JEE Physics Gravitation Notes Set 06

Newton's Law of Gravitation

Every two point masses attract each other with a force along the line joining them, proportional to the product of the masses and inversely proportional to the square of the distance: \( F = \dfrac{Gm_1m_2}{r^2} \), with \( G = 6.67\times10^{-11} \) N m² kg⁻².
Gravity is a central, conservative force. A uniform spherical shell attracts outside points as if its whole mass were at its centre and exerts no net force on a point inside it.

Acceleration due to Gravity

At the surface: \( g = \dfrac{GM}{R^2} = \tfrac43\pi G\rho R \).
At height h: \( g_h = \dfrac{g}{(1+h/R)^2} \approx g\left(1-\dfrac{2h}{R}\right) \) for h << R.
At depth d: \( g_d = g\left(1-\dfrac dR\right) \), zero at the centre.
Due to rotation at latitude \( \lambda \): \( g' = g - \omega^2R\cos^2\lambda \); zero at the equator if \( \omega = \sqrt{g/R} \).

Gravitational Field and Potential

Field of a solid uniform sphere: outside \( E = GM/r^2 \), at the surface \( GM/R^2 \), inside \( E = GMr/R^3 \).
Potential: outside \( V = -GM/r \); inside \( V = -\dfrac{GM}{2R^3}(3R^2-r^2) \); at the centre \( V = -\dfrac{3GM}{2R} \). For a thin shell, the potential is \( -GM/R \) everywhere inside.
Potential energy of two masses: \( U = -\dfrac{Gm_1m_2}{r} \). Work done by an external agent equals the change in U.

Escape Velocity and Satellites

Escape speed: \( v_e = \sqrt{2GM/R} = \sqrt{2gR} = 11.2 \) km/s for Earth. It does not depend on the mass or the direction of projection.
Orbital speed: \( v_o = \sqrt{GM/r} \), near the surface \( \sqrt{gR} = 7.9 \) km/s, and \( v_e = \sqrt2\,v_o \).
Period: \( T = 2\pi\sqrt{r^3/GM} \). Energies in orbit: \( K = \dfrac{GMm}{2r},\ U = -\dfrac{GMm}{r},\ E = -\dfrac{GMm}{2r} \).
A geostationary satellite has a period of 24 h, lies in the equatorial plane and has an orbit radius of about 42,400 km (height about 36,000 km). Objects inside an orbiting satellite are in free fall and appear weightless.

Kepler's Laws

1. Planets move in ellipses with the Sun at one focus.
2. The radius vector sweeps equal areas in equal times; areal velocity \( dA/dt = L/2m \) is constant because angular momentum is conserved.
3. \( T^2\propto a^3 \), where a is the semi-major axis. For a force \( F\propto r^{-n} \) in a circular orbit, \( T\propto r^{(n+1)/2} \).

Worked Examples

Example 1. Find the height above the Earth's surface at which g falls to one fourth of its surface value (R = 6400 km).
Answer:
\( g_h = g/(1+h/R)^2 = g/4 \Rightarrow 1 + h/R = 2 \)
h = R = 6400 km

Example 2. Find the minimum speed to escape from a planet of mass M and radius R from a point at height R above its surface.
Answer:
Total energy must be zero: \( \tfrac12v^2 - \dfrac{GM}{2R} = 0 \)
\( v = \sqrt{GM/R} \)