Gravitation — Revision Notes

⚡ 30-Second Revision

Newton's Law: — $F = G \frac{m_1 m_2}{r^2}$
Universal Gravitational Constant: — $G = 6.67 \times 10^{-11},\text{N m}^2/\text{kg}^2$
Acceleration due to gravity (surface): — $g = \frac{GM_E}{R_E^2} approx 9.8,\text{m/s}^2$
Variation of g (altitude): — g_h = g left(1 - \frac{2h}{R_E}\right) (for $h ll R_E$)
Variation of g (depth): — g_d = g left(1 - \frac{d}{R_E}\right)
Gravitational Potential Energy: — $U = -\frac{GMm}{r}$
Gravitational Potential: — $V = -\frac{GM}{r}$
Escape Velocity: — $v_e = sqrt{\frac{2GM}{R}} = sqrt{2gR}$
Orbital Velocity: — $v_o = sqrt{\frac{GM}{r}} = sqrt{\frac{gR^2}{r}}$
Relationship: — $v_e = sqrt{2} v_o$ (near surface)
Kepler's 3rd Law: — $T^2 propto a^3$

2-Minute Revision

Gravitation is the attractive force between any two masses. Newton's Law states $F = G \frac{m_1 m_2}{r^2}$, where $G$ is the universal constant. The acceleration due to gravity, $g$, is approximately $9.

8, ext{m/s}^2$on Earth's surface. Remember its variations:$g$decreases with altitude ($g_h approx g(1 - 2h/R_E)$) and depth ($g_d = g(1 - d/R_E)$), and is minimum at the equator due to Earth's rotation.

Gravitational potential energy is $U = -\frac{GMm}{r}$, always negative. Escape velocity ($v_e = sqrt{2GM/R}$) is the minimum speed to leave a planet's gravity, while orbital velocity ($v_o = sqrt{GM/r}$) is for stable orbits.

Crucially, $v_e = sqrt{2} v_o$ for orbits near the surface. Kepler's laws describe planetary motion: elliptical orbits, equal areas swept in equal times (due to angular momentum conservation), and $T^2 propto a^3$.

Weightlessness is apparent, not due to zero gravity, but continuous freefall in orbit.

5-Minute Revision

Gravitation, a fundamental force, governs attraction between masses. Newton's Law of Universal Gravitation, $F = G \frac{m_1 m_2}{r^2}$, is central. $G$ is the universal constant ($6.67 \times 10^{-11},\text{N m}^2/\text{kg}^2$), while $g$ is the acceleration due to gravity.

On Earth's surface, $g = \frac{GM_E}{R_E^2} approx 9.8,\text{m/s}^2$. Remember how $g$ varies: it decreases with altitude (g_h = g left(1 + \frac{h}{R_E}\right)^{-2} approx g left(1 - \frac{2h}{R_E}\right) for small $h$) and with depth (g_d = g left(1 - \frac{d}{R_E}\right)).

It's also affected by Earth's rotation, being minimum at the equator. Gravitational potential energy, $U = -\frac{GMm}{r}$, is always negative, indicating an attractive force and a bound system. Gravitational potential is $V = -\frac{GM}{r}$.

Escape velocity ($v_e = sqrt{\frac{2GM}{R}} = sqrt{2gR}$) is the minimum speed to escape a planet's gravitational field. Orbital velocity ($v_o = sqrt{\frac{GM}{r}}$) is the speed required for a stable circular orbit at radius $r = R+h$. For an orbit very close to the surface ($r approx R$), $v_o = sqrt{gR}$. A key relationship is $v_e = sqrt{2} v_o$ (for objects starting from the surface).

Kepler's Laws are crucial: 1) Planets move in elliptical orbits with the Sun at one focus. 2) The line joining a planet and the Sun sweeps equal areas in equal times (conservation of angular momentum). 3) The square of the orbital period ($T$) is proportional to the cube of the semi-major axis ($a$), i.e., $T^2 propto a^3$.

Weightlessness in orbit is an apparent phenomenon, not due to zero gravity, but because the orbiting body and its contents are in continuous freefall. Practice problems involving these formulas and conceptual variations are key to NEET success.

Prelims Revision Notes

1
Newton's Law of Universal Gravitation: — $F = G \frac{m_1 m_2}{r^2}$. Force is always attractive.
2
Universal Gravitational Constant (G): — $6.67 \times 10^{-11},\text{N m}^2/\text{kg}^2$. It's a universal constant, independent of medium or masses.
3
Acceleration due to gravity (g): — $g = \frac{GM_E}{R_E^2}$. Average value on Earth's surface is $9.8,\text{m/s}^2$.

* Variation with Altitude (h): g_h = \frac{GM_E}{(R_E + h)^2} = g left(1 + \frac{h}{R_E}\right)^{-2}. For $h ll R_E$, use approximation g_h approx g left(1 - \frac{2h}{R_E}\right). 'g' decreases with altitude.

* Variation with Depth (d): g_d = g left(1 - \frac{d}{R_E}\right). 'g' decreases with depth, becoming zero at the center of Earth. * **Variation with Latitude ($lambda$):** $g' = g - R_E omega^2 cos^2lambda$.

'g' is minimum at the equator ($lambda=0^circ$) and maximum at the poles ($lambda=90^circ$). * Variation due to shape: Earth is an oblate spheroid; $R_E$ is larger at the equator, so $g$ is slightly less at the equator than at the poles.

1
Gravitational Field Intensity (E): — Force per unit mass, $E = \frac{GM}{r^2}$. Vector quantity.
2
Gravitational Potential (V): — Work done per unit mass from infinity, $V = -\frac{GM}{r}$. Scalar quantity.
3
Gravitational Potential Energy (U): — $U = -\frac{GMm}{r}$. Always negative, indicates a bound system.
4
Escape Velocity ($v_e$): — Minimum velocity to escape gravitational pull. $v_e = sqrt{\frac{2GM}{R}} = sqrt{2gR}$. For Earth, $v_e approx 11.2,\text{km/s}$.
5
Orbital Velocity ($v_o$): — Velocity for stable circular orbit at radius $r = R+h$. $v_o = sqrt{\frac{GM}{r}} = sqrt{\frac{gR^2}{r}}$. For $r approx R$, $v_o = sqrt{gR} approx 7.9,\text{km/s}$.
6
Relationship between $v_e$ and $v_o$ (near surface): — $v_e = sqrt{2} v_o$.
7
Kepler's Laws:

* Law of Orbits: Elliptical orbits with Sun at one focus. * Law of Areas: Equal areas swept in equal times ($rac{dA}{dt} = \text{constant}$), implies conservation of angular momentum. * Law of Periods: $T^2 propto a^3$ (for planets around Sun, or satellites around Earth).

1
Geostationary Satellite: — Orbital period 24 hours, appears stationary, height $approx 36,000,\text{km}$.
2
Weightlessness: — Apparent absence of weight due to continuous freefall, not zero gravity.

Vyyuha Quick Recall

To remember the variations of 'g': All Deep Layers Rotate Slowly.

Altitude: 'g' decreases.
Depth: 'g' decreases.
Latitude: 'g' minimum at equator, maximum at poles.
Rotation: Reduces 'g' at equator.
Shape (oblate spheroid): 'g' less at equator due to larger radius.