 (`θ'' + (g/L) sin θ = 0` as the system θ' = ω, ω' = −(g/L) sin θ; critical points at θ = nπ, even multiples centres and odd multiples saddles; the conserved quantity (1/2)ω² − (g/L) cos θ). https://open.umn.edu/opentextbooks/textbooks/notes-on-diffy-qs-differential-equations-for-engineers [^lebl-period]: Lebl, Jiří (2014). *Notes on Diffy Qs: Differential Equations for Engineers*. Portal Book 031, Chapter 9 Nonlinear systems, pp. 366–367 (release from rest at θ₀ gives ω = ±√((2g/L)(cos θ − cos θ₀)); the period integral has no closed form, T_lin = 2π√(L/g), and T → ∞ as θ₀ → π; printed relative errors 15 %, 3.8 % and 0.048 % at θ₀ = 90°, 45° and 5°). Those printed figures equal (T − T_lin)/T; the 18.0 % and 4.00 % values quoted here against T_lin instead are derived in the Wikitube extract, not printed in the book. [^downey-orbit]: Downey, Allen (2021). *Physical Modeling in MATLAB*, version 4.0. Portal Book 085, Chapter 15 Springs and Things, pp. 159–160 (`F = −G m₁m₂/r² r̂` with the Sun fixed; `KE = mv²/2`, `PE = −G m₁m₂/r`; an ode45 orbit does not conserve energy, and tightening RelTol from 10⁻³ to 10⁻⁵ slows but does not stop the drift). https://open.umn.edu/opentextbooks/textbooks/physical-modeling-in-matlab [^vuik-stability]: Vuik, Kees; Vermolen, Fred; van Gijzen, Martin (2023). *Numerical Methods for Ordinary Differential Equations*. Portal Book 030, Chapter 6 Numerical time integration of ODEs, pp. 79–80, 89, 99–100 (for the undamped oscillator y'' = −y, forward Euler amplifies the energy by 1 + (ωΔt)² per step, backward Euler damps it numerically, and the trapezoidal rule has amplification factor of modulus exactly 1; forward and modified Euler are unstable at every Δt). https://open.umn.edu/opentextbooks/textbooks/numerical-methods-for-ordinary-differential-equations [^murphy-work]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Portal Book 097, Chapter 5 Energy and Fossil Fuels, pp. 88–89 (`W = F d`, valid "when the motion is aligned with the direction of force"; 1 J = 1 N·m; the worked checks 2 N × 0.5 m = 1 J, 0.1 N × 10 m = 1 J, 150 N × 5 m = 750 J). https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^murphy-forms]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Portal Book 097, Chapter 5 Energy and Fossil Fuels, p. 90, Table 5.2 (the energy forms and where each reappears: m g h for hydro and tidal, ½mv² for wind and ocean current, hν for solar, H − TS for chemical, c_p m ΔT for thermal, qV for electric, mc² for nuclear). [^murphy-apple]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Portal Book 097, Chapter 5 Energy and Fossil Fuels, pp. 90–91 (the falling apple carrying 7 J total, passing through (PE, KE) = (7, 0), (5, 2), (3, 4) and (1, 6) J). [^murphy-power]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Portal Book 097, Chapter 5 Energy and Fossil Fuels, pp. 91–93 (a 10 kg box, ≈100 N, lifted 2 m takes ≈200 J: 200 W in 1 s, 50 W in 4 s; 1 W = 1 J/s; 1 kWh = 1,000 J/s × 3,600 s = 3.6 MJ; 1 hp = 745.7 W and 100 hp ≈ 75 kW; "never label anything kW per hour"; some examples use g ≈ 10 m/s²). [^yan-first]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*. Portal Book 115, Chapter 4 The First Law of Thermodynamics for Closed Systems, pp. 127–186 (page to pin) (the closed-system energy balance and its internal, kinetic and potential energy terms). https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Mechanical_energy.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Mechanical energy* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Mechanical_energy.html" data-title="Mechanical energy"></div> *Built from `MICROSIM_GUIDE/specs/sims/Mechanical_energy.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Mechanical_energy) : [Wikitube](https://en.wikitube.io/wiki/Mechanical_energy) · pinned revision [1337580391](https://en.wikipedia.org/w/index.php?oldid=1337580391) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E15 · sim pending (matter/Mechanical_energy).*