Question

A wooden cube floating in water supports a mass \(m = 0.2\,\text{kg}\) on its top. When the mass is removed, the cube rises by \(2\,\text{cm}\). The side of the cube is (density of water \(= 10^3\,\text{kg/m}^3\)):

• A. \(6\,\text{cm}\)
• B. \(12\,\text{cm}\)
• C. \(8\,\text{cm}\)
• D. \(10\,\text{cm}\)

Hint:

In floating body problems: [ textChange in buoyant force = textWeight added or removed ] Always equate displaced fluid weight to the applied load.

Answer 1

The Correct Option is D

Step 1: Use the principle of buoyancy. When the mass is removed, the cube rises, meaning the decrease in buoyant force equals the weight of the removed mass. Step 2: Write the relation. Loss of buoyant force \(=\) weight of mass removed: \[ \rho_{\text{water}} \, g \, A \, h = m g \] where \(A\) = area of the cube’s face, \(h = 2\,\text{cm} = 0.02\,\text{m}\). Step 3: Cancel \(g\) from both sides. \[ \rho_{\text{water}} \, A \, h = m \] Step 4: Substitute given values. \[ 1000 \times A \times 0.02 = 0.2 \] \[ A = \frac{0.2}{1000 \times 0.02} = \frac{0.2}{20} = 0.01\,\text{m}^2 \] Step 5: Find the side of the cube. Since the cube has a square face: \[ A = a^2 \Rightarrow a = \sqrt{0.01} = 0.1\,\text{m} = 10\,\text{cm} \] Step 6: Final conclusion. The side of the cube is \(\boxed{10\,\text{cm}}\).