To determine the minimum edge length of a cubical block of ice that can float while supporting a metal piece weighing 0.5 kg, we need to consider the principles of buoyancy and the specific gravity of the materials involved. Let's break this down step by step.
Understanding Buoyancy
Buoyancy is the upward force exerted by a fluid that opposes the weight of an object submerged in it. According to Archimedes' principle, the buoyant force is equal to the weight of the fluid displaced by the submerged part of the object. For our scenario, the block of ice must displace enough water to support both its own weight and the weight of the metal piece.
Calculating the Weight of the Ice Block
The specific gravity of ice is given as 0.9, which means that ice is 0.9 times as dense as water. The density of water is approximately 1000 kg/m³, so the density of ice can be calculated as follows:
- Density of ice = Specific gravity of ice × Density of water
- Density of ice = 0.9 × 1000 kg/m³ = 900 kg/m³
Next, let's denote the edge length of the cubical block of ice as a. The volume of the ice block can be expressed as:
The weight of the ice block can then be calculated using its density:
- Weight of ice block = Volume × Density = a³ × 900 kg/m³
Weight of the Metal Piece
The weight of the metal piece is straightforward to calculate:
- Weight of metal piece = mass × gravity = 0.5 kg × 9.81 m/s² ≈ 4.905 N
Setting Up the Buoyancy Equation
For the block of ice to float without sinking, the total weight (weight of the ice block plus the weight of the metal piece) must be equal to the buoyant force, which is equal to the weight of the water displaced by the submerged volume of the ice block.
The buoyant force can be expressed as:
- Buoyant force = Volume of water displaced × Density of water × g
- Buoyant force = (Volume of ice block submerged) × 1000 kg/m³ × 9.81 m/s²
Assuming the entire block is submerged (which is the worst-case scenario for our calculation), we can set up the equation:
- a³ × 900 kg/m³ × 9.81 m/s² + 4.905 N ≤ a³ × 1000 kg/m³ × 9.81 m/s²
Simplifying the Equation
Now, we can simplify this equation:
- 900a³ × 9.81 + 4.905 ≤ 1000a³ × 9.81
- 900a³ × 9.81 - 1000a³ × 9.81 ≤ -4.905
- -100a³ × 9.81 ≤ -4.905
- 100a³ × 9.81 ≥ 4.905
Dividing both sides by 9.81 gives:
Finding the Minimum Edge Length
To find the edge length a, we take the cube root of both sides:
Calculating this gives:
- a ≥ 0.17 m (approximately)
Thus, the minimum edge length of the cubical block of ice required to support the 0.5 kg metal piece without sinking is approximately 0.17 meters, or 17 centimeters. This ensures that the block of ice can displace enough water to support both its own weight and the weight of the metal piece.
