Gravitation is a force that exists among all material objects in the universe. For any two objects having nonzero mass, the force of gravity tends to attract them toward each other. Gravity operates on objects of all sizes, from sub atomic particles to clusters of galaxies. It also operates over all distances, no matter how small or great.
The Universe has a lot of forces a lot of pushes and pulls. We are always pushing or pulling something, even if only the ground. But it turns out that in physics, there are really only four fundamental forces from which everything else is derived the strong force, the weak force, the electromagnetic force and the gravitational force.
Newton’s law of Gravitation:
The force of gravitational attraction between two point bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Consider two point bodies of masses m1, m2 are placed at a distance r. The force of gravitational attraction between them
F = G(m1m2/ r2)
G is constant called universal gravitational constant.
The value of G is 6.67 × 10-11 Nm2/kg2
The gravitational force of earth is called gravity i.e. gravity is the force by which earth pulls a body towards its centre.
The acceleration produced in a body due to force of gravity is called acceleration due to gravity and its value is 9.8 m/ s2
Acceleration due to gravity is independent of shape, size and mass of the body.
The gravitational constant “G”:
The value of the gravitational constant G entering the Universal law of gravitation can be determined experimentally and this was first done by scientist Henry Cavendish in 1798.
* Value of G decreases with height or depth from earth’s surface.
* G is maximum at poles.
* G is minimum at equator.
* G decreases due to rotation of earth.
* G decreases if angular speed of earth increases and increases if angular speed of earth decreases.
If angular speed of earth becomes 17 times its present value, a body on the equator becomes weightless.
Acceleration due to gravity of the earth:
The earth can be imagined to be a sphere made of a large number of concentric spherical shells with the smallest one at the centre and the largest at its surface. A point outside the earth is obviously outside all the shells. Thus, all the shells exert a gravitational force at the point outside just as if their masses are concentrated at the point outside just as if their masses are concentrated at their common centre. The total mass of all the shells combined is just the mass of the earth. Hence, at a point outside the earth, the gravitational force is just as if its entire mass of the earth is concentrated at its center.
Weight of the body in lift:
a. If lift is stationary or moving with uniform speed, the apparent weight of a body is equal to its true weight.
b. If lift is going up with acceleration, the apparent weight of a body is more than the true weight.
c. If lift is going down with acceleration, the apparent weight of a body is less than the true weight.
d. If the cord of the lift is broken, it falls freely. In this situation the weight of a body in the lift becomes zero. This is the situation of weightlessness.
e. While going down, if the acceleration of lift is more than acceleration due to gravity, a body in the lift goes in contact of the ceiling of lift.
Escape velocity is the speed at which the sum of an object's kinetic energy and its gravitational potential energy is equal to zero. It is the speed needed to "break free" from the gravitational attraction of a massive body, without further propulsion, i.e., without spending more fuel. For a spherically symmetric massive body such as a (non-rotating) star or planet, the escape velocity at a given distance is calculated by the formula:
where G is the universal gravitational constant (G = 6.67 × 10−11 m3 kg−1 s−2), M the mass of the body, and r the distance from the point in space to its center of mass.
In this equation atmospheric friction is not taken into account. A rocket moving out of gravity well does not actually need to attain escape velocity to do so, but could achieve the same result at any speed with a suitable mode of propulsion and sufficient fuel. Escape velocity only applies to ballistic trajectories.
The term escape velocity is actually a misnomer, and it is often more accurately referred to as escape speed since the necessary speed is a scalar quantity which is independent of direction.
Escape velocity is independent of the mass, shape and size of the body and its direction of projection. Escape velocity is also called second cosmic velocity.
For earth, escape velocity = 11.2 km/ s
For moon, escape velocity = 2.4 km / s
Orbital velocity of a satellite
Where R= radius of earth i.e.
i.e. escape velocity is
Therefore if the orbital velocity of satellite is increased to times the satellite will leave the orbit and escape.
Orbital speed of a satellite:
Orbital speed of a satellite is independent of its mass. Hence satellites of different masses revolving in the orbit of same radius have same orbital speed. Orbit speed of a satellite depends upon the radius of orbit. Greater the radius of orbit, lesser will be orbital speed.
The orbital speed of a satellite revolving near the surface of earth is 7.9km/sec.
Period of revolution of a satellite:
Time taken by a satellite to complete one revolution in its orbit is called its period of revolution.
Period of revolution of a satellite depends upon the height of satellite from the surface of earth. Greater the height, more will be the period of revolution. Period of revolution of a satellite is independent of its mass.
The period of revolution of satellite revolving near the surface of earth is 1 hour 24 minute.
Geostationary orbit is a orbit that fixed with respect to a position of earth.
If a satellite revolves in a equatorial plane in the direction of earth’s rotation i.e. from west to east with a period of revolution equal to time period of rotation of earth on its own axis i.e. 24 hours, then the satellite will appear stationary relative to earth. Such a satellite is called Geo stationary satellite.
i. Law of orbits:
Kepler's first law - sometimes referred to as the law of ellipses - explains that planets are orbiting the sun in a path described as an ellipse. An ellipse can easily be constructed using a pencil, two tacks, a string, a sheet of paper and a piece of cardboard. Tack the sheet of paper to the cardboard using the two tacks. Then tie the string into a loop and wrap the loop around the two tacks. Take your pencil and pull the string until the pencil and two tacks make a triangle.
Then begin to trace out a path with the pencil, keeping the string wrapped tightly around the tacks. The resulting shape will be an ellipse. An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. The two other points are known as the foci of the ellipse. The closer together that these points are, the more closely that the ellipse resembles the shape of a circle. In fact, a circle is the special case of an ellipse in which the two foci are at the same location. Kepler's first law is rather simple - all planets orbit the sun in a path that resembles an ellipse, with the sun being located at one of the foci of that ellipse.
ii. The Law of Areas:
Kepler's second law - sometimes referred to as the law of equal areas - describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. For instance, if an imaginary line were drawn from the earth to the sun, then the area swept out by the line in every 31-day month would be the same.