Study Notes – Gravitation
AIIMS Entrance – Physics
(For quick revision & deep understanding)
1. Simple Explanation
Gravitation is the attractive force that pulls all masses toward one another.
- Every object with mass feels a pull toward every other mass.
- The force is weaker for small masses and stronger for massive bodies.
- It is the same force that keeps planets in orbit, gives us weight, and makes apples fall from trees.
2. Key Points
| Concept | Symbol | Formula / Definition | Notes |
|---|---|---|---|
| Newton’s Universal Law | – | (F = G\frac{m_1m_2}{r^2}) | (G=6.674\times10^{-11},\mathrm{N,m^2,kg^{-2}}) |
| Gravitational Field (Acceleration) | (\vec g) | (\vec g = G\frac{M}{r^2}\hat r) | Direction toward mass (M) |
| Gravitational Potential Energy (GPE) | – | (U = -G\frac{m_1m_2}{r}) | Negative because work needed to separate masses |
| Gravitational Potential (Φ) | – | (\Phi = -G\frac{M}{r}) | (U = m\Phi) |
| Escape Velocity | – | (v_{\text{esc}} = \sqrt{\frac{2GM}{R}}) | Minimum speed to reach infinity |
| Kepler’s 3 Laws | – | 1. Elliptical orbits 2. Equal areas in equal times 3. (T^2 \propto a^3) (period–semi‑major axis) |
Follow from (F = GmM/r^2) and conservation laws |
| Field Inside a Uniform Shell | – | (g = 0) | Newton’s shell theorem |
| Field Inside a Solid Sphere | – | (g(r) = G\frac{M_{\text{enc}}}{r^2} = G\frac{M r}{R^3}) | Linear with (r) |
| Gravitational Redshift (relativistic) | – | (\Delta f / f = \Delta \Phi / c^2) | Small effect, often omitted in AIIMS |
3. Worked Example
Problem:
Find the escape velocity from the surface of Mars (radius (R_{\text{Mars}} = 3.39\times10^6\ \text{m}), mass (M_{\text{Mars}} = 6.42\times10^{23}\ \text{kg})).
Solution Steps
-
Use the escape‑velocity formula
[ v_{\text{esc}} = \sqrt{\frac{2GM}{R}} ] -
Insert known constants
[ G = 6.674\times10^{-11},\text{N m}^2 \text{kg}^{-2} ] -
Compute
[ v_{\text{esc}} = \sqrt{\frac{2 \times 6.674\times10^{-11}\ \times 6.42\times10^{23}} {3.39\times10^{6}}} ] -
Do the arithmetic
[ \begin{aligned} \frac{2GM}{R} &= \frac{2 \times 6.674\times10^{-11} \times 6.42\times10^{23}} {3.39\times10^{6}} \ &\approx \frac{8.58\times10^{13}}{3.39\times10^{6}} \ &\approx 2.53\times10^{7}\ \text{m}^2\text{s}^{-2} \end{aligned} ] -
Take the square root
[ v_{\text{esc}} \approx \sqrt{2.53\times10^{7}} \approx 5.03\times10^{3}\ \text{m/s} ]
Answer:
The escape velocity from Mars is ≈ 5.0 km s⁻¹.
(Hint: Compare with Earth’s escape velocity, 11.2 km s⁻¹; Mars is lighter ⇒ lower escape speed.)
4. Quick Revision Summary
- Newton’s Law
[ F = G\frac{m_1m_2}{r^2} ] - Gravitational Field
[ g = G\frac{M}{r^2} ] - Gravitational Potential Energy
[ U = -G\frac{m_1m_2}{r} ] [ \Phi = -G\frac{M}{r} ] - Escape Velocity
[ v_{\text{esc}} = \sqrt{\frac{2GM}{R}} ] - Kepler’s Laws
- Orbits = ellipses (focus at one focus).
- Equal areas in equal times (⇒ angular momentum conserved).
- (T^2 \propto a^3) (orbital period vs. semi