Why No Mountain on Earth Can Ever Be Taller Than 10 Kilometres

A rigid body, as studied in the chapter on rotational motion, is a useful idealisation: a hard object with a perfectly fixed shape and size, unaffected by any force applied to it. Real solids, though, are never quite this perfect -- even an apparently rigid steel bar genuinely deforms, stretching or bending slightly, when a sufficiently large external force acts on it. Stretching a helical spring by gently pulling its ends makes it lengthen slightly, and releasing the ends lets it spring back to its original size and shape -- this tendency of a body to regain its original size and shape once a deforming force is removed is called elasticity, and the temporary deformation involved is elastic deformation. Applying a comparable force to a lump of putty or mud, by contrast, produces no such recovery at all: the putty simply stays in its new, deformed shape permanently. Materials behaving this way are called plastic, and the property itself is plasticity -- putty and mud are close to ideal plastic materials. The elastic behaviour of ordinary materials is not a minor curiosity; it sits at the very centre of structural and mechanical engineering, since designing anything from a building to a bridge to an aeroplane to an artificial limb genuinely requires precise, quantitative knowledge of exactly how much a chosen material will deform, and how much force it can safely withstand before that deformation becomes permanent or the material fails outright.

When a deforming force is applied to a body, an internal restoring force develops within it, equal in magnitude but opposite in direction to the applied force -- and the restoring force per unit area is called stress: Stress = F/A, measured in newtons per square metre, or pascal (Pa). Three genuinely distinct ways exist for a solid to change its dimensions under an external force. Stretching a cylinder with two equal, opposite forces applied perpendicular to its cross-section produces tensile stress; squeezing it the opposite way produces compressive stress -- both are collectively called longitudinal stress, since both involve a change specifically in the cylinder's length. Applying two equal, opposite forces instead parallel to the cylinder's cross-sectional faces (rather than perpendicular to them) causes one face to slide sideways relative to the other, producing shearing (or tangential) stress -- a genuine deformation possible only in solids, since fluids simply flow rather than resist this kind of sideways force. Finally, submerging a solid sphere in a fluid under high pressure compresses it uniformly from every direction at once, changing its volume but not its shape at all -- this is hydraulic stress, equal in magnitude to the fluid pressure itself, and it is the one type of stress that fluids, as well as solids, can genuinely experience.

Stress describes the internal restoring force a deformed body develops -- strain describes exactly how much that body has actually deformed, always expressed as a dimensionless ratio, a fractional change in some dimension, carrying no units at all. Longitudinal strain, matching tensile or compressive stress, is the ratio of the change in length to the original length, ΔL/L. Shearing strain, matching shearing stress, is the ratio of the sideways displacement between the two faces, Δx, to the cylinder's own length, L, which works out to be exactly tan θ, where θ is the small angle through which the body has tilted from its original position -- and since θ is normally very small (differing from tan θ by only about 1% even at 10 degrees), shearing strain is written simply as tan θ ≈ θ. Volume strain, matching hydraulic stress, is the ratio of the change in volume to the original volume, ΔV/V. In every case, strain measures a pure, relative change -- a strain of 0.01, for instance, always means a 1% change in the relevant dimension, whatever the object's own actual size happens to be.

For small deformations, stress and strain in most materials are directly proportional to one another -- this is Hooke's Law, written stress = k x strain, where k, the proportionality constant, is called the modulus of elasticity. Plotting stress against strain for a real material, by gradually increasing an applied force and recording the resulting strain at each step, reveals a genuinely informative curve. From the origin O up to a point A, the curve is a straight line: this is precisely the region where Hooke's Law holds, and the body still returns fully to its original dimensions once the force is removed -- true elastic behaviour. From A to a further point B, the curve bends away from a straight line, yet the body still recovers its original shape when unloaded; point B itself, called the yield point or elastic limit, marks the boundary of this recoverable region, and the corresponding stress is the material's yield strength. Push the load past B, and something fundamentally changes: the body no longer fully recovers when unloaded, retaining a genuine permanent set even at zero stress -- this irreversible plastic deformation continues until point D, the ultimate tensile strength, beyond which even a reduced force keeps stretching the material further, until it finally snaps at point E, the fracture point. A material whose ultimate strength (D) and fracture point (E) sit close together on the curve is called brittle; one where D and E are far apart, allowing substantial further stretching before fracture, is called ductile. Not every material follows this pattern faithfully: elastomers like rubber, or the elastic tissue lining the aorta, can stretch to several times their original length while still fully recovering, yet noticeably fail to obey Hooke's Law's straight-line proportionality over most of that very large elastic range.

For tensile or compressive stress, the ratio of stress (σ) to longitudinal strain (ε) is called Young's modulus, denoted Y: Y = σ/ε = (F/A)/(ΔL/L). Since strain itself carries no units, Young's modulus shares the same units as stress: N/m², or pascal. Real measured values reveal something genuinely useful: metals generally have large Young's moduli, meaning they need a large force to produce even a small change in length -- increasing the length of a thin steel wire (0.1 cm² cross-section) by just 0.1% requires about 2000 N, while producing the identical strain in aluminium, brass, and copper wires of the same cross-section needs only 690 N, 900 N, and 1100 N respectively, revealing steel to be genuinely more elastic (more strongly resistant to stretching) than any of the other three -- exactly why steel is preferred for heavy-duty machines and structural design, while wood, bone, concrete and glass, all with comparatively small Young's moduli, stretch far more easily under the same load. A structural steel rod of radius 10 mm and length 1.0 m, stretched along its length by a 100 kN force, illustrates the full calculation: the stress works out to F/A = (100x10³)/(π x (0.01)²) ≈ 3.18x10⁸ N/m², and with Y=2.0x10¹¹ N/m² for structural steel, the resulting elongation ΔL=(stress x L)/Y ≈ 1.59x10⁻³ m, just 1.59 mm, a strain of only 0.16%. Even living tissue obeys the identical formula: in a human pyramid at a circus, with a combined mass of 280 kg supported by a 60 kg performer lying at the base, the 220 kg actually borne by that performer's own legs presses down through both thighbones (femurs) with 1078 N each -- and using bone's own Young's modulus (about 9.4x10⁹ N/m²), each thighbone, 0.5 m long with a 2.0 cm radius, compresses by only about 4.55x10⁻⁵ m, a genuinely tiny fractional change of just 0.0091%.

The ratio of shearing stress to shearing strain is called the shear modulus, denoted G (also known as the modulus of rigidity): G = (F/A)/(Δx/L) = (F x L)/(A x Δx), equivalently written G = F/(A x θ), giving shearing stress as σs = G x θ. Shear modulus shares stress's own SI units, N/m² or pascal, and, examining real measured values, is consistently found to be smaller than the same material's Young's modulus -- for most materials, G works out to be roughly Y/3. A square lead slab, 50 cm on each side and 10 cm thick, with its lower edge fixed to the floor and a shearing force of 9.0x10⁴ N applied to its narrow upper face, illustrates the calculation directly: the stress on the 0.05 m² face works out to (9.0x10⁴)/0.05 = 1.8x10⁶ N/m², and using a modulus of 5.6x10⁹ N/m², the resulting sideways displacement of the upper edge, Δx = (stress x L)/G, works out to about 1.6x10⁻⁴ m, just 0.16 mm -- a genuinely small displacement, entirely consistent with lead's real, physical resistance to shearing.

The ratio of hydraulic stress to the resulting volume strain is called the bulk modulus, denoted B: B = -p/(ΔV/V), where the negative sign simply reflects that an increase in pressure p always produces a decrease in volume (a negative ΔV), keeping B itself always positive for any real, stable material. Bulk modulus shares stress's own units, N/m² or pascal -- and, unlike Young's or shear modulus, genuinely applies to solids, liquids and gases alike, since all three states of matter have a volume that can be compressed. The reciprocal of bulk modulus, compressibility (k=1/B), measures the fractional decrease in volume per unit increase in pressure, and real measured values reveal a striking pattern: solids have far larger bulk moduli than liquids, which themselves have far larger bulk moduli than gases -- solids are the very least compressible, gases the very most, with gases being about a million times more compressible than solids. This traces directly back to how tightly a material's own constituent atoms or molecules are bound to their neighbours: solids have their atoms locked in a tight, rigid coupling, liquids have looser but still real molecular bonds, and gas molecules are only very weakly coupled to one another at all. The average depth of the Indian Ocean, about 3000 m, provides a concrete illustration: the pressure at the bottom, from the weight of the water column above, works out to p=hρg=3000 x 1000 x 10 = 3x10⁷ N/m², and dividing by water's own bulk modulus (2.2x10⁹ N/m²) gives a fractional compression of the water at that depth, ΔV/V, of about 1.36% -- a genuinely small but entirely real and measurable effect.

Stretching a wire does real work against the interatomic forces holding it together, and this work does not simply vanish -- it is stored inside the wire itself, as elastic potential energy. Working through this stretching process step by step, adding up the small amount of work done for each tiny further extension as the wire lengthens, gives a clean final result: the elastic potential energy stored per unit volume of the wire is u = ½ x stress x strain, exactly half the product of the two quantities. There is a second, subtler effect too, easy to overlook: stretching a wire lengthwise doesn't just change its length -- it also causes a small change in its width. Simon Poisson pointed out that, within the elastic limit, this sideways (lateral) strain is directly proportional to the lengthwise (longitudinal) strain that caused it, and the ratio between the two, lateral strain divided by longitudinal strain, is called Poisson's ratio, a pure number with no units, depending only on the material itself -- for steels, it typically falls between 0.28 and 0.30, and for aluminium alloys, around 0.33.

Designing a crane rated to lift 10 tonnes (10⁴ kg) safely requires the steel rope's own cross-sectional area to satisfy A ≥ Mg/σy, where σy is mild steel's yield strength (about 300x10⁶ N/m²) -- working through the numbers gives a minimum area of about 3.3x10⁻⁴ m², corresponding to a rope radius of only about 1 cm, and, allowing the standard safety margin (typically about a factor of ten in load), engineers recommend a thicker rope of about 3 cm radius instead; a single wire this thick would be practically rigid, so real ropes are instead braided from many thin wires, exactly like a pigtail, for genuine flexibility alongside strength. A horizontal beam of length l, breadth b and depth d, loaded at its centre and supported near its ends, sags by an amount δ = W l³/(4 b d³ Y) -- since sag depends on depth cubed but breadth only to the first power, increasing a beam's depth is far more effective at resisting bending than increasing its breadth, which is exactly why load-bearing beams in bridges and buildings are commonly built with a cross-section shaped like the letter I: tall enough to strongly resist bending, without the excess weight and cost of making the beam solid all the way through. Even the maximum possible height of a mountain on Earth follows from these same elastic principles: at the base of a mountain of height h, the vertical stress from the mountain's own weight works out to hρg, where ρ is the rock's density -- and since this force acts vertically while the mountain's sides remain genuinely free, this stress is actually a shear stress on the rock, not a simple compression. Setting this shear stress equal to a typical rock's own elastic (shear) limit, about 30x10⁷ N/m², with rock density ρ≈3x10³ kg/m³, gives h = (30x10⁷)/(3x10³ x 10) = 10,000 m, exactly 10 km -- genuinely more than the height of Mount Everest, and a striking confirmation that a mountain very much taller than this would, quite literally, cause the rock at its own base to flow like a fluid under its own crushing weight.