© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Identify the given information
• Side of the cube, $a = 0.5\,\text{m}$
• The cube floats with $30\%$ of its volume under water.
• Density of water, $\rho_w = 1000\,\text{kg/m}^3$.
• We need to determine the maximum additional mass $m$ that can be placed on the cube so that it just remains afloat (i.e., fully submerged but not sinking).
Step 2: Calculate the volume of the cube
The volume of the cube is given by:
$$
V_{\text{cube}} = a^3 = (0.5\,\text{m})^3 = 0.125 \,\text{m}^3.
$$
Step 3: Relate floating condition to buoyant force
The fraction of the volume submerged (initially $30\%$) indicates how much volume of water is displaced. When $30\%$ is submerged, that displaced water weight is equal to the weight of the cube alone (before adding extra mass).
• Volume of water displaced initially is $0.30 \times 0.125 = 0.0375\,\text{m}^3.$
• Mass of water displaced initially is $0.0375 \,\text{m}^3 \times 1000\,\text{kg/m}^3 = 37.5\,\text{kg}.$
Therefore, the cube’s weight (mass $\times$ g) is $37.5\,\text{kg}\,g.$
Step 4: Determine the total buoyant force when the cube is just fully submerged
When the cube is just fully submerged, the entire volume of $0.125\,\text{m}^3$ displaces water. Hence, the buoyant force (equal to the weight of the displaced water) becomes:
$$
F_{\text{buoyant}} = \rho_w \, V_{\text{cube}} \, g
= 1000 \,\text{kg/m}^3 \times 0.125\,\text{m}^3 \times g
= 125\,\text{kg}\,g.
$$
Step 5: Relate extra mass that can be added
Let $M_{\text{cube}}$ be the mass of the cube (which is $37.5\,\text{kg}$ from Step 3). Let $m$ be the maximum mass that can be added to keep the cube just afloat. The total weight when fully submerged is:
$$
W_{\text{total}} = (M_{\text{cube}} + m)\,g.
$$
This must equal the buoyant force $125\,\text{kg}\,g.$
Hence,
$$
(37.5 + m)\,g = 125\,\text{kg} \, g.
$$
Step 6: Solve for the additional mass
Dividing both sides by $g$, we get:
$$
37.5 + m = 125.
$$
Therefore,
$$
m = 125 - 37.5 = 87.5 \,\text{kg}.
$$
Final Answer
The maximum weight that can be added to the cubical block so that it just remains fully submerged without sinking is 87.5 kg.
