Acceleration Due to Gravity: Quick, Friendly Notes

Gravity pulls every object toward Earth’s center. The pull per unit mass is called the acceleration due to gravity, g.

1  – At Earth’s Surface

The familiar value of g comes from Newton’s law of gravitation and Newton’s second law:

\( g \;=\; \dfrac{G\,M_E}{R_E^{\,2}} \) :contentReference[oaicite:0]{index=0}

  • \(G\) – universal gravitational constant.
  • \(M_E\) – mass of Earth.
  • \(R_E\) – radius of Earth.

2  – Above the Surface (height h)

Put a small mass at a height h. Its distance from Earth’s center is \(R_E + h\), so

\( g(h) \;=\; \dfrac{G\,M_E}{\bigl(R_E + h\bigr)^{2}} \) :contentReference[oaicite:1]{index=1}

For small heights (\(h \ll R_E\)) expand the denominator to get a handy approximation:

\( g(h) \;\approx\; g\!\left(1 – \dfrac{2h}{R_E}\right) \) :contentReference[oaicite:2]{index=2}

So g falls roughly 2 h ⁄ RE below its surface value when you go up a modest height h.

3  – Below the Surface (depth d)

Go down a vertical mine shaft of depth d. Only the mass of the inner sphere of radius \(R_E-d\) now attracts you, giving

\( g(d) \;=\; g\!\left(1 – \dfrac{d}{R_E}\right) \) :contentReference[oaicite:3]{index=3}

Here g decreases linearly with depth—halfway to Earth’s center (\(d = 0.5\,R_E\)) you would weigh half as much!

4  – Where Is g Largest?

Right on the surface. Move up (positive h) or move down (positive d) and g always becomes smaller. :contentReference[oaicite:4]{index=4}

5  – Gravitational Potential Energy Refresher

Near the surface (constant g idea) the potential energy at height h is

\( W(h) \;=\; m\,g\,h + W_0 \) :contentReference[oaicite:5]{index=5}

Choose the zero of energy so that \(W_0 = 0\) at the surface. Then \(W = mgh\).

For large changes in distance from Earth’s center the exact work done in lifting a mass m from radius \(r_1\) to \(r_2\) is

\( W_{12} \;=\; -\,G\,M_E\,m \Bigl(\dfrac{1}{r_2} – \dfrac{1}{r_1}\Bigr) \) :contentReference[oaicite:6]{index=6}

High-Yield Ideas for NEET

  1. Surface formula: \( g = {G\,M_E}/{R_E^{2}} \) and how Cavendish’s measurement of G let us “weigh” Earth. :contentReference[oaicite:7]{index=7}
  2. Altitude variation: \( g(h) = {G\,M_E}/{(R_E+h)^{2}} \) and the small-height trick \(g(h)\approx g(1-2h/R_E)\). :contentReference[oaicite:8]{index=8}
  3. Depth variation: \( g(d) = g(1-d/R_E) \); straight-line falloff toward the center. :contentReference[oaicite:9]{index=9}
  4. Maximum at the surface: g drops no matter whether you climb up or dig down. :contentReference[oaicite:10]{index=10}
  5. Exact gravitational potential energy: \( W_{12} = -G\,M_E\,m\bigl(1/r_2 – 1/r_1\bigr) \). :contentReference[oaicite:11]{index=11}

Keep these results on your fingertips, practice plugging in numbers, and you’ll breeze through gravity questions!