A leakproof cylinder of length 1 m, made of a metal which has very low coefficient of expansion, is floating vertically in water at 0 °C such that its height above the water surface is 20 cm. When the temperature of water is increased to 4 °C, the height of the cylinder above the water surface beco…

Question

A leakproof cylinder of length 1 m, made of a metal which has very low coefficient of expansion, is floating vertically in water at 0 °C such that its height above the water surface is 20 cm. When the temperature of water is increased to 4 °C, the height of the cylinder above the water surface becomes 21 cm. The density of water at T=4 °C, relative to the density at T=0 °C, is close to

Accepted Answer

(2) 1.01. Given: the metal barely expands, so the cylinder's own volume and weight are fixed; only the water's density changes. Floating means the weight of water displaced equals the weight of the cylinder, so ρ×ℓ_sub is the same at both temperatures. At 0 °C the submerged length is 100-20=80 cm; at 4 °C it is 100-21=79 cm. ρ₀×80=ρ₄×79, so (ρ₄)/(ρ₀)=(80)/(79)=1.0127. The cylinder floats higher because water at 4 °C is denser — the anomalous expansion of water, with its density maximum at 4 °C. So the ratio is close to 1.01.

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