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27 May, 11:35

A hollow 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.

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  1. 27 May, 11:40
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    -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.
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