Three spherical balls of masses 1kg, 2kg and 3kg are placed at the corners of an equilateral triangle of side 1m. Find the magnitude of the gravitational force experted by the 2kg and 3kg masses on the 1kg mass.

Options

A. 19 G
B. `sqrt17G`
C. `sqrt19G`
D. `G/sqrt17`

Answer

To find the magnitude of the gravitational force exerted by the 2 kg and 3 kg masses on the 1 kg mass, we can follow these steps:

### Step 1: Identify the forces acting on the 1 kg mass
The gravitational force between two masses is given by Newton's law of gravitation:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)),
- \( m_1 \) and \( m_2 \) are the masses,
- \( r \) is the distance between the centers of the two masses.

In this case, we have:
- Mass \( m_1 = 1 \, \text{kg} \) (the mass at the corner of the triangle),
- Mass \( m_2 = 2 \, \text{kg} \) (the second mass),
- Mass \( m_3 = 3 \, \text{kg} \) (the third mass),
- The distance \( r = 1 \, \text{m} \) (the side of the triangle).

### Step 2: Calculate the gravitational force between the 1 kg and 2 kg masses
Using the formula:
\[ F_{12} = \frac{G \cdot m_1 \cdot m_2}{r^2} = \frac{G \cdot 1 \cdot 2}{1^2} = 2G \]

### Step 3: Calculate the gravitational force between the 1 kg and 3 kg masses
Similarly, for the 3 kg mass:
\[ F_{13} = \frac{G \cdot m_1 \cdot m_3}{r^2} = \frac{G \cdot 1 \cdot 3}{1^2} = 3G \]

### Step 4: Determine the direction of the forces
Since the masses are at the corners of an equilateral triangle, the angle between the forces \( F_{12} \) and \( F_{13} \) is \( 60^\circ \).

### Step 5: Calculate the resultant gravitational force
The resultant force \( F_R \) can be calculated using the formula for the resultant of two vectors:
\[ F_R = \sqrt{F_{12}^2 + F_{13}^2 + 2 \cdot F_{12} \cdot F_{13} \cdot \cos(60^\circ)} \]

Substituting the values:
- \( F_{12} = 2G \)
- \( F_{13} = 3G \)
- \( \cos(60^\circ) = \frac{1}{2} \)

So,
\[ F_R = \sqrt{(2G)^2 + (3G)^2 + 2 \cdot (2G) \cdot (3G) \cdot \frac{1}{2}} \]
\[ = \sqrt{4G^2 + 9G^2 + 6G^2} \]
\[ = \sqrt{19G^2} \]
\[ = G\sqrt{19} \]

### Final Result
The magnitude of the gravitational force exerted by the 2 kg and 3 kg masses on the 1 kg mass is:
\[ F_R = G\sqrt{19} \]