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
| Symbol | Value | Units |
|---|---|---|
| $G$ | 6.674×10⁻¹¹ | m³ kg⁻¹ s⁻² |
| $M$ | Mass of the attracting body | kg |
| $r$ | Distance from the centre of mass | m |
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
- Earth’s mass $M_{\oplus}=5.97\times10^{24}\,\text{kg}$.
- Distance from Earth’s centre to the surface $r_{\oplus}=6.37\times10^{6}\,\text{m}$.
- Compute $g$ using the simplified formula: $$g = \dfrac{GM_{\oplus}}{r_{\oplus}} \approx \dfrac{6.674\times10^{-11}\times5.97\times10^{24}}{6.37\times10^{6}} \approx 9.8\,\text{m/s}^2.$$
- ?? 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. 📚🌍
Revision
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