Gravitational Potential Energy of Square System
Example 8.3: Four masses at vertices of a square
m
m
m
m
Results:
System's gravitational potential energy: -5.41 Gm²/l
Gravitational potential at center: -4√2 Gm/l
Numerical values:
Total PE: -5.41 Gm²/l
Center potential: -5.66 Gm/l
Physics Explanation:
This simulation demonstrates the gravitational potential energy of four equal masses arranged at the corners of a square.
System's Gravitational Potential Energy:
For four masses (each mass = m) at the corners of a square (side length = l):
- 4 pairs at distance l (sides of the square)
- 2 pairs at distance √2 l (diagonals of the square)
\[ W(r) = -4 \frac{G m^2}{l} - 2 \frac{G m^2}{\sqrt{2} l} \]
\[ W(r) = -\frac{2 G m^2}{l} \left(2 + \frac{1}{\sqrt{2}}\right) \approx -5.41 \frac{G m^2}{l} \]
Gravitational Potential at Center:
The distance from center to any corner is \( r = \frac{\sqrt{2}}{2} l \):
\[ U(r) = -4 \frac{G m}{r} = -4 \sqrt{2} \frac{G m}{l} \approx -5.66 \frac{G m}{l} \]
