Universal Law of Gravitation — Friendly Notes

1 · How Newton Reached the Law

Newton compared the Moon’s centripetal acceleration \[ a_m = \frac{4\pi^{2}R_m}{T^{2}} \tag{} \] with the familiar surface value \(g\). Because \(R_m \approx 3.84\times10^{8}\,\text{m}\) and \(T\approx27.3\) days, the ratio \(g/a_m\) comes out close to 3600. That is exactly the square of the factor by which Earth’s surface lies closer to Earth’s centre than the Moon does, hinting that gravity falls off as \(1/r^{2}\). :contentReference[oaicite:0]{index=0}

2 · The Law in Words

Every body pulls every other body. The pull is directly proportional to the product of their masses and inversely proportional to the square of the separation. :contentReference[oaicite:1]{index=1}

3 · The Law in Maths

  • Magnitude (for point masses): \[ F = G\frac{m_{1}m_{2}}{r^{2}} \tag{7.5} \] :contentReference[oaicite:2]{index=2}
  • Vector form (arrow points from \(m_1\) to \(m_2\)): \[ \mathbf F_{12}= -\,\mathbf F_{21}= -\,G\,\frac{m_{1}m_{2}}{r^{3}}\, \mathbf r \] (\(\mathbf r\) is the displacement \( \mathbf r_2-\mathbf r_1\), and the minus sign reminds us the force is attractive.) :contentReference[oaicite:3]{index=3}

4 · Working with Many Masses

For several masses you just add individual pulls head-to-tail; this is the principle of superposition. :contentReference[oaicite:4]{index=4}

5 · Two Handy “Shell” Facts

  1. The pull from a uniform spherical shell on an outside point acts as if the whole mass were at the centre.
  2. The pull on a point inside a hollow shell is zero.
Both follow from clever symmetry arguments. :contentReference[oaicite:5]{index=5}

6 · Measuring G

Henry Cavendish (1798) hung tiny lead spheres from a torsion wire and let bigger spheres tug on them. The twist of the wire revealed the value of the gravitational constant \(G\). :contentReference[oaicite:6]{index=6}

7 · Quick Examples You Should Know

  • Planet at perihelion vs. aphelion: conservation of angular momentum gives \(v_{p}r_{p}=v_{A}r_{A}\); the planet moves faster when closer to the Sun. :contentReference[oaicite:7]{index=7}
  • Equilateral-triangle masses: three equal masses at the corners produce zero net force at the centroid; doubling one corner mass spoils the balance. :contentReference[oaicite:8]{index=8}

8 · Links to Kepler’s Third Law

Because the gravitational pull follows \(1/r^{2}\), you can show that \(T^{2}\propto a^{3}\) (square of orbital period vs. cube of semi-major axis), exactly what Kepler’s Law of Periods states and the planetary data confirm. :contentReference[oaicite:9]{index=9}

High-Yield Ideas for NEET

  1. Remember the form \(F=Gm_{1}m_{2}/r^{2}\) and that the force is always attractive.
  2. Inside a hollow sphere gravity is zero; outside, treat the sphere as a point mass at its centre.
  3. Use angular-momentum conservation \(mvr=\text{constant}\) to relate speeds at different orbital points.
  4. Centripetal approach: equate \(\displaystyle \frac{4\pi^{2}R}{T^{2}}\) to \(GM/R^{2}\) to derive orbital relations quickly.
  5. Cavendish torsion-balance idea shows how to measure \(G\); questions often ask what changes the twist if you swap sphere masses or separations.