An ice cube's (1/n)th portion sinks in water, then the density of ice is _______ .

Answer

To find the density of ice when (1/n)th portion of an ice cube sinks in water, we can use the principles of buoyancy and Archimedes' principle. Here's a step-by-step solution:

### Step 1: Understand the situation
When an ice cube is placed in water, a portion of it sinks while the rest floats. According to the problem, (1/n)th of the ice cube sinks. This means that the remaining (n-1)/n portion is floating above the water.

### Step 2: Apply Archimedes' Principle
Archimedes' principle states that the buoyant force (thrust) acting on a submerged object is equal to the weight of the fluid displaced by that object. In this case, the weight of the water displaced by the submerged portion of the ice cube is equal to the weight of the ice cube itself.

### Step 3: Define the variables
Let:
- \( V \) = Total volume of the ice cube
- \( \rho_{ice} \) = Density of ice
- \( \rho_{water} \) = Density of water (approximately \( 1000 \, kg/m^3 \))
- \( g \) = Acceleration due to gravity

### Step 4: Calculate the weight of the ice cube
The weight of the ice cube can be expressed as:
\[ Weight_{ice} = \rho_{ice} \cdot V \cdot g \]

### Step 5: Calculate the volume of the submerged portion
Since (1/n)th of the ice cube sinks, the volume of the submerged portion is:
\[ V_{submerged} = \frac{1}{n} V \]

### Step 6: Calculate the weight of the displaced water
The weight of the water displaced by the submerged portion is:
\[ Weight_{displaced} = \rho_{water} \cdot V_{submerged} \cdot g = \rho_{water} \cdot \left(\frac{1}{n} V\right) \cdot g \]

### Step 7: Set the weights equal according to Archimedes' principle
According to Archimedes' principle:
\[ Weight_{ice} = Weight_{displaced} \]
Substituting the expressions from Steps 4 and 6:
\[ \rho_{ice} \cdot V \cdot g = \rho_{water} \cdot \left(\frac{1}{n} V\right) \cdot g \]

### Step 8: Simplify the equation
We can cancel \( V \) and \( g \) from both sides (assuming \( V \neq 0 \) and \( g \neq 0 \)):
\[ \rho_{ice} = \frac{1}{n} \cdot \rho_{water} \]

### Step 9: Substitute the density of water
Substituting \( \rho_{water} \approx 1000 \, kg/m^3 \):
\[ \rho_{ice} = \frac{1}{n} \cdot 1000 \, kg/m^3 \]

### Final Answer
Thus, the density of ice is:
\[ \rho_{ice} = \frac{1000}{n} \, kg/m^3 \]

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