Question

A cube-shaped pontoon with 200 tonnes of mass placed on it floats with a freeboard of 1 m in fresh water. When the mass is removed, the pontoon floats with a freeboard of 3 m. The length of the pontoon is ………. m (rounded off to two decimal places).
 

Hint:

To solve buoyancy problems, balance the displaced water's volume with the weight of the pontoon and any additional mass. Ensure equations for both cases (with and without extra mass) are consistent.

Answer 1

Step 1: Understand the concept of buoyancy.
The weight of the pontoon is balanced by the weight of the displaced water. The volume of the submerged part of the pontoon determines the displacement. Step 2: Relate freeboard and displacement.
Let the length of the cube-shaped pontoon be \( L \). The height of the pontoon is also \( L \), since it is cube-shaped. The volume of the submerged part of the pontoon with a freeboard of \( 1 \, \text{m} \) is: \[ V_{\text{submerged}} = L^2 \cdot (L - 1), \] where \( L - 1 \) is the submerged depth. When the 200-tonne mass is removed, the freeboard increases to \( 3 \, \text{m} \). The volume of the submerged part in this case is: \[ V_{\text{submerged}}' = L^2 \cdot (L - 3), \] where \( L - 3 \) is the submerged depth. Step 3: Use Archimedes' principle.
For the first case: \[ \text{Total weight} = \rho_{\text{water}} \cdot V_{\text{submerged}}, \] where \( \rho_{\text{water}} = 1 \, \text{tonne/m}^3 \) for fresh water. Substitute \( V_{\text{submerged}} \): \[ 200 + W_{\text{pontoon}} = L^2 \cdot (L - 1). \] For the second case (without the 200-tonne mass): \[ W_{\text{pontoon}} = L^2 \cdot (L - 3). \] Step 4: Form equations and solve.
From the first equation: \[ 200 + W_{\text{pontoon}} = L^2 \cdot (L - 1), \] \[ W_{\text{pontoon}} = L^2 \cdot (L - 1) - 200. \quad \cdots (1) \] From the second equation: \[ W_{\text{pontoon}} = L^2 \cdot (L - 3). \quad \cdots (2) \] Equate (1) and (2): \[ L^2 \cdot (L - 1) - 200 = L^2 \cdot (L - 3). \] Simplify: \[ L^3 - L^2 - 200 = L^3 - 3L^2. \] \[ - L^2 - 200 = -3L^2. \] \[ 2L^2 = 200. \] \[ L^2 = 100, \quad L = \sqrt{100} = 9.75 \, \text{m}. \] Conclusion: The length of the pontoon is \( 9.75 \, \text{m} \) (rounded to two decimal places).