Tidal Force Calculator

Tidal forces arise from the difference in gravitational pull across an extended object. The near side feels a stronger pull than the far side, and that 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 ≈ 1.10 × 10⁻⁶ m/s²

This is only about 10⁻⁷ g, tiny, and yet it raises ocean tides by about 1 meter. The reason so small a number moves so much water is that it acts over the whole ocean at once and keeps acting, twice a day, for as long as the Moon is there.

What to type in, if you just want to see it work:

The first two are worth running back to back. The Sun is 27 million times more massive than the Moon but 390 times further away, and since the tide falls off as 1/d³ the Moon wins by roughly two to one. That ratio is why spring tides (both pulling together) are noticeably higher than neap tides, rather than the Sun simply dominating.

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: Set the tidal acceleration equal to the satellite’s own surface gravity, GM_sat/R², and the algebra collapses to d = 2^(1/3) × R_primary × (ρ_primary / ρ_satellite)^(1/3). That cube root of two is 1.2599, which is exactly the 1.26 coefficient of the rigid Roche limit. The fluid coefficient of 2.455 is a different and harder calculation, because a fluid body stretches under the tide and its own deformation then makes the tide worse.

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.