Answer:
-38°C
Explanation:
When glycerine is heated, it expands from volume V₁ to volume V₂:
V₂ − V₁ = V₁ β (T₂ − T₁)
Since the mass of glycerine is the same before and after:
m = m
ρ₁ V₁ = ρ₂ V₂
V₂ = (ρ₁ / ρ₂) V₁
Substituting:
(ρ₁ / ρ₂) V₁ − V₁ = V₁ β (T₂ − T₁)
(ρ₁ / ρ₂) − 1 = β (T₂ − T₁)
(ρ₁ − ρ₂) / ρ₂ = β (T₂ − T₁)
(ρ₁ − ρ₂) / (ρ₂ β) = T₂ − T₁
T₂ = T₁ + (ρ₁ − ρ₂) / (ρ₂ β)
The glass sphere will float when it has the same density as the glycerine.
Given:
T₁ = 20°C
ρ₁ = 1.26 g/cm³
ρ₂ = 1.30 g/cm³
β = 53×10⁻⁵/°C
T₂ = 20 + (1.26 − 1.30) / (1.30 × 53×10⁻⁵)
T₂ = -38
The glycerine must be cooled to -38°C for the hollow glass sphere to float.
