A metallic sphere weighs `210g` in air, 180 g in water and 120 g in an unknown liquid. Find the density of metal and of liquid.

Explanation

To solve the problem, we will use the principles of buoyancy and the concept of density. The weight of the sphere in different mediums gives us the information we need to find the density of the metal and the unknown liquid.

### Step 1: Calculate the Volume of the Sphere
First, we need to determine the volume of the metallic sphere using its weight in water.

1. **Weight in Air (W_air)** = 210 g
2. **Weight in Water (W_water)** = 180 g
3. **Loss of weight in water (Buoyant Force)** = W_air - W_water = 210 g - 180 g = 30 g

The buoyant force is equal to the weight of the water displaced by the sphere, which can be expressed as:
\[ \text{Buoyant Force} = \text{Volume of Sphere} \times \text{Density of Water} \times g \]

Assuming the density of water is approximately \( 1 \, \text{g/cm}^3 \) and \( g \) cancels out, we can find the volume:
\[ \text{Volume of Sphere} = \frac{\text{Buoyant Force}}{\text{Density of Water}} = \frac{30 \, \text{g}}{1 \, \text{g/cm}^3} = 30 \, \text{cm}^3 \]

### Step 2: Calculate the Density of the Metal
Now that we have the volume of the sphere, we can find the density of the metal.

The density of an object is given by the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]

Using the mass of the sphere in air:
\[ \text{Density of Metal} = \frac{210 \, \text{g}}{30 \, \text{cm}^3} = 7 \, \text{g/cm}^3 \]

### Step 3: Calculate the Density of the Unknown Liquid
Next, we will find the density of the unknown liquid using the weight of the sphere in that liquid.

1. **Weight in Unknown Liquid (W_liquid)** = 120 g
2. **Loss of weight in unknown liquid** = W_air - W_liquid = 210 g - 120 g = 90 g

Using the same principle of buoyancy:
\[ \text{Loss of weight in liquid} = \text{Volume of Sphere} \times \text{Density of Liquid} \]

So we can write:
\[ 90 \, \text{g} = 30 \, \text{cm}^3 \times \text{Density of Liquid} \]

Now, solving for the density of the liquid:
\[ \text{Density of Liquid} = \frac{90 \, \text{g}}{30 \, \text{cm}^3} = 3 \, \text{g/cm}^3 \]

### Final Results
- **Density of Metal** = \( 7 \, \text{g/cm}^3 \)
- **Density of Unknown Liquid** = \( 3 \, \text{g/cm}^3 \)