Gravitational Intensity in Hemispherical Shell
Professional 3D Physics Visualization
Physics of Gravitational Intensity
Core Principle: The gravitational intensity at any point P inside a hemispherical shell always points downward (direction e), demonstrating why option (ii) is correct in the original problem.
Mathematical Derivation
The gravitational intensity E at point P is calculated by integrating contributions from all mass elements dm:
E = ∫ (G·dm/r²) r̂
Where G is the gravitational constant, r is the distance from dm to P, and r̂ is the unit vector pointing from dm to P.
Key Observations
- Complete Shell: For a full spherical shell, symmetry causes all components to cancel out, resulting in zero net intensity inside.
- Hemispherical Shell: The vertical symmetry is broken, resulting in a net downward force component.
- Position Independence: The downward direction of the intensity remains consistent regardless of point P's position within the hemisphere.
Simulation Features: Adjust the position of point P using the slider to observe how the gravitational intensity (shown by the black arrow e) always maintains its downward direction, while reference arrows (d, f, g) demonstrate other possible directions.
