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Most elasticity problems in Class 11 are about stretching wires. Bulk modulus questions look easier, yet students lose marks on them through sign errors, wrong pressure values and confusing volume change with radius change. This post explains what bulk modulus measures, works through three problems with real numbers, and ends with a short practice routine.
What does bulk modulus measure?
When a body is squeezed equally from all sides, as it is when submerged in a fluid, its shape stays the same but its volume shrinks. The stress here is the pressure change Δp, and the strain is the fractional volume change ΔV/V. NCERT's chapter on Mechanical Properties of Solids defines bulk modulus as their ratio:
B = −Δp / (ΔV/V)
The minus sign is there because a rise in pressure causes a fall in volume, so B comes out positive. Its SI unit is the pascal, the same as pressure. The reciprocal, k = 1/B, is the compressibility: the fractional volume change per unit increase in pressure.
The same NCERT table gives typical values: steel 160 GPa, copper 140 GPa, glass 37 GPa, water 2.2 GPa, ethanol 0.9 GPa and air at STP about 1.0 × 10⁻⁴ GPa. The textbook sums it up: gases are about a million times more compressible than solids.
The topic is on the official lists. The JEE (Main) 2026 syllabus names "bulk modulus and modulus of rigidity" in its properties of solids and liquids unit, the NEET (UG) 2026 syllabus lists bulk modulus alongside Young's modulus, and the JEE (Advanced) 2026 brochure includes "bulk modulus in mechanics" and "bulk modulus of gases". Check the current year's documents before your exam.
How much does water shrink deep in the sea?
Example 1. A sealed, flexible bag holds 1.00 litre of seawater at the surface. It is taken down to 1000 m. How much does the water's volume change? Take seawater density 1030 kg/m³, g = 9.8 m/s² and B = 2.2 × 10⁹ Pa.
The pressure increase is the extra fluid column: Δp = ρgh = 1030 × 9.8 × 1000 = 1.01 × 10⁷ Pa. Atmospheric pressure is the same at both points, so it cancels.
Fractional change: ΔV/V = −Δp/B = −(1.01 × 10⁷)/(2.2 × 10⁹) = −4.6 × 10⁻³, a decrease of about 0.46%.
Volume change: ΔV = −4.6 × 10⁻³ × 1.00 L ≈ −4.6 mL.
So even under a hundred atmospheres of extra pressure, the water loses less than half a percent of its volume. That is why we usually treat liquids as incompressible in fluid problems. The density rises in the same proportion, from 1030 to about 1035 kg/m³.
Why does a steel ball shrink so little?
Example 2. A steel ball of radius 1.00 cm goes down to the same depth. By how much does its radius change? B for steel is 1.6 × 10¹¹ Pa.
ΔV/V = −Δp/B = −(1.01 × 10⁷)/(1.6 × 10¹¹) = −6.3 × 10⁻⁵.
For a sphere, V = (4/3)πr³, so for small changes ΔV/V = 3 Δr/r.
Δr/r = −2.1 × 10⁻⁵, so Δr ≈ −2.1 × 10⁻⁷ m, about 0.21 micrometres.
The factor of 3 is the step most students miss. If you write Δr/r = ΔV/V you get an answer three times too large. The same idea for a cube gives ΔV/V = 3 ΔL/L.
What is the bulk modulus of a gas?
For a gas, B depends on how it is compressed. NCERT's Waves chapter derives both cases from the gas laws. For a slow, isothermal compression, pV = constant gives B = p. For a fast, adiabatic compression, pV^γ = constant gives B = γp.
Example 3. Compare the bulk modulus of air at 1 atm with water.
Isothermal: B = p = 1.013 × 10⁵ Pa.
Adiabatic, with γ = 1.4 for air: B = 1.4 × 1.013 × 10⁵ ≈ 1.42 × 10⁵ Pa.
Water's B is 2.2 × 10⁹ Pa, roughly 20,000 times larger than air's isothermal value.
This is exactly what decides the speed of sound, v = √(B/ρ). Newton used B = p for air and got about 280 m/s; Laplace's correction to B = γp gives about 331 m/s, matching measurement, as the NCERT chapter shows. For water, √(2.2 × 10⁹ / 1000) ≈ 1480 m/s, which is why sound travels much faster in water than in air.
What mistakes cost marks in bulk modulus problems?
Using total pressure instead of the change. B uses Δp. If a body goes from the surface to a depth h, Δp = ρgh, not p₀ + ρgh.
Dropping the sign, then getting confused. Keep the minus sign in the definition, then report "volume decreases by" with a positive number.
Mixing up ΔV/V and Δr/r. Use the factor of 3 for spheres and cubes.
Unit slips. Tables list B in GPa (10⁹ Pa). Pressure in atm must be converted: 1 atm = 1.013 × 10⁵ Pa.
Using Young's modulus. Y describes stretching along one direction. Uniform pressure from all sides needs B. Our guide to stress, strain and Young's modulus covers the other two moduli.
Forgetting the gas case. In a gas, B is not a fixed material constant; it equals p or γp depending on the process.
A 30-minute practice routine
Five minutes: from memory, write B and k definitions, the sphere relation ΔV/V = 3Δr/r, and B = p and B = γp for gases.
Ten minutes: solve two depth problems, one with a liquid and one with a solid, using values from the NCERT table. Change the depth and predict how the answer scales before you calculate.
Ten minutes: do one speed-of-sound problem that needs B, for example in water or in a gas at a different temperature.
Five minutes: check each answer for size. A solid should change by parts per million, a liquid by under a percent at ocean depths.
If pressure in fluids is the weaker link, our post on fluid pressure and Bernoulli mistakes covers it. To revise the elastic moduli one idea at a time, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can move from stress and strain to the speed of sound in one place.
Tonight, redo Example 1 for a depth of 4000 m and check that your answer is four times as large.
