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Tidal Force Calculator

Find the tidal acceleration from a massive body.

How it is calculated

The tool uses the first-order tidal approximation a = 2 · G · M · r ÷ d³. Here G = 6.674×10−11 N·m²/kg², M is the mass of the body causing the tide in kilograms, r is the radius of the affected object in metres, and d is the distance between the centres of the two bodies in metres. The result is the difference in gravitational pull between the near side and the centre of the object, in m/s², printed in scientific notation with five significant digits. All three inputs must be above zero. The approximation holds when r is much smaller than d; for close or very large objects it is only an estimate.

Worked example

Moon's pull on Earth: M = 7.342e22 kg, r = 6.371e6 m, d = 3.844e8 m. Compute 2 × 6.674e-11 × 7.342e22 = 9.80e12, and multiply by r to get about 6.244e19. The distance cubed is (3.844e8)³ = 5.680e25. Dividing gives a tidal acceleration of 1.0992 × 10⁻⁶ m/s². This is roughly a ten-millionth of the surface gravity of 9.8 m/s².

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Common mistakes

  • Entering distances in kilometres or masses in tonnes. Every input must be in SI units, metres and kilograms, or the answer is off by large factors.
  • Using the altitude above the surface as d. It has to be the centre-to-centre distance.
  • Putting the diameter in the radius field, which doubles the answer.
  • Forgetting the cube. Halving the distance multiplies the tide by eight, not by two, which is why a nearby small body can beat a distant large one.

Sources

About this calculator

This calculator estimates the differential tidal acceleration, a ≈ 2·G·M·r/d³, the stretching a massive body M causes across an extended object of radius r at distance d. It arises because gravity is stronger on the near side of an object than the far side; the formula captures that difference to first order, and the d³ term shows why tidal effects fall off much faster with distance than ordinary gravity (which falls as 1/d²).

Astrophysicists use this relationship to explain ocean tides raised by the Moon and Sun, to predict tidal locking (why the Moon always shows the same face to Earth), and to find the Roche limit — the distance inside which a body is torn apart by its host's tidal pull, relevant to planetary rings and comets passing close to planets. It's also the physics behind the predicted "spaghettification" of objects falling toward a black hole, where d becomes small enough for the stretching to become extreme.

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Comments (1)

  • Editorial team

    A neat use is comparing two bodies on the same object. Run the Moon (7.342e22 kg at 3.844e8 m) and then the Sun (1.989e30 kg at 1.496e11 m) with Earth's radius of 6.371e6 m. The tool gives about 1.0992 × 10⁻⁶ and 5.0520 × 10⁻⁷ m/s², so the Sun's tide is under half the Moon's even though it is vastly heavier. Which pair of bodies would you like to try, and what did you find?

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