Know that the strength of the gravitational field (a) at the surface of a planet depends on the mass of the planet (b) around a planet decreases as the distance from the planet increases

6.1.2 The Solar System 🚀

Objective

Know that the strength of the gravitational field (a) at the surface of a planet depends on the planet’s mass, and (b) the field around a planet decreases as the distance from the planet increases.

1️⃣ Gravitational Field at a Planet’s Surface

Think of gravity like a giant invisible hand that pulls everything toward the planet’s centre. The closer you are to the centre (i.e., the surface), the stronger the pull.

The formula for the surface gravitational field is:

\$g = \dfrac{G\,M}{r^2}\$

  • \$G\$ is the gravitational constant (\$6.67\times10^{-11}\,\text{N m}^2\text{/kg}^2\$).

  • \$M\$ is the planet’s mass.

  • \$r\$ is the planet’s radius (distance from centre to surface).

Because \$g\$ is directly proportional to \$M\$ and inversely proportional to \$r^2\$, a heavier planet or a smaller radius gives a stronger surface gravity.

PlanetMass (\$M_\oplus\$)Radius (km)Surface \$g\$ (m/s²)
Earth 🌍1.006,3719.81
Moon 🌑0.01231,7371.62
Mars 🪐0.1073,3903.71
Jupiter 🪐317.869,91124.79

Notice how Jupiter’s huge mass gives it a much stronger surface gravity, even though its radius is also large.

2️⃣ Gravitational Field Decreases with Distance

Imagine standing near a magnet. The closer you are, the stronger the pull. Move away, and the pull weakens. The same happens with gravity.

The same formula applies, but now \$r\$ is the distance from the planet’s centre to the point where you’re measuring \$g\$:

\$g = \dfrac{G\,M}{r^2}\$

  1. At the surface, \$r = R\$ (planet’s radius).

  2. At a point 2 × \$R\$ away, \$r = 2R\$ → \$g\$ becomes \$\dfrac{1}{4}\$ of the surface value.

  3. At 3 × \$R\$ away, \$g\$ becomes \$\dfrac{1}{9}\$ of the surface value.

So, the gravitational field falls off with the square of the distance.

Exam Tip 📚

  • Remember: \$g = \dfrac{G\,M}{r^2}\$ – always use the distance from the centre.

  • If you’re given mass and radius, plug them straight into the formula.

  • For distance variation, use the “inverse square” rule: doubling the distance reduces \$g\$ to one‑quarter.

  • Check units: \$G\$ is in N m²/kg², so \$M\$ in kg and \$r\$ in metres give \$g\$ in m/s².

3️⃣ Quick Practice Questions

  • What is the surface gravity of a planet that has half the mass of Earth but the same radius? (Hint: \$g\$ is halved.)

  • If you are 10 000 km above the Earth’s surface, how does the gravitational field compare to that at the surface? (Use the inverse square law.)

  • Which planet would feel the strongest pull on a 70 kg astronaut: Mars or Jupiter? (Consider both mass and radius.)

Use these questions to test your understanding before the exam. Good luck! 🌟