Ahollow glass sphere has a density of 1.30g/cm cubic at 20°c. glycerine has a density of 1.26g/cm cubic at 20°c. at what temperature would the sphere begin to float in glycerine?
given coefficient of volume expansion of glycerine is 53 ×   10^-5/  °c.

-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.