An iron casting containing a number of cavities weight `6000N` in air and `4000N` in water. What is the volume of the cavities in the casting? Density of iron is `7.87g//cm^(3)`. Take `g = 9.8 m//s^(2)` and density of water `= 10^(3) kg//m^(3)`.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We are given:
- Weight of the casting in air (W) = 6000 N
- Weight of the casting in water (W_effective) = 4000 N
- Density of iron (ρ_iron) = 7.87 g/cm³ = 7870 kg/m³ (conversion from g/cm³ to kg/m³)
- Density of water (ρ_water) = 1000 kg/m³
- Acceleration due to gravity (g) = 9.8 m/s²

### Step 2: Calculate the Volume of the Casting
The volume of the casting can be calculated using the buoyant force principle. The effective weight of the casting in water is given by:

\[ W_{effective} = W - F_b \]

Where \( F_b \) is the buoyant force. The buoyant force can be expressed as:

\[ F_b = \rho_{water} \cdot V_{casting} \cdot g \]

Rearranging the equation gives us:

\[ V_{casting} = \frac{W - W_{effective}}{\rho_{water} \cdot g} \]

Substituting the values:

\[ V_{casting} = \frac{6000 \, N - 4000 \, N}{1000 \, kg/m³ \cdot 9.8 \, m/s²} \]

Calculating this:

\[ V_{casting} = \frac{2000 \, N}{1000 \, kg/m³ \cdot 9.8 \, m/s²} \]

\[ V_{casting} = \frac{2000}{9800} \]

\[ V_{casting} = 0.204 \, m³ \]

### Step 3: Calculate the Volume of Iron in the Casting
The volume of iron can be calculated using its weight and density:

\[ V_{iron} = \frac{W}{\rho_{iron} \cdot g} \]

Substituting the values:

\[ V_{iron} = \frac{6000 \, N}{7870 \, kg/m³ \cdot 9.8 \, m/s²} \]

Calculating this:

\[ V_{iron} = \frac{6000}{77146} \]

\[ V_{iron} = 0.078 \, m³ \]

### Step 4: Calculate the Volume of the Cavities
The volume of the cavities (V_cav) can be found by subtracting the volume of iron from the volume of the casting:

\[ V_{cav} = V_{casting} - V_{iron} \]

Substituting the values:

\[ V_{cav} = 0.204 \, m³ - 0.078 \, m³ \]

Calculating this:

\[ V_{cav} = 0.126 \, m³ \]

### Final Answer
The volume of the cavities in the casting is:

\[ \boxed{0.126 \, m³} \]