A `3kg` mass and `4kg` mass are placed on `X` and `Y` axes at a distance of `1` meter from the origin and a `1kg` mass is placed at the origin. Then the resultant gravitational force on `1kg` mass is

Options

A. `7G`
B. `G`
C. `5G`
D. `3G`

Explanation

To find the resultant gravitational force on the `1 kg` mass placed at the origin due to the `3 kg` mass on the `X` axis and the `4 kg` mass on the `Y` axis, we can follow these steps:

### Step 1: Identify the positions of the masses
- The `3 kg` mass is located at `(1, 0)` on the `X` axis.
- The `4 kg` mass is located at `(0, 1)` on the `Y` axis.
- The `1 kg` mass is located at the origin `(0, 0)`.

### Step 2: Calculate the gravitational force due to the `3 kg` mass
The gravitational force \( F_1 \) exerted by the `3 kg` mass on the `1 kg` mass can be calculated using the formula for gravitational force:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
Where:
- \( G \) is the gravitational constant,
- \( m_1 = 3 \, \text{kg} \),
- \( m_2 = 1 \, \text{kg} \),
- \( r = 1 \, \text{m} \) (the distance between the masses).

Substituting the values, we get:
\[ F_1 = \frac{G \cdot 3 \cdot 1}{1^2} = 3G \]

### Step 3: Calculate the gravitational force due to the `4 kg` mass
Similarly, the gravitational force \( F_2 \) exerted by the `4 kg` mass on the `1 kg` mass is:
\[ F_2 = \frac{G \cdot 4 \cdot 1}{1^2} = 4G \]

### Step 4: Determine the direction of the forces
- The force \( F_1 \) due to the `3 kg` mass acts along the negative `X` direction (towards the `3 kg` mass).
- The force \( F_2 \) due to the `4 kg` mass acts along the negative `Y` direction (towards the `4 kg` mass).

### Step 5: Calculate the resultant gravitational force
Since \( F_1 \) and \( F_2 \) are perpendicular to each other (90 degrees apart), we can use the Pythagorean theorem to find the resultant force \( F_R \):
\[ F_R = \sqrt{F_1^2 + F_2^2} \]
Substituting the values:
\[ F_R = \sqrt{(3G)^2 + (4G)^2} = \sqrt{9G^2 + 16G^2} = \sqrt{25G^2} = 5G \]

### Conclusion
The resultant gravitational force on the `1 kg` mass at the origin is \( 5G \).