UKPSC Prelims 2025 Q11: Bulk Modulus for a perfectly rigid body

Q: What is the answer to UKPSC Prelims 2025 Q11?

A. Correct — A, Infinite. Bulk modulus is a ratio: the hydraulic stress applied to a body, divided by the fractional change in volume that stress produces. The stiffer the body, the smaller the volume change and the larger the ratio. A perfectly rigid body is defined as one that does not deform. Squeeze it with any pressure you like and its volume change stays exactly zero, so the denominator of the ratio is zero while the numerator is not. A non-zero quantity divided by a vanishing one grows without bound. That is why the value is taken as **Infinite** — it is the limit the ratio approaches, not a measured number. Carry away the limiting logic: **zero strain in the denominator sends every elastic modulus of a perfectly rigid body to infinity**. The same argument works for Young's modulus and for the modulus of rigidity.

UKPSC Prelims 2025 General Studies Paper-I · Question 11 of 149

Verified against the official question booklet. Answer as per UKPSC's provisional answer key.

Answers from UKPSC's provisional answer key. The Commission publishes a provisional key for UKPSC Prelims 2025 and invites objections before issuing a final key. Every answer shown here is the Commission's own provisional answer (key PDF), not our derivation. If the final key changes an answer, this page is updated to match.

Bulk Modulus for a perfectly rigid body is:

(a)Infinite
(b)Zero
(c)Unity
(d)Some finite low value

Answer

Why

Correct — A, Infinite.

Bulk modulus is a ratio: the hydraulic stress applied to a body, divided by the fractional change in volume that stress produces. The stiffer the body, the smaller the volume change and the larger the ratio.

A perfectly rigid body is defined as one that does not deform. Squeeze it with any pressure you like and its volume change stays exactly zero, so the denominator of the ratio is zero while the numerator is not.

A non-zero quantity divided by a vanishing one grows without bound. That is why the value is taken as Infinite — it is the limit the ratio approaches, not a measured number.

Carry away the limiting logic: zero strain in the denominator sends every elastic modulus of a perfectly rigid body to infinity. The same argument works for Young's modulus and for the modulus of rigidity.

Why the others are wrong

(b)
Zero — Zero is the answer you reach by reading the ratio upside down. It is correct for the compressibility of a perfectly rigid body, since compressibility is defined as one divided by the bulk modulus, and the reciprocal of an unbounded quantity vanishes.
A body whose bulk modulus were itself zero would change volume under the slightest pressure. That describes something infinitely squeezable, which is the opposite of rigid.
(c)
Unity — A modulus has the dimensions of pressure and is quoted in pascals, so a bare number like 1 depends entirely on the unit chosen and describes no limiting behaviour.
Unity is the right answer for the ratio of a perfectly rigid body's volume after loading to its volume before loading, which stays exactly 1 because nothing changes. That dimensionless ratio is not the modulus, and the two must not be swapped.
(d)
Some finite low value — A small finite bulk modulus describes a body that visibly yields to pressure. It fits an easily compressed substance such as a gas, whose bulk modulus at ordinary pressures is far below that of a solid or a liquid.
Rigidity demands the opposite extreme. Since the volume change is zero rather than merely small, the ratio is driven upward without limit, not down to a low figure.

Concept

An elastic modulus measures resistance to deformation. Each one is a stress divided by the strain that stress produces, so a stiffer body — less strain for the same stress — carries the larger modulus.

Bulk modulus is the version for uniform pressure acting on all sides: hydraulic stress over volumetric strain. Young's modulus uses tensile stress over longitudinal strain. The modulus of rigidity uses shearing stress over shearing strain.

'Perfectly rigid' is an idealisation rather than a real material. It names the limiting case in which strain goes to zero for any applied stress, which drives each of these moduli to infinity.

Perfect rigidity is the modelling assumption behind rigid-body mechanics, where bodies are treated as holding their shape and volume exactly while forces and torques act on them.

Elasticity is the topic that relaxes that assumption and lets bodies deform. Sending the strain back to zero recovers the rigid-body picture as a limiting case, and an infinite modulus is the mathematical signature of that limit.

Real materials sit between the extremes. Steel resists compression far more strongly than water does, and water far more than air, but each of them has a finite bulk modulus.

Key facts

Bulk modulus is the applied hydraulic stress divided by the fractional change in volume it produces.
A minus sign is written into the definition so the value stays positive, because raising the pressure reduces the volume.
A perfectly rigid body suffers no change in volume or in shape under load, so its strain stays zero.
As the strain approaches zero for a fixed applied stress, the ratio grows without bound, so the modulus is taken as infinite in the limit.
Compressibility is the reciprocal of bulk modulus, so a perfectly rigid body has zero compressibility.
Bulk modulus carries the dimensions of pressure and is expressed in pascals, as are Young's modulus and the modulus of rigidity.
Solids and liquids resist compression far more strongly than gases do; steel's bulk modulus is much larger than water's.
For an ideal gas compressed isothermally, the bulk modulus works out equal to the pressure itself.

Bulk modulus across the stiffness range

Perfectly rigid body→Modulus infinite — zero volume change

Steel under pressure→Very large but finite modulus

Water under pressure→Modulus far below steel's

Gas at ordinary pressure→Small modulus, readily compressed

Compressibility of a rigid body→Zero — the reciprocal of the modulus

Bulk modulus climbs as the volume change under a given pressure shrinks. The perfectly rigid body is the limiting case where that change is zero and the ratio is unbounded.

Study next

The three elastic moduli — Young's modulus, bulk modulus, modulus of rigidity — and the defining ratio of eachCompressibility as the reciprocal of bulk modulusHooke's law, the proportional limit and the elastic limitThe stress–strain curve: yield point, ultimate strength, fracture pointPoisson's ratio and the relations among the elastic constantsBulk modulus of gases: isothermal versus adiabatic compressionRigid-body mechanics and why perfect rigidity is assumed there

Common traps

Inverting the definition: compressibility is one over the bulk modulus, so a rigid body's compressibility is zero while its bulk modulus is infinite. Both answers sit in the options.
Reading the word 'rigid' as a cue for the modulus of rigidity: in this limiting case bulk modulus, Young's modulus and the modulus of rigidity each run to infinity alike, so that switch changes nothing here — the distinction decides definition-matching items, not a value question like this one.
Treating a modulus as a pure number. It carries units of pressure, so 'unity' means nothing until the unit is fixed.
Swapping the defining ratios: tensile stress over longitudinal strain is Young's modulus; hydraulic stress over volumetric strain is bulk modulus.
Rejecting an infinite answer as a trick option. Here it is the direct consequence of a fixed stress acting while the strain goes to zero.
Confusing 'very stiff' with 'perfectly rigid'. Steel deforms slightly and has a finite modulus; the idealised rigid body does not deform at all.

One form is the limiting case, which is what this stem does: the value of a modulus for a perfectly rigid body, or the compressibility of such a body. The answer follows from the definition rather than from a remembered number.

A second form is definition-matching. CAPF_GAI_2025_Q115 is exactly that — it asks which ratio is called bulk modulus, which Young's modulus and which the modulus of rigidity, with the first two definitions swapped in the statements.

Related PYQs

CAPF_GAI_2025_Q1152025

Same definitional core. That item tests which ratio is the bulk modulus, which is Young's modulus and which is the modulus of rigidity — the very definitions on which this stem's answer rests, and it plants swapped definitions in its first two statements. The demand differs. The CAPF item is a multi-statement correctness question that is settled once the three definitions are recalled correctly. The UKPSC stem asks for the value of one modulus in the limiting case of a perfectly rigid body, so it needs the definition plus the further step of noticing that the strain in the denominator is zero.

Practice

practice — not a real PYQ
The compressibility of a perfectly rigid body is:
- (a)Infinite
- (b)Zero
- (c)Unity
- (d)Equal to its bulk modulus
Answerb — Compressibility is defined as one divided by the bulk modulus. Since the bulk modulus of a perfectly rigid body is unbounded, its reciprocal is zero.
(a) infinite is the value of the bulk modulus itself, not of compressibility. (c) unity would depend on the unit chosen, since compressibility carries units of inverse pressure.
(d) compressibility is the reciprocal of the bulk modulus, so the two are equal only for a body whose modulus is 1 in the chosen units, not for a rigid body.
practice — not a real PYQ
Bulk modulus has the same dimensions as which one of the following?
- (a)Force
- (b)Pressure
- (c)Strain
- (d)Work
Answerb — Bulk modulus is a stress divided by a strain. Strain is a ratio of like quantities and so is dimensionless, which leaves the modulus with the dimensions of stress, that is, force per unit area — pressure.
(a) force lacks the division by area. (c) strain is dimensionless and a modulus is not. (d) work is force times distance, an energy, which differs from pressure by a factor of volume.
practice — not a real PYQ
For a perfectly rigid body, the modulus of rigidity is:
- (a)Zero
- (b)Infinite
- (c)Unity
- (d)A small finite value
Answerb — A perfectly rigid body does not change shape, so the shearing strain produced by any shearing stress is zero. The stress-to-strain ratio therefore grows without bound, exactly as with the bulk modulus.
(a) zero would describe a body offering no resistance to shear, as an ideal fluid at rest does. (c) unity ignores that the modulus carries units of pressure. (d) a small finite value describes an easily sheared material, the opposite of rigid.
practice — not a real PYQ
The ratio of hydraulic stress to the corresponding volumetric strain is called:
- (a)Young's modulus
- (b)Bulk modulus
- (c)Modulus of rigidity
- (d)Poisson's ratio
Answerb — That ratio is the definition of bulk modulus, the measure of a body's resistance to a uniform pressure acting on all sides.
(a) Young's modulus is tensile or compressive stress over longitudinal strain. (c) modulus of rigidity is shearing stress over shearing strain. (d) Poisson's ratio compares lateral strain with longitudinal strain, so it is a ratio of two strains and is dimensionless, not a modulus.