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Sandbox Physics

M071 · Spherical gravity

Gravity Field & Shell Theorem

Edit three spherical density layers, move a radial field probe and release a particle through a diameter tunnel. Compare a hollow cavity with a uniform sphere, then weigh the source using acquired exterior acceleration.

Interactive modelGravity Field & Shell Theorem
Selected timeReady\text{Ready}
Probe radiusReady\text{Ready}
Probe inward accelerationReady\text{Ready}
Probe enclosed massReady\text{Ready}
Probe specific potentialReady\text{Ready}
Total source massReady\text{Ready}
Tunnel particle positionReady\text{Ready}
Tunnel particle speedReady\text{Ready}
Gravity clockReady\text{Ready}
Maximum specific-energy errorReady\text{Ready}
Endpoint refinement differenceReady\text{Ready}
Mass fitted from raw exterior scanReady\text{Ready}
Acquired field stationsReady\text{Ready}

Physics tutorial

Spherical source, field and tunnel

BackgroundA radial force is determined by the source and its geometry.

Why it mattersConnect the scene, the trajectory and a separately acquired sensor record.

Start with the essentials

Focus question
Where does the missing gravity go?
One-sentence intuition
Zero field does not imply zero potential.

Core mathematical model

Enclosed mass

M(<r)=4π∫0rρ(s)s2 dsM(<r)=4\pi\int_0^r\rho(s)s^2\,ds

The editable layers set the enclosed mass; exterior shells cancel in the force.

Field and potential

g(r)=GM(<r)r2Φ(r)=−G[M(<r)r+4π∫rRρ(s)s ds]\begin{aligned}g(r)&=\frac{GM(<r)}{r^2}\\\Phi(r)&=-G\left[\frac{M(<r)}r+4\pi\int_r^R\rho(s)s\,ds\right]\end{aligned}

Outer shells still contribute a constant potential inside their cavity. The central limit is evaluated without dividing by zero.

Uniform sphere benchmark

g(r)=4πGρ3r(r≤R)x(t)=x0cos⁡(4πGρ3 t)\begin{aligned}g(r)&=\frac{4\pi G\rho}{3}r\quad(r\le R)\\x(t)&=x_0\cos\left(\sqrt{\frac{4\pi G\rho}{3}}\,t\right)\end{aligned}

A rest release inside a uniform sphere is harmonic; crossing the centre does not create a singular force.

Specific energy

ε=12x˙2+Φ(∣x∣)\varepsilon=\frac12\dot x^2+\Phi(|x|)

The tunnel follows a diameter and does not remove source mass in this idealization.

Exterior mass inference

gi=GM ri−2g_i=GM\,r_i^{-2}

Fit the slope through the physical zero using exterior raw accelerations and calibrated radii. Source mass is never supplied to the fit.

Common difficulties

Assumption boundary

Typical misconceptionAn ideal model automatically describes a real planet or relaxed star cluster.

Better mental modelThe source is prescribed, the particle has no back reaction, and omitted effects are stated.

Run the experiment

  1. 01

    Move the field probe

    Drag the blue handle from the centre to the exterior. Compare the solid field with the dashed exterior point-mass reference.

    What to observe: Inside the cavity the acceleration is zero while the potential remains negative.
  2. 02

    Change the source

    Try Uniform sphere, Dense core and Rest inside a hollow shell; edit all three densities.

    What to observe: Density boundaries affect the interior field, while all exterior points depend only on total mass.
  3. 03

    Release through the cavity

    Compare the stationary cavity release with Cross the empty cavity. Scrub the gold time handle.

    What to observe: A particle entering the zero-field cavity continues with its acquired velocity.
  4. 04

    Weigh from readings

    Acquire the radial scan, add noise and export the raw CSV.

    What to observe: The exterior fit and exact source mass are separate outputs.