An ornament weighing `50g` in air weighs only `46 g` is water. Assuming that some copper is mixed with gold the prepare the ornament. Find the amount of copper in it. Specific gravity of gold is `20` and that of copper is 10.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given data
- Weight of the ornament in air: \( W_{\text{air}} = 50 \, \text{g} \)
- Weight of the ornament in water: \( W_{\text{water}} = 46 \, \text{g} \)
- Specific gravity of gold: \( SG_{\text{gold}} = 20 \)
- Specific gravity of copper: \( SG_{\text{copper}} = 10 \)

### Step 2: Calculate the loss of weight in water
The loss of weight when the ornament is submerged in water is given by:
\[ \text{Loss of weight} = W_{\text{air}} - W_{\text{water}} = 50 \, \text{g} - 46 \, \text{g} = 4 \, \text{g} \]

### Step 3: Relate the loss of weight to the buoyant force
According to Archimedes' principle, the loss of weight is equal to the buoyant force, which can be expressed as:
\[ \text{Loss of weight} = \text{Upthrust} = V_{\text{total}} \cdot \rho_{\text{water}} \cdot g \]
Where:
- \( V_{\text{total}} \) is the total volume of the ornament.
- \( \rho_{\text{water}} \) is the density of water (approximately \( 1 \, \text{g/cm}^3 \)).
- \( g \) cancels out in the equation.

Thus, we have:
\[ 4 \, \text{g} = V_{\text{total}} \cdot 1 \, \text{g/cm}^3 \]
This implies:
\[ V_{\text{total}} = 4 \, \text{cm}^3 \]

### Step 4: Express the volumes of gold and copper
Let \( m \) be the mass of copper in grams. Then, the mass of gold will be \( 50 - m \) grams.

The volumes can be calculated using the formula:
\[ V = \frac{m}{\text{Specific Gravity}} \]
Thus, we have:
- Volume of copper:
\[ V_{\text{copper}} = \frac{m}{SG_{\text{copper}}} = \frac{m}{10} \]
- Volume of gold:
\[ V_{\text{gold}} = \frac{50 - m}{SG_{\text{gold}}} = \frac{50 - m}{20} \]

### Step 5: Set up the equation for total volume
The total volume of the ornament is the sum of the volumes of copper and gold:
\[ V_{\text{total}} = V_{\text{copper}} + V_{\text{gold}} \]
Substituting the expressions for volumes:
\[ 4 = \frac{m}{10} + \frac{50 - m}{20} \]

### Step 6: Solve the equation
To eliminate the fractions, multiply through by 20:
\[ 80 = 2m + (50 - m) \]
Simplifying gives:
\[ 80 = 2m + 50 - m \]
\[ 80 = m + 50 \]
\[ m = 80 - 50 = 30 \, \text{g} \]

### Step 7: Conclusion
The mass of copper in the ornament is \( 30 \, \text{g} \).