In the study of gravitation, particularly in two dimensions, understanding Newton's law of gravity is essential. This law states that the gravitational force \( F \) between two point masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by the formula:

\( F = \frac{G m_1 m_2}{r^2} \)

where \( G \) is the gravitational constant, approximately \( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \). In practical applications, such as calculating the net gravitational force on a mass in an equilateral triangle configuration, it is crucial to label the forces acting on the mass and calculate their magnitudes.

For instance, consider three 50 kg masses positioned at the vertices of an equilateral triangle. To find the net gravitational force on the bottom mass, we first identify the forces acting on it due to the other two masses. These forces can be labeled as \( F_{13} \) (force between the bottom mass and the top left mass) and \( F_{12} \) (force between the bottom mass and the top right mass). Using the law of gravity, we can calculate both forces, which will yield the same magnitude due to the symmetry of the problem:

\( F_{13} = F_{12} = 4.63 \times 10^{-7} \, \text{N} \)

Next, we decompose these forces into their x and y components. To do this, we need to establish a coordinate system and determine the angles involved. In an equilateral triangle, each angle measures 60 degrees. The components of the forces can be expressed as:

\( F_x = F \cos(\theta) \)

\( F_y = F \sin(\theta) \)

Given that both forces have the same angle with respect to the x-axis, the x-components will cancel each other out due to symmetry, leaving only the y-components contributing to the net force. Thus, the net force can be calculated as:

\( F_{\text{net}} = 2 F_y \)

Substituting the expression for \( F_y \), we find:

\( F_y = F \sin(60^\circ) \)

Calculating this gives:

\( F_y = 4.63 \times 10^{-7} \times \frac{\sqrt{3}}{2} \approx 4 \times 10^{-7} \, \text{N} \)

Thus, the net gravitational force acting on the bottom mass is:

\( F_{\text{net}} = 2 \times 4 \times 10^{-7} = 8 \times 10^{-7} \, \text{N} \)

In terms of direction, this net force points in the positive y-direction. Understanding these steps—labeling forces, calculating their magnitudes, decomposing them into components, and applying symmetry—provides a systematic approach to solving two-dimensional gravitational problems.