3D Gravitational Neutral Point
Example 8.4: Projectile between two massive spheres
Results:
Neutral point location: 2R from M
Minimum launch speed: √(3GM/5R)
Numerical value: 0.0 m/s
Physics Explanation:
This simulation demonstrates the gravitational neutral point between two spheres of mass M and 4M separated by 6R.
Neutral Point Calculation:
The neutral point N is where gravitational forces balance:
\[ \frac{GMm}{r^2} = \frac{4GMm}{(6R-r)^2} \]
Solving gives: \( r = 2R \) (from M)
Minimum Launch Speed:
Using energy conservation between surface of M and neutral point:
Initial energy:
\[ E_i = \frac{1}{2}mv^2 - \frac{GMm}{R} - \frac{4GMm}{5R} \]
Energy at neutral point:
\[ E_N = -\frac{GMm}{2R} - \frac{4GMm}{4R} \]
Solving gives minimum speed:
\[ v = \sqrt{\frac{3GM}{5R}} \]
