There are two bodies of masses `100 kg` and `1000 kg` separated by a distance `1m`. The intensity of gravitational field at the mid point of the line joining them will be

Options

A. `2.4xx10^(-6)N//kg`
B. `2.4xx10^(-7)N//kg`
C. `2.4xx10^(-8) N//kg`
D. `2.4xx10^(-9) N//kg`

Explanation

To find the intensity of the gravitational field at the midpoint between two masses, we can follow these steps:

### Step 1: Understand the Problem
We have two masses, \( m_1 = 100 \, \text{kg} \) and \( m_2 = 1000 \, \text{kg} \), separated by a distance of \( d = 1 \, \text{m} \). We need to find the gravitational field intensity at the midpoint of the line connecting these two masses.

### Step 2: Identify the Midpoint
The midpoint between the two masses is located at a distance of \( \frac{d}{2} = \frac{1}{2} \, \text{m} \) from each mass.

### Step 3: Calculate the Gravitational Field Due to Each Mass
The gravitational field \( E \) due to a mass \( m \) at a distance \( r \) is given by the formula:
\[ E = \frac{G \cdot m}{r^2} \]
where \( G \) is the gravitational constant, \( G \approx 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \).

#### For Mass \( m_1 \):
- Distance from \( m_1 \) to the midpoint \( r_1 = 0.5 \, \text{m} \)
\[ E_1 = \frac{G \cdot m_1}{r_1^2} = \frac{6.674 \times 10^{-11} \cdot 100}{(0.5)^2} = \frac{6.674 \times 10^{-11} \cdot 100}{0.25} = 2.6696 \times 10^{-9} \, \text{N/kg} \]

#### For Mass \( m_2 \):
- Distance from \( m_2 \) to the midpoint \( r_2 = 0.5 \, \text{m} \)
\[ E_2 = \frac{G \cdot m_2}{r_2^2} = \frac{6.674 \times 10^{-11} \cdot 1000}{(0.5)^2} = \frac{6.674 \times 10^{-11} \cdot 1000}{0.25} = 2.6696 \times 10^{-8} \, \text{N/kg} \]

### Step 4: Determine the Direction of the Gravitational Fields
- The gravitational field \( E_1 \) due to \( m_1 \) points towards \( m_1 \).
- The gravitational field \( E_2 \) due to \( m_2 \) points towards \( m_2 \).

Since both fields are directed towards their respective masses, they will add up at the midpoint.

### Step 5: Calculate the Total Gravitational Field Intensity
The total gravitational field intensity \( E \) at the midpoint is the sum of the magnitudes of \( E_1 \) and \( E_2 \):
\[ E = E_1 + E_2 = 2.6696 \times 10^{-9} + 2.6696 \times 10^{-8} = 2.93656 \times 10^{-8} \, \text{N/kg} \]

### Final Answer
The intensity of the gravitational field at the midpoint is approximately:
\[ E \approx 2.94 \times 10^{-8} \, \text{N/kg} \]