Question

A cubical block of ice in water has to support a metal piece weighing 0.5kg. What can be minimum edge of the block so that it does not sink in water. Specific gravity of ice =0.9

• A. The minimum edge of the block is $(0.005)^{1/3}$ meters
• B. The minimum edge of the block is approximately $0.171$ meters
• C. The minimum edge of the block is $(0.005)^{1/3}$ m, which is approximately $0.171$ m
• D. The minimum edge of the block is $0.005$ m

Answer 1

Answer: (C)

The minimum edge of the block is $(0.005)^{1/3}$ meters, which is approximately $0.171$ meters.

Answer 2

Let 'a' be the edge length of the cubical ice block. The volume of the ice block is $V_{ice} = a^3$. The specific gravity of ice is $SG_{ice} = 0.9$. The density of ice is $\rho_{ice} = SG_{ice} \times \rho_{water} = 0.9 \times 1000 \, \text{kg/m}^3 = 900 \, \text{kg/m}^3$. The mass of the ice block is $m_{ice} = \rho_{ice} V_{ice} = 900 a^3$. The weight of the ice block is $W_{ice} = m_{ice} g = 900 a^3 g$.

The mass of the metal piece is $m_{metal} = 0.5 \, \text{kg}$. The weight of the metal piece is $W_{metal} = m_{metal} g = 0.5 g$.

The total downward weight to be supported is $W_{total} = W_{ice} + W_{metal} = (900 a^3 + 0.5) g$.

The buoyant force ($F_B$) is equal to the weight of the water displaced by the submerged volume ($V_{submerged}$) of the ice block: $F_B = \rho_{water} V_{submerged} g$.

For the ice block not to sink, the buoyant force must be greater than or equal to the total weight: $F_B \ge W_{total}$ $\rho_{water} V_{submerged} g \ge (900 a^3 + 0.5) g$

The maximum possible buoyant force occurs when the entire ice block is submerged, i.e., $V_{submerged} = V_{ice} = a^3$. So, $F_{B,max} = \rho_{water} a^3 g = 1000 a^3 g$.

For the block to not sink, this maximum buoyant force must be sufficient to support the total weight: $F_{B,max} \ge W_{total}$ $1000 a^3 g \ge (900 a^3 + 0.5) g$

To find the minimum edge length 'a', we consider the limiting case where the buoyant force exactly balances the total weight, with the ice block just fully submerged: $1000 a^3 g = (900 a^3 + 0.5) g$

Dividing by $g$: $1000 a^3 = 900 a^3 + 0.5$ $100 a^3 = 0.5$ $a^3 = \frac{0.5}{100} = 0.005 \, \text{m}^3$

The minimum edge length is: $a = (0.005)^{1/3} \, \text{m}$

Calculating the value: $a \approx 0.171 \, \text{m}$