Expansion — Set 1

Correct Answer: C. Fractional change in length per unit rise in temperature

• **Fractional change in length per unit rise in temperature** = The linear expansion coefficient α = ΔL/(L₀ΔT) — it gives the fraction by which a solid's length changes for each degree rise in temperature. • **α (unit: K⁻¹)** — For example, steel has α ≈ 12×10⁻⁶ K⁻¹, meaning 1 m of steel grows by 12 μm per kelvin. • 💡 Wrong-option analysis: Fractional change in volume per unit rise in temperature: that defines the cubical expansion coefficient γ; Fractional change in area per unit rise in temperature: that defines the superficial expansion coefficient β; Change in mass per unit rise in temperature: mass does not change with temperature (no material is added or removed).

Correct Answer: B. β = 2α

• **β = 2α** = In an isotropic solid, area expands equally in two perpendicular directions; so β = ΔA/(A₀ΔT) ≈ 2α for small temperature changes. • **γ = 3α, β = 2α, α** — The three expansion coefficients are in the ratio 3:2:1; doubling one direction's expansion coefficient doubles the area expansion. • 💡 Wrong-option analysis: β = α/2: this halves the expected value — incorrect for isotropic expansion; β = 3α: that is the cubical expansion coefficient γ; β = α²: squaring the coefficient gives nonsensical dimensions.

Correct Answer: A. K^-1

• **K⁻¹** = The coefficient of linear expansion α = ΔL/(L₀ΔT) has dimensions of [length/(length × temperature)] = 1/temperature, so the SI unit is per kelvin (K⁻¹). • **Dimensionless/ΔT** — Because ΔL/L is dimensionless and ΔT is in kelvin, α has units of K⁻¹; numerically the same as per °C. • 💡 Wrong-option analysis: N: newton is a unit of force; m: metre is a unit of length; m/s: metre per second is a unit of velocity.

Correct Answer: A. ΔL = αLΔT

• **ΔL = αLΔT** = The extension ΔL of a rod of original length L is directly proportional to both L and the temperature rise ΔT, with α as the proportionality constant. • **α (linear expansion coefficient)** — This formula assumes α is approximately constant over the temperature range; for large ΔT, higher-order terms may be needed. • 💡 Wrong-option analysis: ΔL = L/(αΔT): inverting α gives wrong dependence; ΔL = αΔT/L: dividing by L gives wrong dimensions; ΔL = αL/ΔT: dividing by ΔT inverts the dependence.

Correct Answer: C. γ = 3α

• **γ = 3α** = In an isotropic solid, volume expands equally in all three perpendicular directions; so γ = ΔV/(V₀ΔT) ≈ 3α. • **α, β = 2α, γ = 3α** — These relationships hold for small temperature changes; for large ΔT, the exact expression is (1 + αΔT)³ - 1 ≈ 3αΔT. • 💡 Wrong-option analysis: γ = α³: cubing the coefficient is dimensionally wrong; γ = α/3: this would be smaller than α, which makes no sense for 3D expansion; γ = 2α: that is the superficial (area) expansion coefficient β.

Correct Answer: A. Increases

• **Increases** = When a metal plate with a hole is heated uniformly, every part of the plate — including the rim of the hole — expands outward; the hole enlarges just as if it were made of the same material. • **Scale factor** — The hole expands as if it were a disk of the same metal: its diameter increases by ΔD = αDΔT, where D is the original diameter. • 💡 Wrong-option analysis: Becomes zero: the hole cannot shrink to zero unless the plate melts; Decreases: material expands outward in all directions, not inward to close the hole; Remains unchanged: the hole is part of the plate and scales with the plate's expansion.

Correct Answer: B. The container also expands along with the liquid

• **The container also expands along with the liquid** = When both liquid and container are heated, the container volume also increases; the apparent (observed) rise of liquid is less than its true (real) expansion. • **Apparent expansion = Real expansion − Container expansion** — If the container expands significantly, the apparent expansion can be much less than the real; if the container were infinitely rigid, apparent = real. • 💡 Wrong-option analysis: The liquid changes into gas: at normal heating temperatures liquids don't convert to gas; The container does not expand: all solid containers do expand when heated; The liquid has no real expansion: liquids do expand with heat.

Correct Answer: C. It contracts and becomes denser

• **It contracts and becomes denser** = Water behaves anomalously between 0°C and 4°C — it contracts on heating (unlike most liquids), becoming denser until it reaches maximum density at 4°C. • **Maximum density at 4°C** — This anomalous behavior is due to the breaking of hydrogen-bond structures in ice as it warms from 0°C; above 4°C, normal thermal expansion takes over. • 💡 Wrong-option analysis: It boils slowly in this range: water boils at 100°C, not in the 0–4°C range; It expands and becomes less dense: this is the normal liquid behavior above 4°C; It shows no volume change: volume does change in this range.

Correct Answer: A. Thermostats and automatic switches

• **Thermostats and automatic switches** = A bimetallic strip bends when heated because the two metals have different coefficients of expansion; this bending can open or close an electrical contact, making it ideal for thermostats. • **Bending mechanism** — The metal with higher α (e.g., brass) expands more and forms the outer arc; the strip bends toward the metal with lower α (e.g., invar or steel). • 💡 Wrong-option analysis: Measuring electric current directly: bimetallic strips do not directly measure current (though current heating can make them operate); Producing magnetic fields: bimetallic strips are mechanical, not electromagnetic devices; Measuring atmospheric pressure only: barometers use aneroid capsules, not bimetallic strips.

Correct Answer: B. Allow expansion in hot weather

• **Allow expansion in hot weather** = Steel rails expand significantly in hot weather; without gaps, the rails would buckle under compressive thermal stress, potentially causing derailments. • **ΔL = αLΔT** — A 10-metre steel rail (α ≈ 12×10⁻⁶ K⁻¹) expands by about 1.2 mm per 10 K rise; gaps accommodate this expansion safely. • 💡 Wrong-option analysis: Improve electrical insulation: rails carry electric signals for signalling, but gaps are for thermal expansion, not insulation; Increase friction: gaps do not affect friction between wheel and rail; Reduce the weight of rails: removing material might lighten rails, but that is not the purpose of expansion gaps.