A metal ball weighs `0.096N`.When suspended in water it has an apparent weight of `0.071N`. Find the density of the metal.

Explanation

To find the density of the metal ball, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- Weight of the metal ball (W) = 0.096 N
- Apparent weight when submerged in water = 0.071 N

### Step 2: Calculate the Buoyant Force
The buoyant force (Fb) can be calculated using the formula:
\[ F_b = W - \text{Apparent Weight} \]
Substituting the values:
\[ F_b = 0.096 N - 0.071 N = 0.025 N \]

### Step 3: Relate Weight and Buoyant Force to Density
The weight of the metal ball can be expressed as:
\[ W = \rho_m \cdot V \cdot g \]
Where:
- \( \rho_m \) = density of the metal
- \( V \) = volume of the metal ball
- \( g \) = acceleration due to gravity

The buoyant force can be expressed as:
\[ F_b = \rho_w \cdot V \cdot g \]
Where:
- \( \rho_w \) = density of water (approximately \( 1000 \, \text{kg/m}^3 \))

### Step 4: Set Up the Ratio of Densities
From the above equations, we can set up the ratio:
\[ \frac{W}{F_b} = \frac{\rho_m \cdot V \cdot g}{\rho_w \cdot V \cdot g} \]
Since \( V \) and \( g \) are common in both terms, they can be cancelled out:
\[ \frac{W}{F_b} = \frac{\rho_m}{\rho_w} \]

### Step 5: Substitute the Values
Now substituting the values of \( W \) and \( F_b \):
\[ \frac{0.096 N}{0.025 N} = \frac{\rho_m}{1000 \, \text{kg/m}^3} \]
Calculating the left side:
\[ \frac{0.096}{0.025} = 3.84 \]

### Step 6: Solve for the Density of the Metal
Now we can find \( \rho_m \):
\[ \rho_m = 3.84 \cdot 1000 \, \text{kg/m}^3 = 3840 \, \text{kg/m}^3 \]

### Final Answer
The density of the metal is \( 3840 \, \text{kg/m}^3 \).

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