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WHAT IS GRAVITATION?
WHAT IS GRAVITATION?
We know that the moon goes around the earth. An object when thrown upwards, reaches a certain height and then falls downwards. It is said that when Newton was sitting under a tree, an apple fell on him. The fall of the apple made Newton start thinking. He thought that: if the earth can attract an apple, can it not attract the moon? Is the force the same in both cases? He conjectured that the same type of force is responsible in both cases. He argued that at each point of its orbit, the moon falls towards the earth, instead of going off in a straight line. So, it must be attracted by the earth. But we do not really see the moon falling towards the earth.
Let us try to understand the motion of the moon by recalling activity 8.11.
Activity 8.11:
* Take a piece of thread and tie a small piece of stone at one of its ends. Move the stone to describe a circular path with constant speed by holding the thread at the other end, as shown in Fig. 8.10.
Figure 8.10: A stone describing a circular path with a velocity of constant magnitude
* Now, let the stone go by releasing the thread.
* Can you tell the direction in which the stone moves after it is released?
* By repeating the activity for a few times and releasing the stone at different positions of the circular path, check whether the direction in which the stone moves remains the same or not.
Activity 10.1:
* Take a piece of thread.
* Tie a small stone at one end. Hold the other end of the thread and whirl it round, as shown in Fig. 10.1.
* Note the motion of the stone. Release the thread.
* Again, note the direction of motion of the stone.
Figure 10.1: A stone describing a circular path with a velocity of constant magnitude.
Before the thread is released, the stone moves in a circular path with a certain speed and changes direction at every point. The change in direction involves change in velocity or acceleration. The force that causes this acceleration and keeps the body moving along the circular path is acting towards the centre. This force is called the centripetal (meaning ‘center-seeking’) force. In the absence of this force, the stoneflies off along a straight line. This straight line will be a tangent to the circular path.
Tangent to a circular
A straight line that meets the circle at one and only one point is called a tangent to the circle. Straight-line ABC is a tangent to the circle at point B.
The motion of the moon around the earth is due to the centripetal force. The centripetal force is provided by the force of attraction of the earth. If there were no such force, the moon would pursue a uniform straight-line motion.
It is seen that a falling apple is attracted towards the earth. Does the apple attract the earth? If so, we do not see the earth moving towards an apple. Why?
According to the third law of motion, the apple does attract the earth. But according to the second law of motion, for a given force, acceleration is inversely proportional to the mass of an object [Eq. (9.4)]. The mass of an apple is negligibly small compared to that of the earth. So, we do not see the earth moving towards the apple. Extend the same argument for why the earth does not move towards the moon.
In our solar system, all the planets go around the Sun. By arguing the same way, we can say that there exists a force between the Sun and the planets. From the above facts, Newton concluded that not only does the earth attract an apple and the moon, but all objects in the universe attract each other. This force of attraction between objects is called the gravitational force.
Source: This topic is taken from NCERT TEXTBOOK
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UNIVERSAL LAW OF GRAVITATION
UNIVERSAL LAW OF GRAVITATION
Every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them. The force is along the line joining the centres of two objects.
Let two objects A and B of masses M and m lie at a distance d from each other as shown in Fig. 10.2.
Figure 10.2: The gravitational force between two uniform objects is directed along the line joining their centres.
Let the force of attraction between two objects be F. According to the universal law of gravitation, the force between two objects is directly proportional to the product of their masses. That is,
F \(\propto\) M \(\times\) m ________ (10.1)
And the force between two objects is inversely proportional to the square of the distance between them, that is,
F \(\propto\) \(\frac{1}{d^2}\) ________ (10.2)
Combining Eqs. (10.1) and (10.2), we get
F \(\propto\) \(\frac{M\times m}{d^2}\) ________ (10.3)
F = \(G\frac{M\times m}{d^2}\) ________ (10.4)
where G is the constant of proportionality and is called the universal gravitation constant. By multiplying crosswise, Eq. (10.4) gives
F \(\times\) d2 = G M \(\times\) m
or G = \(\frac{Fd^2}{M\times m}\) ________ (10.5)
The SI unit of G can be obtained by substituting the units of force, distance, and mass in Eq. (10.5) as Nm2kg–2.
The value of G was found out by Henry Cavendish (1731 – 1810) by using a sensitive balance. The accepted value of G is 6.673 \(\times\) 10–11 Nm2kg–2.
We know that there exists a force of attraction between any two objects. Compute the value of this force between you and your friend sitting closeby. Conclude how you do not experience this force!
More to know
Isaac newton was born in Woolsthorpe near Grantham, England. He is generally regarded as the most original and influential theorist in the history of science. He was born in a poor farming family. But he was not good at farming. He was sent to study at Cambridge University in 1661. In 1665 a plague broke out in Cambridge and so Newton took a year off. It was during this year that the incident of the apple falling on him is said to have occurred. This incident prompted Newton to explore the possibility of connecting gravity with the force that kept the moon in its orbit. This led him to the universal law of gravitation. It is remarkable that many great scientists before him knew of gravity but failed to realise it.
Newton formulated the well-known laws of motion. He worked on theories of light and colour. He designed an astronomical telescope to carry out astronomical observations. Newton was also a great mathematician. He invented a new branch of mathematics, called calculus. He used it to prove that for objects outside a sphere of uniform density, the sphere behaves as if the whole of its mass is concentrated at its centre. Newton transformed the structure of physical science with his three laws of motion and the universal law of gravitation. As the keystone of the scientific revolution of the seventeenth century, Newton’s work combined the contributions of Copernicus, Kepler, Galileo, and others into a new powerful synthesis.
It is remarkable that though the gravitational theory could not be verified at that time, there was hardly any doubt about its correctness. This is because Newton based his theory on sound scientific reasoning and backed it with mathematics. This made the theory simple and elegant. These qualities are now recognised as essential requirements of a good scientific theory.
How did Newton guess the Inverse-square rule?
There has always been a great interest in the motion of planets. By the 16th century, a lot of data on the motion of planets had been collected by many astronomers. Based on these data Johannes Kepler derived three laws, which govern the motion of planets. These are called Kepler’s laws.
These are:
1. The orbit of a planet is an ellipse with the Sun at one of the foci, as shown in the figure given below. In this figure O is the position of the Sun.
2. The line joining the planet and the Sun sweep equal areas in equal intervals of time. Thus, if the time of travel from A to B is the same as that from C to D, then the areas OAB and OCD are equal.
3. The cube of the mean distance of a planet from the Sun is proportional to the square of its orbital period T. Or, r3/T2 = constant.
It is important to note that Kepler could not give a theory to explain the motion of planets. It was Newton who showed that the cause of the planetary motion is the gravitational force that the Sun exerts on them. Newton used the third law of Kepler to calculate the gravitational force of attraction. The gravitational force of the earth is weakened by distance. A simple argument goes like this. We can assume that the planetary orbits are circular. Suppose the orbital velocity is v and the radius of the orbit is r. Then the force acting on an orbiting planet is given by F \(\propto\) v2/r.
If T denotes the period, then v = 2\(\pi\)r/T,
so that v2 \(\propto\) r2/T2.
We can rewrite this as v2 \(\propto\) (1/r) \(\times\) (r3/T2). Since r3/T2 is constant by Kepler’s third law, we have v2 \(\propto\) 1/r. Combining this with F \(\propto\) v2/ r, we get F \(\propto\) 1/ r2.
Inverse-square
The law is universal in the sense that it is applicable to all bodies, whether the bodies are big or small, whether they are celestial or terrestrial. Saying that F is inversely proportional to the square of d means, for example, that if d gets bigger by a factor of 6, F becomes \(\frac{1}{36}\) times smaller.
Illustration 10.1:
The mass of the earth is 6 \(\times\)1024 kg and that of the moon is 7.4 × 1022 kg. If the distance between the earth and the moon is 3.84×105 km, calculate the force exerted by the earth on the moon. (Take G = 6.7 × 10–11 Nm2kg-2)
Sol:
The mass of the earth, M = 6 × 1024 kg The mass of the moon,
m = 7.4 × 1022 kg
The distance between the earth and the moon,
d = 3.84 × 105 km
= 3.84 × 105 × 1000 m
= 3.84 × 108 m
G = 6.7 × 10–11 N m2 kg–2
From Eq. (10.4), the force exerted by the earth on the moon is
\(F=G\frac{M\times m}{d^2}\\ =\frac{6.7\times 10^{-11}Nm^2kg^{-2} \times6\times 10^{24}kg\times7.4\times10^{22}kg}{(3.84\times 10^8m)^2}\\=2.02\times 10^{20}N\)
Thus, the force exerted by the earth on the moon is 2.02 \(\times\)1020 N.
Questions
1. State the universal law of gravitation.
2. Write the formula to find the magnitude of the gravitational force between the earth and an object on the surface of the earth
Source: This topic is taken from NCERT TEXTBOOK
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IMPORTANCE OF UNIVERSAL LAW OF GRAVITATION
IMPORTANCE OF UNIVERSAL LAW OF GRAVITATION
The universal law of gravitation successfully explained several phenomena which were believed to be unconnected:
i. the force that binds us to the earth;
ii. the motion of the moon around the earth;
iii. the motion of planets around the Sun; and
iv. the tides due to the moon and the Sun.
Source: This topic is taken from NCERT TEXTBOOK