Solution for Question 12
1. Determining the Dimensions of the Cube (Option a):
Initially, the cube floats half-submerged with the spring relaxed. This means the Buoyant force equals the Weight.
$$ F_B = mg $$ $$ \rho_{water} \cdot V_{sub} \cdot g = m \cdot g $$ $$ 1000 \cdot \frac{L^3}{2} = 2.048 $$ $$ L^3 = \frac{2.048 \times 2}{1000} = 0.004096 \text{ m}^3 $$ $$ L = \sqrt[3]{0.004096} = 0.16 \text{ m} = 16 \text{ cm} $$Statement (a) is correct.
2. Determining Spring Force when Fully Submerged (Option c):
When additional water is added to fully submerge the cube, the new Buoyant force is:
$$ F_B’ = \rho_{water} \cdot L^3 \cdot g = 1000 \cdot 0.004096 \cdot 10 = 40.96 \text{ N} $$The weight of the cube is $mg = 2.048 \times 10 = 20.48 \text{ N}$.
The spring force $F_s$ balances the difference:
$$ F_B’ = mg + F_s \implies F_s = 40.96 – 20.48 = 20.48 \text{ N} $$Statement (c) is correct.
3. Determining Displacement and Added Water (Options b and d):
Let $x$ be the extension of the spring (displacement of the cube upwards). The water level rises to exactly cover the top of the cube.
Geometric Constraint:
Initially, water level was at $L/2$ (8 cm) on the cube. Finally, it is at $L$ (16 cm) on the cube. This means the water level rose relative to the cube by 8 cm. However, the cube itself rose by distance $x$. Therefore, the absolute rise in water level $\Delta h_{water}$ is:
$$ \Delta h_{water} = x + 8 \text{ cm} $$Volume Conservation:
The volume of added water $V_{add}$ fills the space created by the rise in water level, minus the change in the submerged volume of the cube.
$$ V_{add} = A_{vessel} \cdot \Delta h_{water} – \Delta V_{submerged} $$$\Delta V_{submerged} = V_{total} – V_{initial} = L^3 – \frac{L^3}{2} = \frac{L^3}{2} = \frac{4096}{2} = 2048 \text{ cm}^3$.
Using the values from option (b) ($V_{add} = 10.24 \text{ L} = 10240 \text{ cm}^3$) and option (d) ($x = 12.48 \text{ cm}$), let’s verify consistency:
$$ V_{add} = 600(x + 8) – 2048 $$ $$ 10240 \stackrel{?}{=} 600(12.48 + 8) – 2048 $$ $$ 10240 \stackrel{?}{=} 600(20.48) – 2048 $$ $$ 10240 \stackrel{?}{=} 12288 – 2048 $$ $$ 10240 = 10240 $$The calculations are consistent. Thus, both statements are correct.