Mechanical Equilibrium

"Point of equilibrium" redirects here; not to be confused with Equilibrium point (mathematics), which, for those keeping score, is a slightly different flavour of stasis.

An object resting on a surface and the corresponding free body diagram showing the forces acting on the object. The normal force N is equal, opposite, and collinear to the gravitational force mg so the net force and moment is zero. Consequently, the object is in a state of static mechanical equilibrium. A quaint illustration of the universe's inherent laziness, wouldn't you say?

In the grand, often predictable, theatre of classical mechanics, a particle achieves a state of mechanical equilibrium when the sum of all forces acting upon it – the dreaded net force – precisely cancels itself out, resulting in a grand total of zero. This isn't merely a suggestion; it's a fundamental condition. [1] : 39 Extending this rather straightforward concept, a more intricate physical system, composed of numerous interacting components, finds itself in mechanical equilibrium only when this zero-net-force condition applies to each and every one of its individual constituents. [1] : 45–46 [2] One might wonder why such an obvious point requires such explicit delineation, but here we are.

Beyond the rather blunt declaration of forces cancelling each other out, mechanical equilibrium can be articulated through several mathematically equivalent lenses, each offering a slightly different perspective on the same fundamental state of unchanging motion – or, more often, unchanging stillness.

In the realm of momentum, a system is deemed to be in equilibrium if the momentum of all its constituent parts remains utterly constant over time. No sudden surges, no inexplicable lurches; just a steadfast, unwavering trajectory.
Shifting focus to velocity, the system is in equilibrium if the velocity of its components also remains constant. This is where the common misconception often arises: constant velocity does not necessarily mean zero velocity. It simply means unchanging velocity. A rather crucial distinction, one would think.
For objects that prefer to spin rather than merely translate, a state of rotational mechanical equilibrium dictates that the angular momentum of the object must be conserved, and, consequently, the net torque acting upon it must be precisely zero. [2] No unwelcome twists or turns, just a serene, unperturbed rotation, or, more likely, no rotation at all.
More broadly, within conservative systems – those delightful constructs where energy is conserved and path independence reigns supreme – equilibrium manifests at a specific point within the system's configuration space. At this precise juncture, the gradient of the system's potential energy, when measured against its various generalized coordinates, registers as zero. It's where the system feels most "comfortable," or perhaps, least inclined to change.

Should a particle in equilibrium possess the rather unexciting characteristic of zero velocity, it is then bestowed with the title of static equilibrium. [3] [4] Given that all particles in equilibrium maintain a constant velocity, it is always theoretically possible to identify an inertial reference frame in which said particle appears utterly stationary relative to that chosen frame. A matter of perspective, as always.

Stability

A particularly fascinating, and often overlooked, characteristic of systems residing in mechanical equilibrium is their inherent stability. Does a tiny nudge send them spiralling into chaos, or do they stubbornly return to their initial state, like a cat to its preferred napping spot? This is where stability theory steps in, to quantify the system's resilience, or lack thereof.

Potential energy stability test

When a system's behaviour can be elegantly described by a function representing its potential energy, the system's various equilibrium points can be meticulously identified and scrutinized using the powerful tools of calculus. These critical points, where the system is momentarily at rest or in unchanging motion, are precisely where the first derivative of the potential energy function is zero. Once these points of apparent calm are located, the true character of their stability – whether they are stable, unstable, or merely indifferent – is unveiled by applying the second derivative test.

Let's assume, for simplicity's sake, that V represents the static equation of motion for a system endowed with a single degree of freedom. The following calculations then illuminate the nature of equilibrium:

Diagram of a ball placed in an unstable equilibrium. A poignant metaphor for many human endeavors.

Second derivative < 0: In this rather precarious scenario, the potential energy function has reached a local maximum. This unequivocally signals an unstable equilibrium state. Imagine a ball perched precariously atop a hill; the slightest disturbance, an arbitrarily minuscule displacement from this point, will cause the system's intrinsic forces to relentlessly propel it further and further away, ensuring its swift departure from equilibrium. It’s the universe’s way of saying, "You were never meant to stay here."

Diagram of a ball placed in a stable equilibrium. The universe's preferred state, perhaps.

Second derivative > 0: Conversely, when the potential energy resides at a local minimum, we are graced with a stable equilibrium. Here, the system exhibits a commendable resilience. A small perturbation, a gentle nudge, will be met with restorative forces that diligently work to bring the system back to its original equilibrium position. If a system boasts more than one stable equilibrium state, those whose potential energy is higher than the absolute minimum are designated as metastable states. They are stable, yes, but only until a sufficiently large disturbance knocks them into a lower, more fundamentally stable configuration. Like a grudging truce.
Second derivative = 0: This is where things become delightfully ambiguous. When the second derivative is zero, the state is considered neutral to the lowest order of approximation. It implies that if the system is displaced by a small amount, it will essentially remain in equilibrium, neither returning nor fleeing. To truly ascertain the precise stability of such an enigmatic system, one must delve deeper, examining higher order derivatives. If the lowest nonzero derivative encountered is of an odd order, or if it's an even order with a negative value, the state is ultimately unstable. However, if the lowest nonzero derivative is both of an even order and possesses a positive value, then the state is indeed stable. Should all derivatives mysteriously vanish to zero, deriving conclusions from derivatives alone becomes, shall we say, utterly impossible. Consider the function e^(-1/x^2) (defined as 0 at x=0). All its derivatives at x=0 are zero, yet it clearly has a local minimum at x=0, making it a stable equilibrium. Now, multiply this function by the Sign function, and while all derivatives at x=0 still remain zero, the equilibrium transforms into an unstable one. A subtle trap, for the unwary.

Diagram of a ball placed in a neutral equilibrium. The ultimate indifference.

Function is locally constant: In a truly neutral state, the potential energy simply does not vary over a finite range. The state of equilibrium, therefore, possesses a finite width. This particular condition is sometimes referred to as being marginally stable, or existing in a state of utter indifference, or even an astable equilibrium. The system simply doesn't care where it is within that range.

When one dares to consider more than a single dimension – because, let's face it, the universe is rarely that simple – it becomes entirely plausible to encounter varying stability characteristics across different directions. For instance, a system might exhibit robust stability with respect to displacements along the x-direction, yet simultaneously be profoundly unstable along the y-direction. This rather inconvenient scenario is elegantly termed a saddle point. Generally speaking, an equilibrium is only truly deemed "stable" if it demonstrates stability in all possible directions. Anything less is merely a temporary reprieve.

Statically indeterminate system

• Main article: Statically indeterminate

Occasionally, the standard equilibrium equations – those foundational conditions for force and moment equilibrium – prove woefully insufficient to definitively determine all the unknown forces and reactions within a system. Such a frustrating and incomplete scenario is precisely what is described as a statically indeterminate system. It's as if the system is holding back information, deliberately complicating matters.

Fortunately, these statically indeterminate situations are not insurmountable. They can often be resolved, albeit with a bit more effort, by incorporating additional information that lies beyond the confines of the basic equilibrium equations. This might include material properties, deformation characteristics, or compatibility conditions, effectively forcing the system to reveal its hidden truths.

Ship stability illustration explaining the stable and unstable dynamics of buoyancy (B), center of buoyancy (CB), center of gravity (CG), and weight (W). A rather critical concern, unless you enjoy unexpected aquatic adventures.

Examples

A stationary object (or an entire collection of objects) is classified as being in "static equilibrium," which, as previously noted, is merely a specialized instance of the broader concept of mechanical equilibrium. A humble paperweight steadfastly holding down documents on a desk provides an exemplary, if unremarkable, illustration of static equilibrium. More visually intriguing examples include an artfully constructed rock balance sculpture, defying gravity with casual elegance, or a meticulously stacked tower of blocks in the game of Jenga – provided, of course, that the sculpture or block stack has not yet succumbed to the inevitable, teetering on the brink of collapsing. The moment of collapse, after all, is decidedly not equilibrium.

It's crucial to remember that objects in motion are not necessarily excluded from the esteemed state of equilibrium. A child, for instance, gleefully sliding down a playground slide at a perfectly constant speed, would technically be in mechanical equilibrium. However, they would decidedly not be in static equilibrium, at least not in the chosen reference frame of the earth or the slide itself. Their velocity, though constant, is not zero.

Another common and rather tangible example of mechanical equilibrium involves a person applying force to a spring to compress it to a specific, defined point. If they hold it precisely there, the compressive load they exert and the opposing reaction force from the spring will be exactly equal. In this frozen moment, the system achieves mechanical equilibrium. The moment that compressive force is released, of course, the spring, with its inherent stubbornness, will dutifully return to its original, uncompressed state.

Of particular theoretical interest is the minimal number of static equilibria possessed by homogeneous, convex bodies when they are left to rest under the influence of gravity on a perfectly horizontal surface. In the simpler, two-dimensional planar case, the absolute minimal number of such equilibria is four. However, in the full three dimensions, one can ingeniously construct an object that possesses only a single stable and a single unstable balance point. [5] Such a remarkably unique object has been christened a gömböc – a testament to the elegant complexities that even seemingly simple physics can yield.

Notes and references

[1] ^{^} ^a ^b John L Synge & Byron A Griffith (1949). Principles of Mechanics (2nd ed.). McGraw-Hill.
[2] ^{^} ^a ^b Beer FP, Johnston ER, Mazurek DF, Cornell PJ, and Eisenberg, ER (2009). Vector Mechanics for Engineers: Statics and Dynamics (9th ed.). McGraw-Hill. p. 158. {{cite book}} : CS1 maint: multiple names: authors list (link)
[3] ^{^} Herbert Charles Corben & Philip Stehle (1994). Classical Mechanics (Reprint of 1960 second ed.). Courier Dover Publications. p. 113. ISBN 0-486-68063-0.
[4] ^{^} Lakshmana C. Rao; J. Lakshminarasimhan; Raju Sethuraman; Srinivasan M. Sivakumar (2004). Engineering Mechanics. PHI Learning Pvt. Ltd. p. 6. ISBN 81-203-2189-8.
[5] ^{^} "Mathematics". Gömböc. 2021. Retrieved 12 November 2023.