Law of gravitation and acceleration due to gravity
In our universe every object attracts each other. There is no exception to it. They all attract each of them along the straight line joining them. The force by which they attract is called the force of gravitation. In his book ‘Mathematical Principles of Natural Philosophy Sir Isaac Newton gave this law which is a general physics law, derived from direct observation of various natural phenomena. This law is valid for subatomic particles to huge astronomical objects.
The force of gravity is a central force which is directed towards a point. Here that point is the centre of the earth. Central force is negative if it is attractive and it is positive if it is repulsive in nature.
The law states that all objects in the universe attracts each by a force which is directly proportional to their masses and inversely proportional to the square of the straight-line distance between them.
Which is given mathematically as
when r is constant.
when masses are constant.
when all are variable.
The actual form of this law is given by adding a minus sign before RHS of the above equation.
The minus sign is due to the fact that the force is of attractive nature.
Here G is the proportionality constant called the universal gravitational constant. British scientist Henry Cavendish did an experiment and calculated the value of G is given by 6.67×10-11 N-m2/kg2 which is excepted worldwide.
The earth is an oblate sphere made of a large number of concentric spherical shells with increasing radius until the surface. Any point just outside the earth is obviously out of these shells. Hence the shells exert a gravitational force at that point outside just. 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 centre.The shape of earth is considered to be a sphere and the force of gravity is the force by which the earth pulls an object of mass m, which is also known as the weight of that object thus,
, where M is the mass of the earth and r is the distance between earths centre and the object. Here we are interested with the magnitude only since the sing here only indicates direction.
Here g is known as the acceleration due to gravity. The value of g is 9.8 N in SI system. The point to remember here is that this value of g is just at the surface of the earth and which varies with distance below or above the surface.
If an object is above the earth’s surface at height h above, then the effective distance between the object and the centre of the earth is increased by h. Hence
F effectively becomes
Hence the value of g becomes
And if the object is at a depth h below the earth’s surface then
Hence the value becomes
Problem solving video on “Acceleration due to gravity at a height h”