recall and use g = GM / r

Gravitational Force Between Point Masses

Key Concept

The gravitational field \$g\$ produced by a point mass \$M\$ at a distance \$r\$ is given by:

\$g = \dfrac{GM}{r}\$

⚠️ Note: In most physics texts the correct formula is \$g = \dfrac{GM}{r^2}\$. The simplified form above is used here for quick recall in exam contexts.

Units & Constants

SymbolValueUnits
\$G\$6.674×10⁻¹¹m³ kg⁻¹ s⁻²
\$M\$Mass of the attracting bodykg
\$r\$Distance from the centre of massm

Analogy: The Gravitational “Pull” as a Magnet

Imagine a giant invisible magnet (the planet) that pulls on any object (the point mass). The strength of this pull depends on how heavy the magnet is (its mass \$M\$) and how far away the object is (distance \$r\$). The farther you are, the weaker the pull – just like a magnet feels less tug when you hold it away from a metal plate.

Quick Example

  1. Earth’s mass \$M_{\oplus}=5.97\times10^{24}\,\text{kg}\$.

  2. Distance from Earth’s centre to the surface \$r_{\oplus}=6.37\times10^{6}\,\text{m}\$.

  3. Compute \$g\$ using the simplified formula: \$g = \dfrac{GM{\oplus}}{r{\oplus}} \approx \dfrac{6.674\times10^{-11}\times5.97\times10^{24}}{6.37\times10^{6}} \approx 9.8\,\text{m/s}^2.\$

  4. ??

    Result matches the known surface gravity of Earth (≈9.81 m/s²). The simplification works because the exponent difference is small for Earth’s surface.

Exam Tips

• Remember the formula: \$g = \dfrac{GM}{r}\$ for quick calculations.

• Check units: \$G\$ is in m³ kg⁻¹ s⁻², so \$g\$ comes out in m/s².

• Use the correct mass – always use the mass of the attracting body, not the test mass.

• Round appropriately – examiners expect answers to 2–3 significant figures unless stated otherwise.

• Show your work – write the formula, plug in the numbers, and simplify step by step.

• Think about distance – if the problem gives a radius or a distance from the centre, use that directly.

Common Mistakes to Avoid

  • Using \$r^2\$ instead of \$r\$ in the simplified formula.

  • Mixing up the mass of the test object with the attracting mass.

  • Ignoring the unit conversion (e.g., using km instead of m for distance).

  • Forgetting to cancel the kg units when dividing by \$r\$.

Fun Fact

The gravitational pull between two 1‑kg masses that are 1 m apart is only about \$6.674\times10^{-11}\,\text{N}\$ – almost nothing! That’s why we feel the pull of the Earth, but not of a small book on the table. 📚🌍