Tidal Force Calculator

Tidal forces arise from the gradient of gravity: the near side of an object feels stronger gravity than the far side. This stretches objects radially and squeezes them laterally.

🌍 Astrophysics📐 Fᵗ = 2GMmΔr/r³💧 Tides

Massive Body Mass M (kg)

Test mass m (kg)

Distance r (m)

Object size Δr (m)

Please enter valid values.

Formula & Reference

Variable	Symbol	Formula	Units
Tidal Force	F_t	2GMmΔr/r³	N
Tidal Acceleration	a_t	2GMΔr/r³	m/s²
Roche Limit	d	d = 2.46 R_M(M/m)^1/3	m
Earth-Moon Tidal Acc	—	~1.1 × 10⁻⁶ m/s² at Earth surface	m/s²

Step-by-Step Examples

Example 1

Moon Tidal Force on Earth

M(Moon)=7.342e22, m=1 kg, r=3.844e8 m, dr=6.371e6 m (Earth radius).

F_t = 2*6.674e-11*7.342e22*1*6.371e6/(3.844e8)^3
a_t = 1.12e-6 m/s^2

✓ Tidal acceleration ~1.1 um/s^2 — stretches Earth 0.5 m

Example 2

Sun vs Moon Tides

Sun: M=2e30, r=1.5e11; Moon: M=7.3e22, r=3.84e8. Compare a_t.

a_t(Sun) = 2*G*M_sun*R_Earth/r_sun^3 = 5.05e-7
a_t(Moon) = 1.12e-6
Ratio = 2.2: Moon tides are 2.2x stronger than Sun tides

✓ Moon tides dominate despite Sun being far more massive

Example 3

Roche Limit Example

For Moon-like satellite near Earth.

d_Roche = 2.46 * R_Earth * (M_Earth/M_moon)^(1/3)
d = 2.46 * 6.371e6 * (81.3)^(1/3) = 9.49e6 m = 9490 km

✓ Roche limit ~9490 km for Moon-density satellite

Real-World Applications

🌊

Ocean Tides

Moon and Sun gravity gradient stretches Earth, raising ocean tides on near and far sides.

⚪

Saturn Rings

Tidal forces inside Saturn Roche limit shredded a moon, creating the ring system.

⚫

Spaghettification

Near a black hole, tidal forces stretch infalling objects into thin streams.

🌋

Tidal Heating

Io is squeezed by Jupiter tidal forces, generating internal heat and volcanic activity.

Common Mistakes to Avoid

⚠️

Tidal force scales as 1/r^3

Much stronger dependence than gravity (1/r^2). Tidal forces fall off faster with distance.

⚠️

Moon tides stronger than Sun despite smaller mass

Tidal force ~ M/r^3. Moon (small mass, close) wins over Sun (large mass, far): a_t Moon/Sun = 2.2.

⚠️

Delta_r is object size, not distance

The tidal force stretches the object of size delta_r located at distance r from the tidal source.

Frequently Asked Questions

Why are there two high tides per day? ▾

The near side is pulled toward the Moon (tidal bulge). The far side is also pulled outward because the Moon pull is less than at Earth center. Two bulges, two tides as Earth rotates.

What is the Roche limit? ▾

Distance inside which tidal forces overcome a satellite self-gravity, ripping it apart. d = 2.46*R_M*(M_central/m_satellite)^(1/3) for fluid bodies.

How do tidal forces heat Io? ▾

Jupiter tidal forces flex Io as it orbits in an elliptical path (perturbed by other moons). Flexing dissipates energy as heat. Io has 100x more volcanic activity than Earth.

What is spaghettification? ▾

Near a stellar-mass black hole, tidal forces on a human body exceed 10^12 g. The body is stretched into a thin stream of particles.

How do binary stars tidally interact? ▾

Close binaries are tidally locked (same face always toward each other). Tidal dissipation circularizes orbits and synchronizes rotation over millions of years.

Related Physics Calculators

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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