A copper cube of mass `0.50kg` is weighed in water. The mass comes out to be `0.40 kg.` Is the cube hollower solid ? Given density of copper `=8.96 xx 10^(3)"kgm"^(-3)` and density of water `=10^(3) "kgm"^(-3).`

Answer

To determine whether the copper cube is hollow or solid, we can follow these steps:

### Step 1: Calculate the loss of weight in water
The weight of the copper cube in air is given as \(0.50 \, \text{kg}\) and its weight in water is \(0.40 \, \text{kg}\). The loss of weight when submerged in water can be calculated as:
\[ \text{Loss of weight} = \text{Weight in air} - \text{Weight in water} = 0.50 \, \text{kg} - 0.40 \, \text{kg} = 0.10 \, \text{kg} \]

### Step 2: Relate the loss of weight to the volume of water displaced
According to Archimedes' principle, the loss of weight in water is equal to the weight of the water displaced. The weight of the water displaced can be expressed as:
\[ \text{Weight of water displaced} = \text{Volume of water displaced} \times \text{Density of water} \times g \]
Where \(g\) is the acceleration due to gravity (which will cancel out later). The density of water is given as \(1000 \, \text{kg/m}^3\).

### Step 3: Set up the equation
From the loss of weight, we can set up the equation:
\[ 0.10 \, \text{kg} = V \times 1000 \, \text{kg/m}^3 \]
Where \(V\) is the volume of the cube in cubic meters.

### Step 4: Solve for the volume \(V\)
Rearranging the equation gives:
\[ V = \frac{0.10 \, \text{kg}}{1000 \, \text{kg/m}^3} = 0.0001 \, \text{m}^3 \]

### Step 5: Calculate the density of the cube
The density of the cube can be calculated using the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
Substituting the mass of the cube and the volume we found:
\[ \text{Density of cube} = \frac{0.50 \, \text{kg}}{0.0001 \, \text{m}^3} = 5000 \, \text{kg/m}^3 \]

### Step 6: Compare the density of the cube with the density of copper
The density of copper is given as \(8960 \, \text{kg/m}^3\). Now we compare:
\[ 5000 \, \text{kg/m}^3 < 8960 \, \text{kg/m}^3 \]
Since the density of the cube (5000 kg/m³) is less than the density of solid copper (8960 kg/m³), we conclude that the cube must be hollow.

### Conclusion
The copper cube is hollow.

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