Madhava Publications
Introduction
In our daily life, we all have noticed that everything that is thrown up will fall back to earth. Going uphill is a lot more tiring than going downhills and raindrops from clouds move towards the earth. There are a lot more such phenomena. An Italian scientist Galileo recognized every object experiences a constant acceleration toward the earth, irrespective of their masses.
The stars, moon and planets have been observed since ancient times. The motion of the moon around the earth and the motion of the earth around the sun were some phenomena that needed to be explained.
In early times it was believed that the earth is the center of the universe and everything revolves around the earth. This was known as the geocentric theory. This prevails for a very long time and at that time there was not much advancement in this subject. Later Galileo and other astronomers found that the Sun is the center and not the earth, and all the planets including the earth move around the Sun. This theory was known as the heliocentric theory. After the establishment of the heliocentric theory, the rapid advancement of various subjects of Science happens. Therefore Galileo is known as the Father of Modern Science.
In this chapter, we will discuss the laws that will explain all the phenomena discussed above.
Kepler’s Law
A nobleman Tycho Brahe from Denmark spent his entire life recording observations for the planets with naked eyes. His compiled data was analyzed later by his assistant Johannes Kepler. He could extract from the data three law’s called Kepler’s Laws.
Three Laws of Kepler
- Law of the orbit: All Planets move around the sun in an elliptical orbit with the sun at one of its foci.
- Law of Area: The line that joins any planet to the sun sweeps equal areas in equal intervals of time. This law comes from the observations that planets appear to move slower when they are farther from the sun than when they are nearer.
- Law of period: The Square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse traced out by the planet. Mathematically,
The graph between the Time period and their semi-major axis is drawn below
This law is also consistent with the conservation of angular momentum.
Since the angular momentum of the planet revolving around the sun at any point of time is conserved.
The universal law of Gravitation
Kepler’s laws were known to Newton and enabled him to make a great scientific leap in proposing his universal law of gravitation.
Statement of Universal law of gravitation:
Every object in the Universe attracts every other object with a force directed along the line of the center for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects.
Where G= universal gravitational constant
The Universal Gravitational Law can explain almost anything, right from how an apple falls from a tree to why the moon revolves around the earth.
- The gravitational force is always attractive. Also the magnitude of gravitational force on mass 1 due to mass 2 is equal to the magnitude of the force on mass 2 due to mass 1.
- This law refers to point masses whereas we deal with extended objects which have finite size, so we use the concept of center of mass here. For example, The force of attraction between a hollow spherical shell of uniform density and a point mass of the shell is just as if the entire mass is concentrated at the center of the shell.
- If we have a collection of point masses the force on any one of them is the vector sum of the gravitational force exerted by the other point masses. This is actually the principle of superposition.
- The force of attraction due to a hollow spherical shell of uniform density, on a point mass situated inside it is zero. Qualitatively, we can again understand this result. Various regions of the spherical shell attract the point mass inside it in various directions. These forces cancel each other completely.
Acceleration Due to Gravity
When you throw something in the air, you notice that it goes up and then it comes down toward the earth and finally lands on the ground. The same thing will happen if you drop an object from some height. It will also land up on the ground. This happens because of the gravitational pull of the earth.
Now let's take two objects of mass m1 and m2 (where m1>m2). When we drop these masses from some height you will notice that both the masses hit the ground at the same time irrespective of their masses. So we can conclude from here that the gravitational pull of the earth on various objects near it is independent of the masses of the objects.
The conclusion from the above discussion is that earth pulls every object towards it with a force that is independent of the mass of the object. This pull is due to the gravitational force between the earth and the object.
Let the acceleration due to gravity on the object be ‘g’ , so gravitational force due to earth on an object of mass ‘m’ would be ‘mg’ which is actually equal to the weight of the object.
Let the mass of the earth is me and the radius of the earth is R, gravitational force between the earth and the object near the earth would be
From the above-derived formula of acceleration due to gravity ‘g’ is it clear that it is independent of the mass of the object and only depends on the mass of the earth and the distance of the object and the center of the earth.
When the objects are near the earth when (h<<R), then we take the radius of earth ‘R’ as the distance between the earth and the object.
But when we move to a greater height from the earth or at a certain depth inside the earth then the value of g would be different. In the next section, we would see the variation of ‘g’ with height and depth.
Variation of g with height and depth
The value of g varies with altitude and depth from the surface of the earth. We will discuss the derivation of the variation of g with depth and height. After that, we will see from the results that the value of g is maximum at the surface of the earth and decreases with both altitude and depth.
Variation of acceleration due to gravity with depth.
Suppose we have to calculate ‘g’ at the depth‘d’ from the surface of the earth. Suppose M be the mass of the whole earth and M1 be the mass of the inner sphere of the earth below the depth d as shown in the figure.
While calculating the value of g at the depth‘d’ we have to consider the gravitational force due to inner mass M1 and not the mass of the whole earth.
The acceleration due to gravity at depth d is gd, the mass of the inner sphere is M1 and the distance of it from the center of the earth is (Re-d) so we can write the value of acceleration due to gravity at depth d as
….1
Now simplify the value of M1.
Now put the value of M1 in equation 1.
So we have finally the result.
The acceleration due to gravity varies with depth as 
)
Variation of acceleration due to gravity with height ‘h’
When we want to calculate the value of acceleration due to gravity at an altitude ‘h’ where
Acceleration due to gravity at height h go is given as
So now we have results,
Acceleration due to gravity varies with height as 
The graph given below concludes the above discussion. It is clear that the value of g is maximum at the surface of the earth and decreases with both altitude and depth.
A fun thing to do: Virtual lab
Below is the link to understand the universal law of gravitation.
What can we do in this virtual lab simulation?
- We can change the masses of the objects
- We can change the distance between them
What can we observe?
- We can see the direction of the gravitational force on each body
- We can also see the magnitude of the gravitational force both in decimal and scientific notations.
