Which one of the following is dimensionless quantity?
- (a)Stress
- (b)Strain
- (c)Pressure
- (d)Force
Correct — B, Strain. Strain is the ratio of a change in dimension to the original dimension — for example, the extension of a wire divided by its original length. Because it divides a length by a length, the units cancel and strain is a pure number with no dimensions. Every other option carries dimensions.
- (a)Stress — Stress is force per unit area, with dimensions of [M L⁻¹ T⁻²] and the SI unit pascal (N/m²) — it is not dimensionless.
- (c)Pressure — Pressure is also force per unit area, dimensionally identical to stress ([M L⁻¹ T⁻²], unit pascal), so it has dimensions.
- (d)Force — Force equals mass × acceleration, with dimensions [M L T⁻²] and the SI unit newton — clearly a dimensional quantity.
A physical quantity is dimensionless when its defining ratio cancels all units. Strain is defined as change in length over original length (or change in volume over original volume), so it reduces to a pure number. Quantities formed as force divided by area (stress, pressure) or as mass times acceleration (force) retain dimensions.
The examiner groups strain with stress, pressure and force to tempt you into thinking all elasticity terms carry units. The discriminator is simple — a quantity built purely as a ratio of like quantities has no dimensions.
- Strain (linear, volumetric or shear) is a ratio of two like quantities, so it is dimensionless and unit-less.
- Stress and pressure are both force per unit area, dimensions [M L⁻¹ T⁻²], SI unit pascal.
- Force has dimensions [M L T⁻²], SI unit newton.
- Other common dimensionless quantities include refractive index, relative density, angle (radian) and strain.
Only strain reduces to a pure number, so option (b) is the dimensionless quantity.
- Assuming every elasticity term (stress, strain, modulus) carries units — strain does not.
- Treating angle or refractive index as dimensional; they are also pure numbers.
Asked as 'which quantity is dimensionless' or by giving a dimensional formula and asking which quantity it belongs to.
No directly related past PYQ was found.
- practice — not a real PYQ
Which one of the following pairs consists only of dimensionless quantities?
- (a)Strain and refractive index
- (b)Stress and pressure
- (c)Force and momentum
- (d)Work and energy
Answer(a) Strain and refractive index — both are ratios of like quantities and carry no dimensions.
- practice — not a real PYQ
The dimensional formula [M L⁻¹ T⁻²] represents
- (a)Force
- (b)Strain
- (c)Pressure
- (d)Work
Answer(c) Pressure — force per unit area (same dimensions as stress and the modulus of elasticity).