A body weighs 100 N in air. When immersed in water, its weight decreases by 5 N. Calculate the density of the body.

Answer

To solve the problem step by step, we need to determine the density of the body based on the information provided.

### Step 1: Understand the Given Information
- The weight of the body in air (W_air) = 100 N
- The decrease in weight when immersed in water (buoyant force, F_b) = 5 N

### Step 2: Calculate the Weight of the Body in Water
The weight of the body when immersed in water (W_water) can be calculated as:
\[ W_{water} = W_{air} - F_b \]
Substituting the values:
\[ W_{water} = 100 \, \text{N} - 5 \, \text{N} = 95 \, \text{N} \]

### Step 3: Calculate the Mass of the Body
Using the weight of the body in air, we can find the mass (m) of the body using the formula:
\[ W_{air} = m \cdot g \]
Where \( g \) (acceleration due to gravity) is approximately \( 10 \, \text{m/s}^2 \). Rearranging the formula gives:
\[ m = \frac{W_{air}}{g} \]
Substituting the values:
\[ m = \frac{100 \, \text{N}}{10 \, \text{m/s}^2} = 10 \, \text{kg} \]

### Step 4: Relate Buoyant Force to Volume and Density of Water
The buoyant force can be expressed as:
\[ F_b = V \cdot \rho_{water} \cdot g \]
Where:
- \( V \) = Volume of the body
- \( \rho_{water} \) = Density of water (approximately \( 1000 \, \text{kg/m}^3 \))

Since the buoyant force is given as 5 N, we can rearrange the equation to find the volume:
\[ V = \frac{F_b}{\rho_{water} \cdot g} \]
Substituting the values:
\[ V = \frac{5 \, \text{N}}{1000 \, \text{kg/m}^3 \cdot 10 \, \text{m/s}^2} = \frac{5}{10000} = 0.0005 \, \text{m}^3 \]

### Step 5: Calculate the Density of the Body
The density (\( \rho_{body} \)) of the body can be calculated using the formula:
\[ \rho_{body} = \frac{m}{V} \]
Substituting the values we found:
\[ \rho_{body} = \frac{10 \, \text{kg}}{0.0005 \, \text{m}^3} = 20000 \, \text{kg/m}^3 \]

### Final Answer
The density of the body is:
\[ \rho_{body} = 2 \times 10^4 \, \text{kg/m}^3 \]

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