An irregular piece of metal weighs 10.00g in air and 8.00g when submerged in water
(a) Find the volume of the metal and its density.
(b) If the same piece of metal weighs 8.50g when immersed in a particular oil. What is the density of the oil?

Explanation

To solve the problem step by step, let's break it down into parts (a) and (b).

### Part (a): Finding the Volume and Density of the Metal

1. **Identify the weights**:
- Weight of the metal in air (W_air) = 10.00 g
- Weight of the metal when submerged in water (W_water) = 8.00 g

2. **Calculate the loss of weight in water**:
\[ \text{Loss of weight} = W_{\text{air}} - W_{\text{water}} = 10.00 \, \text{g} - 8.00 \, \text{g} = 2.00 \, \text{g} \]

3. **Use Archimedes' principle to find the volume of the metal**:
The loss of weight in water is equal to the weight of the water displaced, which can be expressed as:
\[ \text{Loss of weight} = \text{Volume} \times \text{Density of water} \times g \]
Since the density of water is approximately \(1 \, \text{g/cm}^3\) or \(1000 \, \text{kg/m}^3\), we can convert the loss of weight from grams to kilograms:
\[ 2.00 \, \text{g} = 0.002 \, \text{kg} \]
Therefore, the volume (V) of the metal can be calculated as:
\[ V = \frac{\text{Loss of weight}}{\text{Density of water} \times g} = \frac{0.002 \, \text{kg}}{1000 \, \text{kg/m}^3} = 2 \times 10^{-6} \, \text{m}^3 \]

4. **Calculate the density of the metal**:
The density (ρ) of the metal can be calculated using the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{10.00 \, \text{g}}{2 \times 10^{-6} \, \text{m}^3} \]
Converting mass to kg:
\[ \text{Density} = \frac{0.010 \, \text{kg}}{2 \times 10^{-6} \, \text{m}^3} = 5000 \, \text{kg/m}^3 \]

### Summary of Part (a):
- Volume of the metal = \(2 \times 10^{-6} \, \text{m}^3\)
- Density of the metal = \(5000 \, \text{kg/m}^3\)

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### Part (b): Finding the Density of the Oil

1. **Identify the weight of the metal in oil**:
- Weight of the metal in oil (W_oil) = 8.50 g

2. **Calculate the loss of weight in oil**:
\[ \text{Loss of weight in oil} = W_{\text{air}} - W_{\text{oil}} = 10.00 \, \text{g} - 8.50 \, \text{g} = 1.50 \, \text{g} \]

3. **Use the relationship between densities**:
The density of the oil can be found using the formula:
\[ \frac{\text{Loss of weight in oil}}{\text{Loss of weight in water}} = \frac{\text{Density of oil}}{\text{Density of water}} \]
Rearranging gives:
\[ \text{Density of oil} = \frac{\text{Loss of weight in oil}}{\text{Loss of weight in water}} \times \text{Density of water} \]
Substituting the known values:
\[ \text{Density of oil} = \frac{1.50 \, \text{g}}{2.00 \, \text{g}} \times 1000 \, \text{kg/m}^3 = 0.75 \times 1000 \, \text{kg/m}^3 = 750 \, \text{kg/m}^3 \]

### Summary of Part (b):
- Density of the oil = \(750 \, \text{kg/m}^3\)

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