A body of mass 100 kg and density 500 kg `m^(-3)` floats in water. The additional mass should be added to the body so that the body will shink is

Options

A. 80 kg
B. 100 kg
C. 150 kg
D. 200 kg

Explanation

To solve the problem of determining how much additional mass should be added to a body of mass 100 kg and density 500 kg/m³ so that it sinks in water, we can follow these steps:

### Step 1: Calculate the Volume of the Body
The volume \( V \) of the body can be calculated using the formula:
\[ V = \frac{m}{\rho} \]
where \( m \) is the mass of the body and \( \rho \) is the density of the body.

Given:
- Mass \( m = 100 \, \text{kg} \)
- Density \( \rho = 500 \, \text{kg/m}^3 \)

Substituting the values:
\[ V = \frac{100 \, \text{kg}}{500 \, \text{kg/m}^3} = 0.2 \, \text{m}^3 \]

### Step 2: Understand the Condition for Sinking
For the body to sink, the total weight of the body plus the additional mass \( m \) must be greater than the buoyant force acting on it. The buoyant force is equal to the weight of the water displaced by the submerged volume of the body.

### Step 3: Write the Equation for Sinking Condition
The weight of the body plus the additional mass is given by:
\[ \text{Weight of the system} = (m_s + m)g \]
where \( m_s \) is the mass of the body and \( g \) is the acceleration due to gravity.

The buoyant force \( F_b \) can be calculated as:
\[ F_b = \text{Density of water} \times \text{Volume displaced} \times g \]
Given that the density of water is \( 1000 \, \text{kg/m}^3 \) and the volume displaced is equal to the volume of the body when it is fully submerged:
\[ F_b = 1000 \, \text{kg/m}^3 \times 0.2 \, \text{m}^3 \times g = 200g \]

### Step 4: Set Up the Equation
For the body to sink, we need:
\[ (m_s + m)g > F_b \]
Substituting the values:
\[ (100 \, \text{kg} + m)g > 200g \]

### Step 5: Simplify the Equation
Dividing both sides by \( g \) (assuming \( g \neq 0 \)):
\[ 100 \, \text{kg} + m > 200 \, \text{kg} \]

### Step 6: Solve for the Additional Mass \( m \)
Rearranging the equation gives:
\[ m > 200 \, \text{kg} - 100 \, \text{kg} \]
\[ m > 100 \, \text{kg} \]

### Conclusion
Thus, the additional mass that should be added to the body for it to sink is:
\[ m = 100 \, \text{kg} \]

### Final Answer
The correct option is **100 kg**.

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