Page 1 In Chapters 8 and 9, we have learnt about the motion of objects and force as the cause of motion. We have learnt that a force is needed to change the speed or the direction of motion of an object. We always observe that an object dropped from a height falls towards the earth. We know that all the planets go around the Sun. The moon goes around the earth. In all these cases, there must be some force acting on the objects, the planets and on the moon. Isaac Newton could grasp that the same force is responsible for all these. This force is called the gravitational force. In this chapter we shall learn about gravitation and the universal law of gravitation. We shall discuss the motion of objects under the influence of gravitational force on the earth. We shall study how the weight of a body varies from place to place. We shall also discuss the conditions for objects to float in liquids. 10.1Gravitation 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 the 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 _____________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. Fig. 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 ‘centreseeking’) force. In the absence of this 10 G G G G GRAVITATION RAVITATION RAVITATION RAVITATION RAVITATION Chapter 202021 Page 2 In Chapters 8 and 9, we have learnt about the motion of objects and force as the cause of motion. We have learnt that a force is needed to change the speed or the direction of motion of an object. We always observe that an object dropped from a height falls towards the earth. We know that all the planets go around the Sun. The moon goes around the earth. In all these cases, there must be some force acting on the objects, the planets and on the moon. Isaac Newton could grasp that the same force is responsible for all these. This force is called the gravitational force. In this chapter we shall learn about gravitation and the universal law of gravitation. We shall discuss the motion of objects under the influence of gravitational force on the earth. We shall study how the weight of a body varies from place to place. We shall also discuss the conditions for objects to float in liquids. 10.1Gravitation 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 the 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 _____________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. Fig. 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 ‘centreseeking’) force. In the absence of this 10 G G G G GRAVITATION RAVITATION RAVITATION RAVITATION RAVITATION Chapter 202021 SCIENCE 132 10.1.1 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. force, the stone flies off along a straight line. This straight line will be a tangent to the circular path. More to know Tangent to a circle 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. G 2 Mm F = d Fig. 10.2: The gravitational force between two uniform objects is directed along the line joining their centres. 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. 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 ? M × m (10.1) And the force between two objects is inversely proportional to the square of the distance between them, that is, ? 2 1 F d (10.2) Combining Eqs. (10.1) and (10.2), we get F ? 2 × M m d (10.3) or, G 2 M × m F = d (10.4) where G is the constant of proportionality and is called the universal gravitation constant. By multiplying crosswise, Eq. (10.4) gives F × d 2 = G M × m 202021 Page 3 In Chapters 8 and 9, we have learnt about the motion of objects and force as the cause of motion. We have learnt that a force is needed to change the speed or the direction of motion of an object. We always observe that an object dropped from a height falls towards the earth. We know that all the planets go around the Sun. The moon goes around the earth. In all these cases, there must be some force acting on the objects, the planets and on the moon. Isaac Newton could grasp that the same force is responsible for all these. This force is called the gravitational force. In this chapter we shall learn about gravitation and the universal law of gravitation. We shall discuss the motion of objects under the influence of gravitational force on the earth. We shall study how the weight of a body varies from place to place. We shall also discuss the conditions for objects to float in liquids. 10.1Gravitation 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 the 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 _____________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. Fig. 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 ‘centreseeking’) force. In the absence of this 10 G G G G GRAVITATION RAVITATION RAVITATION RAVITATION RAVITATION Chapter 202021 SCIENCE 132 10.1.1 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. force, the stone flies off along a straight line. This straight line will be a tangent to the circular path. More to know Tangent to a circle 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. G 2 Mm F = d Fig. 10.2: The gravitational force between two uniform objects is directed along the line joining their centres. 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. 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 ? M × m (10.1) And the force between two objects is inversely proportional to the square of the distance between them, that is, ? 2 1 F d (10.2) Combining Eqs. (10.1) and (10.2), we get F ? 2 × M m d (10.3) or, G 2 M × m F = d (10.4) where G is the constant of proportionality and is called the universal gravitation constant. By multiplying crosswise, Eq. (10.4) gives F × d 2 = G M × m 202021 GRAVITATION 133 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 wellknown 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. Isaac Newton (1642 – 1727) How did Newton guess the inversesquare 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, r 3 /T 2 = 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 ? v 2 /r. If T denotes the period, then v = 2pr/T, so that v 2 ? r 2 /T 2 . We can rewrite this as v 2 ? (1/r) × ( r 3 /T 2 ). Since r 3 /T 2 is constant by Kepler’s third law, we have v 2 ? 1/r. Combining this with F ? v 2 / r, we get, F ? 1/ r 2 . A B C D O 202021 Page 4 In Chapters 8 and 9, we have learnt about the motion of objects and force as the cause of motion. We have learnt that a force is needed to change the speed or the direction of motion of an object. We always observe that an object dropped from a height falls towards the earth. We know that all the planets go around the Sun. The moon goes around the earth. In all these cases, there must be some force acting on the objects, the planets and on the moon. Isaac Newton could grasp that the same force is responsible for all these. This force is called the gravitational force. In this chapter we shall learn about gravitation and the universal law of gravitation. We shall discuss the motion of objects under the influence of gravitational force on the earth. We shall study how the weight of a body varies from place to place. We shall also discuss the conditions for objects to float in liquids. 10.1Gravitation 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 the 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 _____________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. Fig. 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 ‘centreseeking’) force. In the absence of this 10 G G G G GRAVITATION RAVITATION RAVITATION RAVITATION RAVITATION Chapter 202021 SCIENCE 132 10.1.1 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. force, the stone flies off along a straight line. This straight line will be a tangent to the circular path. More to know Tangent to a circle 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. G 2 Mm F = d Fig. 10.2: The gravitational force between two uniform objects is directed along the line joining their centres. 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. 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 ? M × m (10.1) And the force between two objects is inversely proportional to the square of the distance between them, that is, ? 2 1 F d (10.2) Combining Eqs. (10.1) and (10.2), we get F ? 2 × M m d (10.3) or, G 2 M × m F = d (10.4) where G is the constant of proportionality and is called the universal gravitation constant. By multiplying crosswise, Eq. (10.4) gives F × d 2 = G M × m 202021 GRAVITATION 133 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 wellknown 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. Isaac Newton (1642 – 1727) How did Newton guess the inversesquare 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, r 3 /T 2 = 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 ? v 2 /r. If T denotes the period, then v = 2pr/T, so that v 2 ? r 2 /T 2 . We can rewrite this as v 2 ? (1/r) × ( r 3 /T 2 ). Since r 3 /T 2 is constant by Kepler’s third law, we have v 2 ? 1/r. Combining this with F ? v 2 / r, we get, F ? 1/ r 2 . A B C D O 202021 SCIENCE 134 From Eq. (10.4), the force exerted by the earth on the moon is G 2 M × m F = d 11 2 2 24 22 8 2 6.7 10 N m kg 6 10 kg 7.4 10 kg (3.84 10 m)  × × × × × = × = 2.02 × 10 20 N. Thus, the force exerted by the earth on the moon is 2.02 × 10 20 N. uestions 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. 10.1.2 IMPORTANCE OF THE 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. 10.2 Free Fall Let us try to understand the meaning of free fall by performing this activity. Activity _____________10.2 • Take a stone. • Throw it upwards. • It reaches a certain height and then it starts falling down. We have learnt that the earth attracts objects towards it. This is due to the gravitational force. Whenever objects fall towards the earth under this force alone, we say that the objects are in free fall. Is there any or 2 G= × F d M 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 N m 2 kg –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 × 10 –11 N m 2 kg –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! 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. Inversesquare 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 1 36 times smaller. Example 10.1 The mass of the earth is 6 × 10 24 kg and that of the moon is 7.4 × 10 22 kg. If the distance between the earth and the moon is 3.84×10 5 km, calculate the force exerted by the earth on the moon. (Take G = 6.7 × 10 –11 N m 2 kg 2 ) Solution: The mass of the earth, M = 6 × 10 24 kg The mass of the moon, m = 7.4 × 10 22 kg The distance between the earth and the moon, d = 3.84 × 10 5 km = 3.84 × 10 5 × 1000 m = 3.84 × 10 8 m G = 6.7 × 10 –11 N m 2 kg –2 More to know Q 202021 Page 5 In Chapters 8 and 9, we have learnt about the motion of objects and force as the cause of motion. We have learnt that a force is needed to change the speed or the direction of motion of an object. We always observe that an object dropped from a height falls towards the earth. We know that all the planets go around the Sun. The moon goes around the earth. In all these cases, there must be some force acting on the objects, the planets and on the moon. Isaac Newton could grasp that the same force is responsible for all these. This force is called the gravitational force. In this chapter we shall learn about gravitation and the universal law of gravitation. We shall discuss the motion of objects under the influence of gravitational force on the earth. We shall study how the weight of a body varies from place to place. We shall also discuss the conditions for objects to float in liquids. 10.1Gravitation 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 the 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 _____________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. Fig. 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 ‘centreseeking’) force. In the absence of this 10 G G G G GRAVITATION RAVITATION RAVITATION RAVITATION RAVITATION Chapter 202021 SCIENCE 132 10.1.1 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. force, the stone flies off along a straight line. This straight line will be a tangent to the circular path. More to know Tangent to a circle 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. G 2 Mm F = d Fig. 10.2: The gravitational force between two uniform objects is directed along the line joining their centres. 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. 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 ? M × m (10.1) And the force between two objects is inversely proportional to the square of the distance between them, that is, ? 2 1 F d (10.2) Combining Eqs. (10.1) and (10.2), we get F ? 2 × M m d (10.3) or, G 2 M × m F = d (10.4) where G is the constant of proportionality and is called the universal gravitation constant. By multiplying crosswise, Eq. (10.4) gives F × d 2 = G M × m 202021 GRAVITATION 133 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 wellknown 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. Isaac Newton (1642 – 1727) How did Newton guess the inversesquare 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, r 3 /T 2 = 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 ? v 2 /r. If T denotes the period, then v = 2pr/T, so that v 2 ? r 2 /T 2 . We can rewrite this as v 2 ? (1/r) × ( r 3 /T 2 ). Since r 3 /T 2 is constant by Kepler’s third law, we have v 2 ? 1/r. Combining this with F ? v 2 / r, we get, F ? 1/ r 2 . A B C D O 202021 SCIENCE 134 From Eq. (10.4), the force exerted by the earth on the moon is G 2 M × m F = d 11 2 2 24 22 8 2 6.7 10 N m kg 6 10 kg 7.4 10 kg (3.84 10 m)  × × × × × = × = 2.02 × 10 20 N. Thus, the force exerted by the earth on the moon is 2.02 × 10 20 N. uestions 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. 10.1.2 IMPORTANCE OF THE 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. 10.2 Free Fall Let us try to understand the meaning of free fall by performing this activity. Activity _____________10.2 • Take a stone. • Throw it upwards. • It reaches a certain height and then it starts falling down. We have learnt that the earth attracts objects towards it. This is due to the gravitational force. Whenever objects fall towards the earth under this force alone, we say that the objects are in free fall. Is there any or 2 G= × F d M 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 N m 2 kg –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 × 10 –11 N m 2 kg –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! 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. Inversesquare 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 1 36 times smaller. Example 10.1 The mass of the earth is 6 × 10 24 kg and that of the moon is 7.4 × 10 22 kg. If the distance between the earth and the moon is 3.84×10 5 km, calculate the force exerted by the earth on the moon. (Take G = 6.7 × 10 –11 N m 2 kg 2 ) Solution: The mass of the earth, M = 6 × 10 24 kg The mass of the moon, m = 7.4 × 10 22 kg The distance between the earth and the moon, d = 3.84 × 10 5 km = 3.84 × 10 5 × 1000 m = 3.84 × 10 8 m G = 6.7 × 10 –11 N m 2 kg –2 More to know Q 202021 GRAVITATION 135 calculations, we can take g to be more or less constant on or near the earth. But for objects far from the earth, the acceleration due to gravitational force of earth is given by Eq. (10.7). 10.2.1 TO CALCULATE THE VALUE OF g To calculate the value of g, we should put the values of G, M and R in Eq. (10.9), namely, universal gravitational constant, G = 6.7 × 10 –11 N m 2 kg 2 , mass of the earth, M = 6 × 10 24 kg, and radius of the earth, R = 6.4 × 10 6 m. G 2 M g = R 11 2 2 24 6 2 6.7 10 N m kg 6 10 kg = (6.4 10 m) × × × × = 9.8 m s –2 . Thus, the value of acceleration due to gravity of the earth, g = 9.8 m s –2 . 10.2.2 MOTION OF OBJECTS UNDER THE INFLUENCE OF GRAVITATIONAL FORCE OF THE EARTH Let us do an activity to understand whether all objects hollow or solid, big or small, will fall from a height at the same rate. Activity _____________10.3 • Take a sheet of paper and a stone. Drop them simultaneously from the first floor of a building. Observe whether both of them reach the ground simultaneously. • We see that paper reaches the ground little later than the stone. This happens because of air resistance. The air offers resistance due to friction to the motion of the falling objects. The resistance offered by air to the paper is more than the resistance offered to the stone. If we do the experiment in a glass jar from which air has been sucked out, the paper and the stone would fall at the same rate. change in the velocity of falling objects? While falling, there is no change in the direction of motion of the objects. But due to the earth’s attraction, there will be a change in the magnitude of the velocity. Any change in velocity involves acceleration. Whenever an object falls towards the earth, an acceleration is involved. This acceleration is due to the earth’s gravitational force. Therefore, this acceleration is called the acceleration due to the gravitational force of the earth (or acceleration due to gravity). It is denoted by g. The unit of g is the same as that of acceleration, that is, m s –2 . We know from the second law of motion that force is the product of mass and acceleration. Let the mass of the stone in activity 10.2 be m. We already know that there is acceleration involved in falling objects due to the gravitational force and is denoted by g. Therefore the magnitude of the gravitational force F will be equal to the product of mass and acceleration due to the gravitational force, that is, F = m g (10.6) From Eqs. (10.4) and (10.6) we have 2 = G × M m m g d or G 2 M g = d (10.7) where M is the mass of the earth, and d is the distance between the object and the earth. Let an object be on or near the surface of the earth. The distance d in Eq. (10.7) will be equal to R, the radius of the earth. Thus, for objects on or near the surface of the earth, G 2 M × m mg = R (10.8) G 2 M g = R (10.9) The earth is not a perfect sphere. As the radius of the earth increases from the poles to the equator, the value of g becomes greater at the poles than at the equator. For most 202021Read More
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