Key Points

Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The formula is $F = G \frac{m_1 m_2}{r^2}$.

G is the universal gravitational constant, which is a fundamental constant in physics. Its accepted value is $G = 6.67 \times 10^{-11} \text{ N m}^2\text{/kg}^2$.

All planets move in elliptical orbits with the Sun situated at one of the foci of the ellipse. This law defines the shape of planetary paths.

The line that joins any planet to the sun sweeps out equal areas in equal intervals of time. This implies that a planet moves faster when it is closer to the sun and slower when it is farther away.

The square of the time period of revolution of a planet ($T^2$) is proportional to the cube of the semi-major axis of its orbit ($a^3$). Mathematically, $T^2 \propto a^3$.

The acceleration experienced by an object due to Earth's gravity is denoted by g. On the surface, it is calculated as $g = \frac{GM_E}{R_E^2}$, where $M_E$ is the mass of the Earth and $R_E$ is its radius.

Acceleration due to gravity decreases with height h above the Earth's surface. The value is given by $g(h) = \frac{GM_E}{(R_E+h)^2}$. For $h \ll R_E$, this can be approximated as $g(h) \approx g(1 - \frac{2h}{R_E})$.

Acceleration due to gravity also decreases with depth d below the Earth's surface. The value is given by the linear relation $g(d) = g(1 - \frac{d}{R_E})$.

The gravitational potential energy (V) of two masses $m_1$ and $m_2$ separated by a distance r is given by $V = -\frac{Gm_1 m_2}{r}$. The negative sign indicates an attractive force, and the zero of potential energy is taken at infinite separation.

Escape speed is the minimum speed required for an object to escape the gravitational influence of a celestial body. For Earth, it is calculated as $v_e = \sqrt{\frac{2GM_E}{R_E}} = \sqrt{2gR_E}$, which is approximately $11.2 \text{ km/s}$.

An object revolving around the Earth is a satellite. The speed of a satellite in a circular orbit of radius $r = R_E + h$ is given by $v_o = \sqrt{\frac{GM_E}{r}}$.

The time period (T) for a satellite to complete one orbit of radius r is $T = \frac{2\pi r}{v_o} = 2\pi \sqrt{\frac{r^3}{GM_E}}$. Squaring this gives $T^2 = (\frac{4\pi^2}{GM_E})r^3$, which is Kepler's third law.

The total energy (E) of a satellite in a circular orbit is negative, indicating a bound system. It is given by $E = K + V = -\frac{GM_E m}{2r}$, where the kinetic energy is exactly half the magnitude of the potential energy.

The gravitational force on a point mass situated anywhere inside a uniform hollow spherical shell is zero. This is a key result of the shell theorem.

For a point mass situated outside a uniform spherical shell or solid sphere, the gravitational force is the same as if the entire mass of the sphere were concentrated at its center.