Short Notes: Gravitation
7.1 Universal Law of Gravitation
Definition: Every two point masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
| Formula | Details / Meaning |
|---|---|
| \(F = G\frac{m_1 m_2}{r^2}\) | G is the universal gravitational constant, \(G = 6.67\times 10^{-11}\,\text{N m}^2\text{kg}^{-2}\). Force acts along the line joining the centres. |
| \(\vec{F}_{12} = -\vec{F}_{21}\) | Newton's third law: forces are equal and opposite. |
Important concepts and consequences
- Inverse-square nature: If the separation doubles, force becomes one-fourth.
- Superposition principle: Net gravitational force on a mass is vector sum of forces due to individual masses.
- Shell theorem: A uniform spherical shell of mass exerts no net gravitational force on an internal point; for external points it behaves as if all mass is concentrated at the centre.
7.2 Gravitational Field and Potential
Gravitational field (intensity)
Definition: Gravitational field at a point is the force experienced per unit mass placed at that point.
| Quantity | Expression |
|---|---|
| Field intensity at distance r from mass M | \(\vec{g}(r) = -\,G\frac{M}{r^2}\,\hat{r}\) |
| Acceleration due to gravity at Earth's surface | \(g = \dfrac{GM}{R^2}\), where R is Earth's radius. |
Gravitational potential and potential energy
Gravitational potential (scalar): Work done per unit mass in bringing a test mass from infinity to a point in the field (zero at infinity).
| Quantity | Expression |
|---|---|
| Potential at distance r from mass M | \(V(r) = -\,\dfrac{GM}{r}\) |
| Potential energy of mass m at r | \(U(r) = mV(r) = -\,\dfrac{GMm}{r}\) |
| Potential energy at infinity | \(U(\infty)=0\) |
Sign convention: gravitational potential and potential energy are negative for bound systems because work must be done against gravity to move mass to infinity.
7.2.1 Variation of g with altitude and depth
At height h above the surface:
\(g' = \dfrac{GM}{(R+h)^2}\)
For small heights where \(h \ll R\), use binomial approximation to write:
\(g' \approx g\left(1 - \dfrac{2h}{R}\right)\)
Derivation (brief):
\(g' = g\left(\dfrac{R^2}{(R+h)^2}\right)\)
\( = g\left(1 + \dfrac{h}{R}\right)^{-2}\)
\( \approx g\left(1 - 2\dfrac{h}{R}\right)\) for \(h \ll R\).
At depth d below the surface (assuming uniform Earth density):
\(g' = g\left(1 - \dfrac{d}{R}\right)\)
Reason: by shell theorem only the mass enclosed within radius \(R-d\) contributes. The effective mass varies as \((R-d)^3\), leading to linear decrease of g with depth for Earth model of uniform density.
7.3 Orbital Motion
Condition for circular orbit: Centripetal force required for circular motion is provided by gravitational attraction.
Equate centripetal force and gravitational force:
\( \dfrac{mv^2}{r} = G\dfrac{Mm}{r^2} \)
\( \Rightarrow v_{\text{o}} = \sqrt{\dfrac{GM}{r}} \)
Alternative form using surface gravity \(g\) and Earth radius \(R\):
\( v_{\text{o}} = \sqrt{g}\,\dfrac{R}{\sqrt{r}} = \sqrt{\dfrac{gR^2}{r}}\)
Orbital speed at Earth's surface (approximate for low orbit):
\(v_{\text{o}} = \sqrt{gR} \approx 7.9\ \text{km s}^{-1}\)
Time period of circular orbit:
\(T = \dfrac{2\pi r}{v_{\text{o}}}\)
\( \Rightarrow T = 2\pi\sqrt{\dfrac{r^3}{GM}} \)
Escape velocity
Definition: Minimum speed required at the surface (or at distance r) to move to infinity with zero residual kinetic energy, neglecting other forces.
Using energy conservation between surface (or radius r) and infinity:
\(-\dfrac{GMm}{r} + \dfrac{1}{2}mv^2 = 0\)
\(\Rightarrow v_{\text{e}} = \sqrt{\dfrac{2GM}{r}} \)
From surface values:
\(v_{\text{e}} = \sqrt{2gR} \approx 11.2\ \text{km s}^{-1}\)
Relation between escape velocity and circular orbital velocity at same radius:
\(v_{\text{e}} = \sqrt{2}\,v_{\text{o}}\)
7.4 Satellite Energy
For a satellite of mass m in a circular orbit of radius r about mass M:
- Kinetic energy: \( \text{KE} = \dfrac{1}{2}mv_{\text{o}}^{2} = \dfrac{GMm}{2r}\)
- Potential energy: \( \text{PE} = -\,\dfrac{GMm}{r}\)
- Total (mechanical) energy: \( E = \text{KE} + \text{PE} = -\,\dfrac{GMm}{2r}\)
- Binding energy: Energy required to remove the satellite to infinity = \(|E| = \dfrac{GMm}{2r}\)
Note: The total energy is negative for a bound circular orbit; for a parabolic (escape) trajectory total energy is zero.
7.5 Kepler's Laws of Planetary Motion
| Law | Statement and consequence |
|---|---|
| First law (Law of Ellipses) | Planets move in elliptical orbits with the Sun at one focus. |
| Second law (Law of Areas) | Radius vector from the Sun to a planet sweeps out equal areas in equal times; this implies areal velocity is constant and angular momentum about the Sun is conserved. |
| Third law (Harmonic law) | \(T^{2} \propto r^{3}\). For a small mass orbiting a central mass M, \( \dfrac{T^{2}}{r^{3}} = \dfrac{4\pi^{2}}{GM}\), a constant for that central mass. |
Kepler's third law follows directly from Newton's law of gravitation and the circular-orbit expression for period.
Example applications and typical values
- Surface gravity of Earth: \(g \approx 9.8\ \text{m s}^{-2}\) (derived from \(g = GM/R^{2}\)).
- Low Earth orbital speed (approximate): \(7.9\ \text{km s}^{-1}\).
- Escape speed from Earth's surface (approximate): \(11.2\ \text{km s}^{-1}\).
- Geostationary orbit: a circular orbit in Earth's equatorial plane with period equal to Earth's rotation (sidereal day); radius is obtained from \(T = 2\pi\sqrt{r^{3}/GM}\).
Useful reminders and tips
- Always check whether r denotes distance from the centre or from the surface; use \(r=R+h\) when height above surface is involved.
- Use binomial approximation only when the fractional quantity is much less than 1 (for example \(h/R \ll 1\)).
- For problems involving Earth's interior, note the assumption of uniform density if using the linear depth relation; actual Earth density varies with radius and results differ slightly.
- Gravitational potential is a scalar and adds algebraically; gravitational field is a vector and must be added vectorially.
FAQs on Short Notes: Gravitation
| 1. What is the law of universal gravitation? |  |
| 2. How does gravitational force affect objects on Earth? | |
| 3. What is the significance of the gravitational constant (G)? | |
| 4. What is the concept of gravitational potential energy? | |
| 5. How does gravity affect the motion of planets in the solar system? | |
