A leak proof cylinder of length 1m, 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 :

Solution

Physics sir · 3 steps

The cylinder's weight never changes, so the buoyant force must stay the same at both temperatures.

1. Fixed weight. The metal barely expands, so the cylinder keeps the same size and weight. While it floats, the buoyant force always equals this weight.
2. Submerged lengths. The cylinder is 100 cm long. At 0 °C, 100 - 20 = 80 cm is under water. At 4 °C, 100 - 21 = 79 cm is under water.

   ρ₀ (80A) g = ρ₄ (79A) g ⇒ ρ₄/ρ₀ = 80/79

3. So. ρ₄/ρ₀ = 80/79 ≈ 1.013, which is close to 1.01, option (c). Denser water needs less submerged volume to give the same upthrust, so the cylinder rises.

The idea to keep. For a floating body of fixed weight, the density of the liquid times the submerged volume stays constant. Use the submerged length, not the length above the surface.

Answer from the NTA final answer key. Solution by PrepWiser.