GRAVITATION — Complete JEE Mains Revision.
Every Formula | Every Trap | Every Mains Question Type
1. Universal Law of Gravitation
Newton’s Law:
$$ F = G \frac{m_1 m_2}{r^2} $$G = $6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$
Key Results
- Force ∝ product of masses
- Force ∝ 1 / r² (inverse square)
- Force always attractive
Vector Form:
$$ \vec{F} = - G \frac{m_1 m_2}{r^2} \hat{r} $$Mains Trap: Distance is center-to-center, NOT surface distance.
2. Gravitational Field & Acceleration Due to Gravity
Field Definition:
$$ g = \frac{F}{m} $$ For planet of mass M: $$ g = \frac{GM}{R^2} $$Variation with height (h):
$$ g_h = \frac{GM}{(R+h)^2} $$ For small h: $$ g_h = g \left(1 - \frac{2h}{R}\right) $$Variation with depth (d):
$$ g_d = g \left(1 - \frac{d}{R}\right) $$Mains Trap: g decreases linearly with depth but inverse square with height.
3. Gravitational Potential & Potential Energy
Potential:
$$ V = -\frac{GM}{r} $$Potential Energy:
$$ U = -\frac{GMm}{r} $$ Relation: $$ g = - \frac{dV}{dr} $$Near Earth surface:
$$ U = mgh $$Mains Trap: Potential is negative because reference is at infinity.
4. Escape Velocity
Using energy conservation: $$ \frac{1}{2}mv^2 - \frac{GMm}{R} = 0 $$ $$ v_e = \sqrt{\frac{2GM}{R}} $$ Also: $$ v_e = \sqrt{2gR} $$
Mains Trap: Escape velocity independent of mass of body.
5. Orbital Motion
For satellite: Centripetal force = Gravitational force $$ \frac{mv^2}{r} = \frac{GMm}{r^2} $$ $$ v_o = \sqrt{\frac{GM}{r}} $$ Relation: $$ v_e = \sqrt{2} v_o $$Time Period:
$$ T = 2\pi \sqrt{\frac{r^3}{GM}} $$ For Earth satellite: $$ T^2 \propto r^3 $$Total Energy in Orbit:
$$ E = -\frac{GMm}{2r} $$Mains Trap: Total energy negative → bound system.
6. Kepler’s Laws
1st Law: Elliptical orbit
2nd Law: Equal areas in equal times
3rd Law:
$$ \frac{T^2}{r^3} = \text{constant} $$
Mains Trap: r = semi-major axis for ellipse.
7. Geo-stationary Satellite
Conditions:- Orbit in equatorial plane
- Time period = 24 hr
- Same direction as Earth rotation
JEE Mains Question Profiling
| Topic | Type | Weight |
|---|---|---|
| Escape velocity | Direct formula | High |
| Orbital energy | Conceptual + numerical | High |
| Variation of g | Approximation based | Medium |
| Kepler’s law | Ratio problems | Medium |
Final Strategic Insight
Gravitation = Energy + Circular Motion + Ratio Logic Master these → 1 guaranteed Mains question solved.
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