Tidal Force Calculator
Calculate the tidal force and tidal acceleration exerted by a massive body across an extended object.
Understand tides, Roche limits, and spaghettification.
Tidal forces arise from the difference in gravitational pull across an extended object. The near side of an object feels a stronger pull than the far side — this difference is the tidal force.
Tidal force formula:
F_tidal = 2GMmR / d³
Tidal acceleration (per unit mass):
a_tidal = 2GM × R / d³
Where:
- M = mass of the source body (the tidal body, e.g., Moon)
- m = mass of the test object (e.g., ocean water)
- R = size (radius) of the test object
- d = distance between centers
Earth’s tides:
The Moon (M ≈ 7.34 × 10²² kg) at 384,400 km creates tidal accelerations on Earth (R = 6,371 km):
a_tidal ≈ 2 × 6.674×10⁻¹¹ × 7.34×10²² × 6.371×10⁶ / (3.844×10⁸)³ ≈ 1.13 × 10⁻⁶ m/s²
This is only about 10⁻⁷ g — tiny, yet it raises ocean tides by about 1 meter.
Spaghettification: Near a stellar-mass black hole, the tidal force on a human body becomes enormous. An astronaut falling into a 10 M☉ black hole would experience tidal stretching (spaghettification) well before crossing the event horizon. For a supermassive black hole (billions of M☉), spaghettification happens inside the horizon.
Roche limit connection: When tidal acceleration a_tidal ≈ surface gravity of the satellite (GM_sat/R²), the satellite is disrupted. This gives the Roche limit formula.
Tidal heating: Jupiter’s moon Io is the most volcanically active body in the Solar System due to tidal heating from Jupiter and gravitational interactions with Europa and Ganymede.