Project: Hypergravity Habitat
Document type: physics and notation reference
Status: working reference
Scope: effective gravity, circular motion, bank angle, angular rate, Coriolis effects, projectile deflection, gravity gradients, and scaling relationships
This document collects the core physics used throughout the Hypergravity Habitat repository. It is intended to make assumptions transparent and prevent inconsistent calculations across documents.
The reference is first-order. It is suitable for concept screening and requirements discussion, not for detailed engineering, safety certification, or human-subject protocol approval.
Reader-facing documents should avoid raw equation markup. Formulas should be written in readable plain text or Unicode notation and explained immediately.
Example:
g_eff = √(g² + a_c²)
Meaning: the resultant effective gravity is the vector combination of Earth gravity and generated centripetal acceleration.
| Symbol | Meaning | Unit |
|---|---|---|
| g | Earth-normal gravitational acceleration | m/s² |
| g_eff | resultant effective gravity | m/s² |
| g_rel | resultant effective gravity relative to 1 g | dimensionless |
| a_c | centripetal acceleration | m/s² |
| v | tangential speed | m/s |
| r | radius | m |
| ω or Ω | angular velocity | rad/s |
| T | time or flight time, depending on context | s |
| L | distance or circumference, depending on context | m |
| θ | bank angle or resultant-vector angle | degrees or radians |
| y | lateral projectile deflection | m |
Use ω for platform angular velocity in general calculations and Ω when emphasizing rotating-frame effects such as Coriolis acceleration.
Centripetal acceleration:
a_c = v² / r = ω² r
Tangential speed:
v = √(a_c r)
Angular velocity:
ω = v / r = √(a_c / r)
Rotations per minute:
rpm = (ω / 2π) × 60
Circumference:
L = 2πr
Lap time:
T_lap = 2πr / v
For a terrestrial circular platform, Earth gravity acts vertically and centripetal acceleration acts horizontally. The resultant effective gravity magnitude is:
g_eff = √(g² + a_c²)
If the target resultant gravity is expressed as a multiple of Earth gravity:
g_eff = g_rel × g
then the required lateral acceleration is:
a_c = g × √(g_rel² − 1)
This equation is central. A target of 1.10 g resultant effective gravity does not require only 0.10 g lateral acceleration. It requires approximately 0.458 g lateral acceleration.
The resultant load vector is tilted relative to vertical by:
θ = arctan(a_c / g)
If a platform floor is aligned with the resultant vector, this is the approximate required floor or vehicle bank angle.
| Target resultant gravity | Lateral acceleration | Bank angle |
|---|---|---|
| 1.05 g | 0.320 g | 17.8° |
| 1.10 g | 0.458 g | 24.6° |
| 1.20 g | 0.663 g | 33.6° |
| 1.25 g | 0.750 g | 36.9° |
| 1.50 g | 1.118 g | 48.2° |
For fixed target resultant gravity, a_c is fixed. Therefore:
v is proportional to √r
ω is proportional to 1 / √r
Increasing radius reduces angular rate and Coriolis effects, but increases circumference, land use, guideway length, and likely cost. It also increases required speed for the same lateral acceleration.
In rotating systems, effective acceleration can vary with radius:
a_c(r) = ω²r
For small radial height or payload dimension Δr:
Δa_c ≈ ω² × Δr
Gravity gradients matter for:
Large radii reduce relative gradients across a given payload size.
In a rotating frame, a moving object experiences Coriolis acceleration.
Vector form:
a_cor_vector = −2 Ω × v_rel
Magnitude:
a_cor = 2 Ω v_rel sin(φ)
where φ is the angle between the rotation vector and the relative velocity vector.
For worst-case screening:
a_cor ≈ 2 Ω v_rel
Coriolis effects are relevant for:
For a projectile moving over distance L with speed v, approximate flight time is:
T ≈ L / v
First-order lateral deflection:
y ≈ Ω v T²
or:
y ≈ Ω × L² / v
This is a simplified screening model. It does not include drag, spin, Magnus effect, launch angle, target height, release height, curved cabin geometry, direction relative to platform rotation, or athlete compensation.
Use calculations/coriolis_projectile_deflection.py for first-order estimates.
For a platform moving at speed v and decelerating at a_b:
s = v² / (2a_b)
t = v / a_b
This matters for emergency stop, safe evacuation, passenger comfort, payload integrity, braking heat, and energy recovery.
Vehicle or rotating-platform kinetic energy:
E_k = 0.5 × m × v²
Large moving systems store substantial kinetic energy. This affects emergency braking, containment, collision risk, power recovery, and safety-case requirements.
First-order propulsion power depends on losses such as rolling resistance, aerodynamic drag, bearing losses, guideway losses, levitation or control power, HVAC, and laboratory loads.
Aerodynamic drag force:
F_d = 0.5 × ρ × C_d × A × v²
Drag power:
P_d = F_d × v = 0.5 × ρ × C_d × A × v³
This shows why high speeds can strongly affect operating cost.
Current scripts:
calculations/hypergravity_sizing.pycalculations/coriolis_projectile_deflection.pycalculations/railway_g_envelope.pyThese scripts should be treated as concept tools.
The basic physics of a hypergravity platform are simple, but their design consequences are severe. Small increases in resultant effective gravity can imply substantial lateral acceleration, banking, speed, angular-rate, and safety constraints.
All future documents should link back to this reference or to reproducible calculation scripts when presenting numerical claims.
Project: Hypergravity Habitat · Status: exploratory research documentation · License: see repository license and file-level notes