The volume of a 1 cm ^3 metal ball increases by 0.0018 cm ^3 when heated through temperature θ. If the linear…

Analyze the question: A metal ball of volume 1 cm³ expands by 0.0018 cm³ when heated through a temperature change θ. We are given the linear expansivity α = 2.0 × 10⁻⁵ K⁻¹ and must find θ.

Concept: For volume expansion of a solid, the cubic (volume) expansivity is approximately three times the linear expansivity:

γ = 3α

and

ΔV = V₁ γ θ

Solution:

1. Cubic expansivity: γ = 3 × 2.0 × 10⁻⁵ = 6.0 × 10⁻⁵ K⁻¹

2. Use ΔV = V₁γθ: 0.0018 = 1 × 6.0 × 10⁻⁵ × θ

3. Solve for θ: θ = 0.0018 / (6.0 × 10⁻⁵) θ = 1.8 × 10⁻³ / 6.0 × 10⁻⁵ θ = 30 K

Since a change of 1 K equals a change of 1°C, θ = 30°C.

Check: 6.0 × 10⁻⁵ × 30 = 1.8 × 10⁻³ = 0.0018 ✓

Exam tip: Remember the chain: α (linear) → 2α (area) → 3α (volume). Whenever a question gives linear expansivity but asks about volume, multiply by 3 first. Also, a temperature change in kelvin equals the same change in °C, so no conversion is needed.