Gravitational Forces at Triangle Centroid
Example 8.2: Three masses at vertices of an equilateral triangle
m
m
m
2m
Results:
Net force on centroid mass (2m): 0 N
Force components:
x-component: 0 N
y-component: 0 N
Physics Explanation:
This simulation demonstrates the gravitational forces acting on a mass placed at the centroid of an equilateral triangle with masses at its vertices.
Case (a): All masses equal (m)
The gravitational force from each vertex mass cancels out at the centroid due to symmetry. The net force is zero.
Case (b): Mass at vertex A doubled (2m)
When the mass at vertex A is doubled, the forces no longer cancel completely. The resultant force is:
\[ F_R = F_{GA} + F_{GB} + F_{GC} \]
Where:
- \( F_{GA} = \frac{G(2m)(2m)}{1^2} \hat{j} = 4Gm^2 \hat{j} \)
- \( F_{GB} = \frac{Gm(2m)}{1^2} (-\cos30^\circ \hat{i} - \sin30^\circ \hat{j}) \)
- \( F_{GC} = \frac{Gm(2m)}{1^2} (\cos30^\circ \hat{i} - \sin30^\circ \hat{j}) \)
The x-components from B and C cancel out, while the y-components add up to:
\[ F_R = 2Gm^2 \hat{j} \]
