A raft of wood of density `0.8 xx 10^3` kg `m^(-3)` and mass 120 kg floats in water. How much weight should be put on the raft to make it just sink?

Options

A. 20 kg
B. 30 kg
C. 40 kg
D. 80 kg

Answer

To solve the problem of how much weight should be put on a raft to make it just sink, we can follow these steps:

### Step 1: Calculate the volume of the raft
The volume of the raft can be calculated using the formula:

\[ V = \frac{m}{\rho} \]

where:
- \( m \) is the mass of the raft (120 kg)
- \( \rho \) is the density of the raft (0.8 x \( 10^3 \) kg/m³ = 800 kg/m³)

Substituting the values:

\[ V = \frac{120 \text{ kg}}{800 \text{ kg/m}^3} = 0.15 \text{ m}^3 \]

### Step 2: Calculate the weight of the water displaced by the raft
The weight of the water displaced by the raft is equal to the buoyant force acting on it when it is floating. This can be calculated using the formula:

\[ \text{Weight of water displaced} = V \times \rho_{\text{water}} \times g \]

where:
- \( \rho_{\text{water}} \) is the density of water (1000 kg/m³)
- \( g \) is the acceleration due to gravity (approximately 9.81 m/s², but we can ignore it for weight comparison)

Thus, the weight of the water displaced is:

\[ \text{Weight of water displaced} = 0.15 \text{ m}^3 \times 1000 \text{ kg/m}^3 = 150 \text{ kg} \]

### Step 3: Determine the total weight when the raft is just about to sink
For the raft to just sink, the total weight (raft + additional weight) must equal the weight of the water displaced:

\[ \text{Total weight} = \text{Weight of raft} + \text{Additional weight} \]

Let \( M \) be the additional weight that needs to be added. Therefore:

\[ 120 \text{ kg} + M = 150 \text{ kg} \]

### Step 4: Solve for the additional weight \( M \)
Rearranging the equation gives:

\[ M = 150 \text{ kg} - 120 \text{ kg} = 30 \text{ kg} \]

### Conclusion
Thus, the weight that should be put on the raft to make it just sink is **30 kg**.

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