Two masses `90kg` and `160 kg` are `5m` apart. The gravitational field intensity at a point `3m` from `90kg` and `4m` from `160 kg` is

Options

A. `10G`
B. `5G`
C. `5sqrt(2)G`
D. `10sqrt(2)G`

Explanation

To find the gravitational field intensity at a point that is 3 m from a mass of 90 kg and 4 m from a mass of 160 kg, we can follow these steps:

### Step 1: Understand the Setup
We have two masses:
- Mass \( m_1 = 90 \, \text{kg} \)
- Mass \( m_2 = 160 \, \text{kg} \)

The distance between the two masses is \( 5 \, \text{m} \). We need to find the gravitational field intensity at a point that is \( 3 \, \text{m} \) from \( m_1 \) and \( 4 \, \text{m} \) from \( m_2 \).

### Step 2: Calculate the Gravitational Field Intensity from Each Mass
The gravitational field intensity \( E \) due to a mass \( m \) at a distance \( r \) is given by the formula:
\[ E = \frac{Gm}{r^2} \]
where \( G \) is the gravitational constant.

#### For mass \( m_1 \):
- Distance \( r_1 = 3 \, \text{m} \)
\[ E_1 = \frac{G \cdot 90}{3^2} = \frac{G \cdot 90}{9} = 10G \]

#### For mass \( m_2 \):
- Distance \( r_2 = 4 \, \text{m} \)
\[ E_2 = \frac{G \cdot 160}{4^2} = \frac{G \cdot 160}{16} = 10G \]

### Step 3: Determine the Direction of the Gravitational Fields
Both gravitational fields \( E_1 \) and \( E_2 \) will point towards their respective masses. Since the point of interest is between the two masses, we can consider the directions:
- \( E_1 \) points towards \( m_1 \) (90 kg).
- \( E_2 \) points towards \( m_2 \) (160 kg).

### Step 4: Calculate the Resultant Gravitational Field Intensity
Since both fields are acting perpendicular to each other (as per the given distances), we can use the Pythagorean theorem to find the resultant gravitational field intensity \( E \):
\[ E = \sqrt{E_1^2 + E_2^2} = \sqrt{(10G)^2 + (10G)^2} = \sqrt{100G^2 + 100G^2} = \sqrt{200G^2} = 10G\sqrt{2} \]

### Final Answer
The gravitational field intensity at the point is:
\[ E = 10G\sqrt{2} \]