when a metal ball is heated through 30°c,it volume becomes 1.0018cm^3 if the linear expansivity of the material of the ball is 2.0×10^-5k^-1

Question

when a metal ball is heated through 30°c,it volume becomes 1.0018cm^3 if the linear expansivity of the material of the ball is 2.0×10^-5k^-1, calculate its original volume​

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Xavia 2 weeks 2021-07-22T16:58:07+00:00 1 Answers 0 views 0

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    2021-07-22T16:59:54+00:00

    Answer:

    The original volume of the metal sphere is approximately 1 cm³

    Explanation:

    The given parameters are;

    The temperature change of the metal ball, ΔT = 30°C = 30 K

    The new volume of the metal ball, V₂  = 1.0018 cm³

    The linear expansivity of the material ball, α = 2.0 × 10⁻⁵ K⁻¹

    We have;

    d₂ = d₁·(1 + α·ΔT)

    Where;

    d₁ = The original diameter of the metal ball

    d₂ = The new diameter of the ball

    From the volume of the ball, V₂, we have;

    V₂ = 1.0018 cm³ = (4/3)×π×r₂³

    Where;

    r₂ = The new radius = d₂/2

    ∴ V₂ = 1.0018 cm³ = (4/3)×π×(d₂/2)³

    ∴ d₂ = ∛(2³ × 1.0018 cm³/((4/3) × π)) ≈ 1.241445 cm

    d₁ = d₂/(1 + α·ΔT)

    ∴ d₁ ≈ 1.241445 cm/(1 + 2.0 × 10⁻⁵·K × 30 K) ≈ 1.24070058 cm

    The original volume of the metal ball, V₁ = (4/3)×π×(d₁/2)³

    ∴ V₁ = (4/3)×π×(1.24070058/2)³ ≈ 0.99999902845 cm³ ≈ 1 cm³

    The original volume of the metal sphere, V₁ ≈ 1 cm³.

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