A glass beaker is placed partially filled with water in a sink it has a mass of 390 gm and an interior volume of `500 cm^(2)` when water starts filling the sink, it is found that if beaker is less than half full it will float but if it is more than half full it remains on the bottom of the sink as the water rises to its rim, what is the density of the material of which the beaker is made?

Answer

To solve the problem, we need to determine the density of the material of the glass beaker based on the information provided. Let's break down the solution step by step.

### Step 1: Understand the Problem
The beaker has a mass of 390 grams and an interior volume of 500 cm³. It floats when it is less than half full of water (which is 250 cm³) and sinks when it is more than half full.

### Step 2: Calculate the Mass of Water in the Beaker
When the beaker is half full, it contains:
\[ \text{Volume of water} = \frac{500 \, \text{cm}^3}{2} = 250 \, \text{cm}^3 \]

The density of water is approximately 1 g/cm³, so the mass of the water in the beaker is:
\[ \text{Mass of water} = \text{Volume} \times \text{Density} = 250 \, \text{cm}^3 \times 1 \, \text{g/cm}^3 = 250 \, \text{g} \]

### Step 3: Apply the Principle of Buoyancy
When the beaker is half full, the buoyant force acting on it must equal the total weight of the beaker plus the weight of the water inside it.

The buoyant force (F_b) can be calculated as:
\[ F_b = \text{Density of water} \times \text{Volume of beaker} \times g \]
Where:
- Density of water = 1000 g/m³ (or 1 g/cm³)
- Volume of beaker = 500 cm³
- g = acceleration due to gravity (approximately 10 m/s²)

So,
\[ F_b = 1000 \, \text{g/m}^3 \times 500 \, \text{cm}^3 \times 10 \, \text{m/s}^2 \]

### Step 4: Calculate the Total Weight
The total weight (W) of the beaker and the water when the beaker is half full is:
\[ W = \text{Mass of beaker} + \text{Mass of water} \]
\[ W = 390 \, \text{g} + 250 \, \text{g} = 640 \, \text{g} \]

### Step 5: Set Up the Equation
At equilibrium (when the beaker floats), the buoyant force equals the total weight:
\[ F_b = W \]
\[ 1000 \, \text{g/m}^3 \times 500 \, \text{cm}^3 \times 10 \, \text{m/s}^2 = 640 \, \text{g} \]

### Step 6: Calculate the Volume of the Beaker
Since the beaker is hollow, we need to find the volume of the material of the beaker. The total volume of the beaker is 500 cm³, and we need to find the volume of the material.

The volume of the beaker (V_total) when it is fully submerged is:
\[ V_{\text{total}} = \frac{640 \, \text{g}}{1000 \, \text{g/m}^3} = 640 \, \text{cm}^3 \]

### Step 7: Calculate the Volume of the Material of the Beaker
The volume of the material of the beaker (V_material) is:
\[ V_{\text{material}} = V_{\text{total}} - V_{\text{interior}} = 640 \, \text{cm}^3 - 500 \, \text{cm}^3 = 140 \, \text{cm}^3 \]

### Step 8: Calculate the Density of the Material of the Beaker
Now we can find the density (ρ) of the material of the beaker:
\[ \rho = \frac{\text{Mass of beaker}}{\text{Volume of material}} \]
\[ \rho = \frac{390 \, \text{g}}{140 \, \text{cm}^3} \approx 2.79 \, \text{g/cm}^3 \]

### Final Answer
The density of the material of which the beaker is made is approximately **2.79 g/cm³**.

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