Gravitation – Smart, Student-Friendly Notes
1. Why We Care
We notice gravity every time a ball drops or we puff while climbing uphill. Galileo showed that, near Earth, every freely falling object speeds up at the same rate. Later, careful sky-watching by Tycho Brahe and the brilliant pattern-spotting of Johannes Kepler linked that everyday pull to the majestic dance of planets around the Sun. :contentReference[oaicite:0]{index=0}
2. Kepler’s Three Planet Rules
- Orbit Rule: Every planet loops around the Sun on an ellipse; the Sun sits at one focus. (A circle is just a “perfect” ellipse.)
- Area Rule: A line from the planet to the Sun sweeps out equal areas in equal times. Farther from the Sun ⇒ slower sweep; closer ⇒ faster.
- Time-Period Rule: The square of a planet’s “year” (\(T\)) is proportional to the cube of its orbit’s semi-major axis (\(a\)): \(T^{2}\;\propto\;a^{3}\).
These rules hinted that one simple force controls all orbital motion. :contentReference[oaicite:1]{index=1}
3. Newton’s Universal Gravity Law
Any two point masses pull on each other with a force \[F = G \frac{m_{1} m_{2}}{r^{2}}\] where \(m_{1}\) and \(m_{2}\) are the masses, \(r\) is the centre-to-centre distance, and \(G\) is the gravitational constant. This single rule explains Kepler’s patterns and everyday falling apples alike. :contentReference[oaicite:2]{index=2}
4. The Gravitational Constant (\(G\))
\(G = 6.67 \times 10^{-11}\;{\rm N\,m^{2}\,kg^{-2}}\). It is tiny, so noticeable gravity needs either huge masses (planets) or very small separations (standing on Earth’s surface). :contentReference[oaicite:3]{index=3}
5. Acceleration Due to Gravity (\(g\)) at Earth’s Surface
Treat Earth as a sphere of mass \(M\) and radius \(R\). Plug the planet and a 1 kg object into the universal law and you get \[g \;=\; \frac{GM}{R^{2}}\;\approx\;9.8\;{\rm m/s^{2}}\] This is the “speed-gain-per-second” every freely falling body feels near sea level. :contentReference[oaicite:4]{index=4}
5.1 How \(g\) Changes
- Height (\(h << R\)): \(g_h \approx g\!\left(1 – \dfrac{2h}{R}\right)\)
- Depth (\(d < R\)): \(g_d \approx g\!\left(1 – \dfrac{d}{R}\right)\)
6. Gravitational Potential Energy (GPE)
Bringing two masses together (against gravity) stores energy: \[U = -\,\frac{G m_{1} m_{2}}{r}\] The minus sign means energy is lower (more negative) when the masses are closer—they “want” to be together. :contentReference[oaicite:5]{index=5}
7. Escape Speed
Kick an object hard enough and it will never fall back. The minimum launch speed from Earth’s surface (ignoring air drag) is \[v_{\text{esc}} = \sqrt{2 g R}\;\approx\;11.2\;{\rm km/s}\]. No further push is needed once this speed is reached because gravity’s pull weakens with distance faster than the object slows. :contentReference[oaicite:6]{index=6}
8. Satellites in Circular Orbits
A satellite of mass \(m\) circling a planet of mass \(M\) at radius \(r\) balances gravity with the required centripetal pull, leading to the orbital speed \[v = \sqrt{\frac{GM}{r}}\] and time period \[T = 2\pi\sqrt{\frac{r^{3}}{GM}}\].
8.1 Total Mechanical Energy
The sum of kinetic and potential energy is negative: \[E = -\,\frac{G M m}{2r}\] A more distant orbit (larger \(r\)) is less negative, so the satellite is “less tightly bound.” :contentReference[oaicite:7]{index=7}
9. High-Yield NEET Takeaways
- Recognise and apply Kepler’s laws to orbital speed and period questions.
- Use the universal gravitation formula to link force, mass, and distance.
- Relate \(g\) to planet mass/radius and predict how it varies with height or depth.
- Calculate escape speed and understand why it is independent of mass.
- Compute satellite energy and speed in circular orbits, spotting negative total energy answers.
Keep practicing with real numbers—muscle memory with the formulas makes NEET gravitation problems feel light!
