# Summary

- Newton's law of universal gravitation states that the gravitational forceof attraction between any two particles of masses m
_{1}and m_{2}separated by a distance r has the magnitude- F=

- Where G is the universal gravitational constant, which has the value 6.672 x 10 -11 Nm2 kg-2.
- If we have to find the resultant gravitational force acting on the particle m due to a number of masses M1,M2.....Mn each given by the law of gravitation. From the principle of superposition each force acts indpendently and uninfluenced by the other bodies. The resultatnt force FR is then found by vecot addition
- Where the symbol âˆ‘ stands for summation.

- Kepler's laws of planetary motion state that
- all planets move in elliptical orbits with the sun at one of the focal points
- The radius vector drawn from the sun to a planet sweetps out equal areas in equal time intervals. This follows from the fact that the force of gravitation on the planet is central and hence angular momentum is conserved.
- The square of the orbital period of a planet is proportional to the cube of the semi-major axis of the elliptical oribit of the planet
- The period T and radius R of the circular orbit of a planet about the sun are related by
- Where is the mass of the sun. Most planets have nearly circular orbits about the sun. For elliptical orbits, the above equation is valid if R is replaced by the semi-major axis, a.
- The acceleration due to gravity
- at a height h above the Earth's surface
- B) at depth d below the Earth's surface
- The gravitation force is a conservative force, and therefore a potential energy function can be defined. The gravitational potential energy associated with two particles seperated by a distance r given by
- Where V is taken to be zero at r->âˆž. The total potential energy for a system of particles is the sum of energies for all pairs of particles, with each pair represented by a term of the form given by above equation. This prescription follows from the principle of superposition.
- If an isolated system consists of a particle of mass m moving with speed v in the vicinity of a massive body of mass M,the total mechanical energy of the particle is given by
- That is, the total mechanical energy is the sum of the kinetic and potential energies, the total energy is negative for any bound system, that is, one in which the orbit is closed. Such as an elliptical orbit. The kinetic and potential energies are
- The escape speed from the surface of the earth is
- If a particle is outside a uniform spherical shell or solid sphere with a spherically symmetric internal mass distribution, the sphere attrats the particle as thought the mass of the sphere or shell were concentrated at the centre of the sphere.,
- If a particle is inside a uniform spherical shell, the gravitational force on the particle is zero. If a particle is inside a homogeneous solid sphere, the force on the particle acts toward the cen tre of the sphere. This force is exerted by the spehrical mass interior to the particle.
- A geostationary (geosynchronous communication) satellite moves in a circular orbit in the equatorial plane at a approximate distance of 4.22 x 10 4 km from the earth's centre.