[Concept] : Gravitation

“Everything that goes up always comes down”

You might of heard of this quote. But this quote is not entirely true!!!

If you throw an object with velocity of 11.2km/s it’ll never come back!!!! This velocity is called the escape velocity of the earth. In fact every planet has a escape velocity which depends on its gravitational pull.

Gravitation is one of the most important branch of physics. It is due to gravitational pull  of sun that we live today. Gravity is everywhere on universe. Gravitational force is one of the basic force of physics.

English physicist Sir Isaac Newton discovered the laws of gravitation.(He had discovered the laws of motion earlier and argued that there must be some force acting on moon by earth else it would follow a straight line path( from the first law of motion) rather than a circular path around the earth). He stated the universal law of gravitation as:

Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to

the square of the distance between them.

Mathematically if two masses of mass m1 and m2 are separated by a distance r with each other the force of gravitation between them is

F = (Gm1m2)/ r2

where G is the universal gravitation constant. It has the value 6.67x 10-11 m3kg-1s-2. It is interesting to note that the universal gravitational constant was discovered much later in the eighteenth century by Henry Cavendish. Before that the constant was used as unity( same as the second law of motion) but for that gravitational mass was used in place of real mass.

To calculate the the gravitational constant of the earth Cavendish used two small spheres with same mass fixed to the ends of a light horizontal rod suspended by thin wire. Then he placed two large sphere with similar mass near to the small spheres. The attractive

force between smaller and larger spheres causes the rod to rotate and twist the

wire suspension. He measured the angle of rotation by the deflection of a light beam reflected from a mirror attached to the vertical suspension wire. With repeated attempts and proper calculations he calculated the gravitational constant of the earth. (Actually his actual  experiment was to measure the density of the earth with respect to water with the help of the gravitational interactions he measured).

Calculating the acceleration due to gravity of the earth:

After calculating the force due to the earth now lets calculate the acceleration with which the earth attracts every body:

from Newton’s second law

F = ma   where m is the mass of the body and a is acceleration

therefore

F = ma = (GMem)/R2

where, Me is the mass of the earth & R is the radius of earth.

therefore                                                                                                a = (GMe)/R2

putting values of the gravitational constant, mass of the earth and the radius of the earth

we get the acceleration due to gravity of earth as  9.81 ms-2. Please note here that this value is valid for objects on or near to the surface of the earth.

You might think that the astronaut in the space station are in gravity free state. This is not true. They are in state of free- fall due to the centrifugal force acting on them as they are orbiting around the earth which cancels the gravitational force.

Also there is a common misconception between the mass and weight of an object. Mass is an inherent virtue of an object while weight is the product of  acceleration due to gravity and mass. Due to this your weight will be different on different planet. Also your weight will be slightly different on equator in comparison to that of on the poles as earth is slightly flattened near the pole. (Hence the radius slightly decreases thereby increasing the acceleration due to gravity.

Lets calculate the escape velocity of the earth:

By using the law of  conservation of energy  one can easily calculate the escape velocity of the earth. Let v be the escape velocity of the earth, m be the mass of the object thrown and Me the mass of the earth

The kinetic energy of the object at earth is mv2/2. Its potential energy is (-GMem)/R where R is the radius of the earth. Its final potential energy is zero and we take the final kinetic energy also be zero. (If we throw by velocity greater than the escape velocity final kinetic will not be zero.

therefore

mv2/2  +  (-GMem)/r  = 0 + 0

v =  after putting the values the escape velocity comes out to be 11.2 km per sec.

It is to be noted here that the rockets launched in the space are not thrown with the escape velocity but they have fuel which with which they are able to move up even with less velocity.

To spaceships (like the voyager) rockets first take the ship in upper atmosphere and then give them the boost to escape velocity as there the escape velocity is less than that on earth.

Laws of Planetary motion (Kepler’s laws) :

Kepler gave the laws of planetary motion with the help of the data collected by Tycho Brahe (under whom he worked ). It is really amazing to know that these people collected data without the help of any telescope . Lets see the laws:

• Law of orbits :All planets rotate around the sun in elliptical orbits with sun at any one of the focal points. The closest distance between the planet and sun is called perihelion and the farthest distance is called aphelion.
  
• Law of equal areas: The radius vector drawn from the Sun to the planet sweeps out equal area in equal interval of time.
  With this law we can conclude that the speed with which the planet revolves around the sun is not constant. When it is nearer to the sun its speed is more than when it is far away from the sun. This is to keep the area swept by the line constant. (see the figure).
  
• Law of harmonies: The square of time period of revolution of a planet is proportional to cube of the semi-major axis of its elliptical orbit.
  From this law we can conclude that  the ratio T2/R3 is a constant for any planet or satellite