Page 1 CHAPTER EIGHT GRAVITATION 8.1 INTRODUCTION Early in our lives, we become aware of the tendency of all material objects to be attracted towards the earth. Anything thrown up falls down towards the earth, going uphill is lot more tiring than going downhill, raindrops from the clouds above fall towards the earth and there are many other such phenomena. Historically it was the Italian Physicist Galileo (1564-1642) who recognised the fact that all bodies, irrespective of their masses, are accelerated towards the earth with a constant acceleration. It is said that he made a public demonstration of this fact. To find the truth, he certainly did experiments with bodies rolling down inclined planes and arrived at a value of the acceleration due to gravity which is close to the more accurate value obtained later. A seemingly unrelated phenomenon, observation of stars, planets and their motion has been the subject of attention in many countries since the earliest of times. Observations since early times recognised stars which appeared in the sky with positions unchanged year after year. The more interesting objects are the planets which seem to have regular motions against the background of stars. The earliest recorded model for planetary motions proposed by Ptolemy about 2000 years ago was a ‘geocentric’ model in which all celestial objects, stars, the sun and the planets, all revolved around the earth. The only motion that was thought to be possible for celestial objects was motion in a circle. Complicated schemes of motion were put forward by Ptolemy in order to describe the observed motion of the planets. The planets were described as moving in circles with the centre of the circles themselves moving in larger circles. Similar theories were also advanced by Indian astronomers some 400 years later. However a more elegant model in which the Sun was the centre around which the planets revolved – the ‘heliocentric’ model – was already mentioned by Aryabhatta (5 th century A.D.) in his treatise. A thousand years later, a Polish monk named Nicolas 8.1 Introduction 8.2 Kepler’s laws 8.3 Universal law of gravitation 8.4 The gravitational constant 8.5 Acceleration due to gravity of the earth 8.6 Acceleration due to gravity below and above the surface of earth 8.7 Gravitational potential energy 8.8 Escape speed 8.9 Earth satellites 8.10 Energy of an orbiting satellite 8.11 Geostationary and polar satellites 8.12 Weightlessness Summary Points to ponder Exercises Additional exercises 2020-21 Page 2 CHAPTER EIGHT GRAVITATION 8.1 INTRODUCTION Early in our lives, we become aware of the tendency of all material objects to be attracted towards the earth. Anything thrown up falls down towards the earth, going uphill is lot more tiring than going downhill, raindrops from the clouds above fall towards the earth and there are many other such phenomena. Historically it was the Italian Physicist Galileo (1564-1642) who recognised the fact that all bodies, irrespective of their masses, are accelerated towards the earth with a constant acceleration. It is said that he made a public demonstration of this fact. To find the truth, he certainly did experiments with bodies rolling down inclined planes and arrived at a value of the acceleration due to gravity which is close to the more accurate value obtained later. A seemingly unrelated phenomenon, observation of stars, planets and their motion has been the subject of attention in many countries since the earliest of times. Observations since early times recognised stars which appeared in the sky with positions unchanged year after year. The more interesting objects are the planets which seem to have regular motions against the background of stars. The earliest recorded model for planetary motions proposed by Ptolemy about 2000 years ago was a ‘geocentric’ model in which all celestial objects, stars, the sun and the planets, all revolved around the earth. The only motion that was thought to be possible for celestial objects was motion in a circle. Complicated schemes of motion were put forward by Ptolemy in order to describe the observed motion of the planets. The planets were described as moving in circles with the centre of the circles themselves moving in larger circles. Similar theories were also advanced by Indian astronomers some 400 years later. However a more elegant model in which the Sun was the centre around which the planets revolved – the ‘heliocentric’ model – was already mentioned by Aryabhatta (5 th century A.D.) in his treatise. A thousand years later, a Polish monk named Nicolas 8.1 Introduction 8.2 Kepler’s laws 8.3 Universal law of gravitation 8.4 The gravitational constant 8.5 Acceleration due to gravity of the earth 8.6 Acceleration due to gravity below and above the surface of earth 8.7 Gravitational potential energy 8.8 Escape speed 8.9 Earth satellites 8.10 Energy of an orbiting satellite 8.11 Geostationary and polar satellites 8.12 Weightlessness Summary Points to ponder Exercises Additional exercises 2020-21 184 PHYSICS B A C P S S' 2b 2a Copernicus (1473-1543) proposed a definitive model in which the planets moved in circles around a fixed central sun. His theory was discredited by the church, but notable amongst its supporters was Galileo who had to face prosecution from the state for his beliefs. It was around the same time as Galileo, a nobleman called Tycho Brahe (1546-1601) hailing from Denmark, spent his entire lifetime recording observations of the planets with the naked eye. His compiled data were analysed later by his assistant Johannes Kepler (1571- 1640). He could extract from the data three elegant laws that now go by the name of Kepler’s laws. These laws were known to Newton and enabled him to make a great scientific leap in proposing his universal law of gravitation. 8.2 KEPLER’S LAWS The three laws of Kepler can be stated as follows: 1. Law of orbits : All planets move in elliptical orbits with the Sun situated at one of the foci Fig. 8.1(a) An ellipse traced out by a planet around the sun. The closest point is P and the farthest point is A, P is called the perihelion and A the aphelion. The semimajor axis is half the distance AP. Fig. 8.1(b) Drawing an ellipse. A string has its ends fixed at F 1 and F 2 . The tip of a pencil holds the string taut and is moved around. of the ellipse (Fig. 8.1a). This law was a deviation from the Copernican model which allowed only circular orbits. The ellipse, of which the circle is a special case, is a closed curve which can be drawn very simply as follows. Select two points F 1 and F 2 . Take a length of a string and fix its ends at F 1 and F 2 by pins. With the tip of a pencil stretch the string taut and then draw a curve by moving the pencil keeping the string taut throughout.(Fig. 8.1(b)) The closed curve you get is called an ellipse. Clearly for any point T on the ellipse, the sum of the distances from F 1 and F 2 is a constant. F 1 , F 2 are called the focii. Join the points F 1 and F 2 and extend the line to intersect the ellipse at points P and A as shown in Fig. 8.1(b). The midpoint of the line PA is the centre of the ellipse O and the length PO = AO is called the semi- major axis of the ellipse. For a circle, the two focii merge into one and the semi-major axis becomes the radius of the circle. 2. Law of areas : The line that joins any planet to the sun sweeps equal areas in equal intervals of time (Fig. 8.2). This law comes from the observations that planets appear to move slower when they are farther from the sun than when they are nearer. Fig. 8.2 The planet P moves around the sun in an elliptical orbit. The shaded area is the area ?A swept out in a small interval of time ?t. 3. Law of periods : 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. Table 8.1 gives the approximate time periods of revolution of eight* planets around the sun along with values of their semi-major axes. * Refer to information given in the Box on Page 182 2020-21 Page 3 CHAPTER EIGHT GRAVITATION 8.1 INTRODUCTION Early in our lives, we become aware of the tendency of all material objects to be attracted towards the earth. Anything thrown up falls down towards the earth, going uphill is lot more tiring than going downhill, raindrops from the clouds above fall towards the earth and there are many other such phenomena. Historically it was the Italian Physicist Galileo (1564-1642) who recognised the fact that all bodies, irrespective of their masses, are accelerated towards the earth with a constant acceleration. It is said that he made a public demonstration of this fact. To find the truth, he certainly did experiments with bodies rolling down inclined planes and arrived at a value of the acceleration due to gravity which is close to the more accurate value obtained later. A seemingly unrelated phenomenon, observation of stars, planets and their motion has been the subject of attention in many countries since the earliest of times. Observations since early times recognised stars which appeared in the sky with positions unchanged year after year. The more interesting objects are the planets which seem to have regular motions against the background of stars. The earliest recorded model for planetary motions proposed by Ptolemy about 2000 years ago was a ‘geocentric’ model in which all celestial objects, stars, the sun and the planets, all revolved around the earth. The only motion that was thought to be possible for celestial objects was motion in a circle. Complicated schemes of motion were put forward by Ptolemy in order to describe the observed motion of the planets. The planets were described as moving in circles with the centre of the circles themselves moving in larger circles. Similar theories were also advanced by Indian astronomers some 400 years later. However a more elegant model in which the Sun was the centre around which the planets revolved – the ‘heliocentric’ model – was already mentioned by Aryabhatta (5 th century A.D.) in his treatise. A thousand years later, a Polish monk named Nicolas 8.1 Introduction 8.2 Kepler’s laws 8.3 Universal law of gravitation 8.4 The gravitational constant 8.5 Acceleration due to gravity of the earth 8.6 Acceleration due to gravity below and above the surface of earth 8.7 Gravitational potential energy 8.8 Escape speed 8.9 Earth satellites 8.10 Energy of an orbiting satellite 8.11 Geostationary and polar satellites 8.12 Weightlessness Summary Points to ponder Exercises Additional exercises 2020-21 184 PHYSICS B A C P S S' 2b 2a Copernicus (1473-1543) proposed a definitive model in which the planets moved in circles around a fixed central sun. His theory was discredited by the church, but notable amongst its supporters was Galileo who had to face prosecution from the state for his beliefs. It was around the same time as Galileo, a nobleman called Tycho Brahe (1546-1601) hailing from Denmark, spent his entire lifetime recording observations of the planets with the naked eye. His compiled data were analysed later by his assistant Johannes Kepler (1571- 1640). He could extract from the data three elegant laws that now go by the name of Kepler’s laws. These laws were known to Newton and enabled him to make a great scientific leap in proposing his universal law of gravitation. 8.2 KEPLER’S LAWS The three laws of Kepler can be stated as follows: 1. Law of orbits : All planets move in elliptical orbits with the Sun situated at one of the foci Fig. 8.1(a) An ellipse traced out by a planet around the sun. The closest point is P and the farthest point is A, P is called the perihelion and A the aphelion. The semimajor axis is half the distance AP. Fig. 8.1(b) Drawing an ellipse. A string has its ends fixed at F 1 and F 2 . The tip of a pencil holds the string taut and is moved around. of the ellipse (Fig. 8.1a). This law was a deviation from the Copernican model which allowed only circular orbits. The ellipse, of which the circle is a special case, is a closed curve which can be drawn very simply as follows. Select two points F 1 and F 2 . Take a length of a string and fix its ends at F 1 and F 2 by pins. With the tip of a pencil stretch the string taut and then draw a curve by moving the pencil keeping the string taut throughout.(Fig. 8.1(b)) The closed curve you get is called an ellipse. Clearly for any point T on the ellipse, the sum of the distances from F 1 and F 2 is a constant. F 1 , F 2 are called the focii. Join the points F 1 and F 2 and extend the line to intersect the ellipse at points P and A as shown in Fig. 8.1(b). The midpoint of the line PA is the centre of the ellipse O and the length PO = AO is called the semi- major axis of the ellipse. For a circle, the two focii merge into one and the semi-major axis becomes the radius of the circle. 2. Law of areas : The line that joins any planet to the sun sweeps equal areas in equal intervals of time (Fig. 8.2). This law comes from the observations that planets appear to move slower when they are farther from the sun than when they are nearer. Fig. 8.2 The planet P moves around the sun in an elliptical orbit. The shaded area is the area ?A swept out in a small interval of time ?t. 3. Law of periods : 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. Table 8.1 gives the approximate time periods of revolution of eight* planets around the sun along with values of their semi-major axes. * Refer to information given in the Box on Page 182 2020-21 GRAVITATION 185 t Table 8.1Data from measurement of planetary motions given below confirm Kepler’s Law of Periods (a = Semi-major axis in units of 10 10 m. T = Time period of revolution of the planet in years(y). Q = The quotient ( T 2 /a 3 ) in units of 10 -34 y 2 m -3 .) Planet a T Q Mercury 5.79 0.24 2.95 Venus 10.8 0.615 3.00 Earth 15.0 1 2.96 Mars 22.8 1.88 2.98 Jupiter 77.8 11.9 3.01 Saturn 143 29.5 2.98 Uranus 287 84 2.98 Neptune 450 165 2.99 Pluto* 590 248 2.99 The law of areas can be understood as a consequence of conservation of angular momentum whch is valid for any central force . A central force is such that the force on the planet is along the vector joining the Sun and the planet. Let the Sun be at the origin and let the position and momentum of the planet be denoted by r and p respectively. Then the area swept out by the planet of mass m in time interval ?t is (Fig. 8.2) ?A given by ?A = ½ (r × v?t) (8.1) Hence ?A /?t =½ (r × p)/m, (since v = p/m) = L / (2 m) (8.2) where v is the velocity, L is the angular momentum equal to ( r × p). For a central force, which is directed along r, L is a constant as the planet goes around. Hence, ? A /?t is a constant according to the last equation. This is the law of areas. Gravitation is a central force and hence the law of areas follows. Example 8.1 Let the speed of the planet at the perihelion P in Fig. 8.1(a) be v P and the Sun-planet distance SP be r P . Relate {r P , v P } to the corresponding quantities at the aphelion {r A, v A }. Will the planet take equal times to traverse BAC and CPB ? Answer The magnitude of the angular momentum at P is L p = m p r p v p , since inspection tells us that r p and v p are mutually perpendicular. Similarly, L A = m p r A v A . From angular momentum conservation m p r p v p = m p r A v A or v v p A = r r A p t Since r A > r p , v p > v A . The area SBAC bounded by the ellipse and the radius vectors SB and SC is larger than SBPC in Fig. 8.1. From Kepler’s second law, equal areas are swept in equal times. Hence the planet will take a longer time to traverse BAC than CPB. 8.3 UNIVERSAL LAW OF GRAVITATION Legend has it that observing an apple falling from a tree, Newton was inspired to arrive at an universal law of gravitation that led to an explanation of terrestrial gravitation as well as of Kepler’s laws. Newton’s reasoning was that the moon revolving in an orbit of radius R m was subject to a centripetal acceleration due to earth’s gravity of magnitude 2 2 2 4 m m m R V a R T p = = (8.3) where V is the speed of the moon related to the time period T by the relation 2 / m V R T p = . The time period T is about 27.3 days and R m was already known then to be about 3.84 × 10 8 m. If we substitute these numbers in Eq. (8.3), we get a value of a m much smaller than the value of acceleration due to gravity g on the surface of the earth, arising also due to earth’s gravitational attraction. Johannes Kepler (1571–1630) was a scientist of German origin. He formulated the three laws of planetary motion based on the painstaking observations of Tycho Brahe and coworkers. Kepler himself was an assistant to Brahe and it took him sixteen long years to arrive at the three planetary laws. He is also known as the founder of geometrical optics, being the first to describe what happens to light after it enters a telescope. * Refer to information given in the Box on Page 182 2020-21 Page 4 CHAPTER EIGHT GRAVITATION 8.1 INTRODUCTION Early in our lives, we become aware of the tendency of all material objects to be attracted towards the earth. Anything thrown up falls down towards the earth, going uphill is lot more tiring than going downhill, raindrops from the clouds above fall towards the earth and there are many other such phenomena. Historically it was the Italian Physicist Galileo (1564-1642) who recognised the fact that all bodies, irrespective of their masses, are accelerated towards the earth with a constant acceleration. It is said that he made a public demonstration of this fact. To find the truth, he certainly did experiments with bodies rolling down inclined planes and arrived at a value of the acceleration due to gravity which is close to the more accurate value obtained later. A seemingly unrelated phenomenon, observation of stars, planets and their motion has been the subject of attention in many countries since the earliest of times. Observations since early times recognised stars which appeared in the sky with positions unchanged year after year. The more interesting objects are the planets which seem to have regular motions against the background of stars. The earliest recorded model for planetary motions proposed by Ptolemy about 2000 years ago was a ‘geocentric’ model in which all celestial objects, stars, the sun and the planets, all revolved around the earth. The only motion that was thought to be possible for celestial objects was motion in a circle. Complicated schemes of motion were put forward by Ptolemy in order to describe the observed motion of the planets. The planets were described as moving in circles with the centre of the circles themselves moving in larger circles. Similar theories were also advanced by Indian astronomers some 400 years later. However a more elegant model in which the Sun was the centre around which the planets revolved – the ‘heliocentric’ model – was already mentioned by Aryabhatta (5 th century A.D.) in his treatise. A thousand years later, a Polish monk named Nicolas 8.1 Introduction 8.2 Kepler’s laws 8.3 Universal law of gravitation 8.4 The gravitational constant 8.5 Acceleration due to gravity of the earth 8.6 Acceleration due to gravity below and above the surface of earth 8.7 Gravitational potential energy 8.8 Escape speed 8.9 Earth satellites 8.10 Energy of an orbiting satellite 8.11 Geostationary and polar satellites 8.12 Weightlessness Summary Points to ponder Exercises Additional exercises 2020-21 184 PHYSICS B A C P S S' 2b 2a Copernicus (1473-1543) proposed a definitive model in which the planets moved in circles around a fixed central sun. His theory was discredited by the church, but notable amongst its supporters was Galileo who had to face prosecution from the state for his beliefs. It was around the same time as Galileo, a nobleman called Tycho Brahe (1546-1601) hailing from Denmark, spent his entire lifetime recording observations of the planets with the naked eye. His compiled data were analysed later by his assistant Johannes Kepler (1571- 1640). He could extract from the data three elegant laws that now go by the name of Kepler’s laws. These laws were known to Newton and enabled him to make a great scientific leap in proposing his universal law of gravitation. 8.2 KEPLER’S LAWS The three laws of Kepler can be stated as follows: 1. Law of orbits : All planets move in elliptical orbits with the Sun situated at one of the foci Fig. 8.1(a) An ellipse traced out by a planet around the sun. The closest point is P and the farthest point is A, P is called the perihelion and A the aphelion. The semimajor axis is half the distance AP. Fig. 8.1(b) Drawing an ellipse. A string has its ends fixed at F 1 and F 2 . The tip of a pencil holds the string taut and is moved around. of the ellipse (Fig. 8.1a). This law was a deviation from the Copernican model which allowed only circular orbits. The ellipse, of which the circle is a special case, is a closed curve which can be drawn very simply as follows. Select two points F 1 and F 2 . Take a length of a string and fix its ends at F 1 and F 2 by pins. With the tip of a pencil stretch the string taut and then draw a curve by moving the pencil keeping the string taut throughout.(Fig. 8.1(b)) The closed curve you get is called an ellipse. Clearly for any point T on the ellipse, the sum of the distances from F 1 and F 2 is a constant. F 1 , F 2 are called the focii. Join the points F 1 and F 2 and extend the line to intersect the ellipse at points P and A as shown in Fig. 8.1(b). The midpoint of the line PA is the centre of the ellipse O and the length PO = AO is called the semi- major axis of the ellipse. For a circle, the two focii merge into one and the semi-major axis becomes the radius of the circle. 2. Law of areas : The line that joins any planet to the sun sweeps equal areas in equal intervals of time (Fig. 8.2). This law comes from the observations that planets appear to move slower when they are farther from the sun than when they are nearer. Fig. 8.2 The planet P moves around the sun in an elliptical orbit. The shaded area is the area ?A swept out in a small interval of time ?t. 3. Law of periods : 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. Table 8.1 gives the approximate time periods of revolution of eight* planets around the sun along with values of their semi-major axes. * Refer to information given in the Box on Page 182 2020-21 GRAVITATION 185 t Table 8.1Data from measurement of planetary motions given below confirm Kepler’s Law of Periods (a = Semi-major axis in units of 10 10 m. T = Time period of revolution of the planet in years(y). Q = The quotient ( T 2 /a 3 ) in units of 10 -34 y 2 m -3 .) Planet a T Q Mercury 5.79 0.24 2.95 Venus 10.8 0.615 3.00 Earth 15.0 1 2.96 Mars 22.8 1.88 2.98 Jupiter 77.8 11.9 3.01 Saturn 143 29.5 2.98 Uranus 287 84 2.98 Neptune 450 165 2.99 Pluto* 590 248 2.99 The law of areas can be understood as a consequence of conservation of angular momentum whch is valid for any central force . A central force is such that the force on the planet is along the vector joining the Sun and the planet. Let the Sun be at the origin and let the position and momentum of the planet be denoted by r and p respectively. Then the area swept out by the planet of mass m in time interval ?t is (Fig. 8.2) ?A given by ?A = ½ (r × v?t) (8.1) Hence ?A /?t =½ (r × p)/m, (since v = p/m) = L / (2 m) (8.2) where v is the velocity, L is the angular momentum equal to ( r × p). For a central force, which is directed along r, L is a constant as the planet goes around. Hence, ? A /?t is a constant according to the last equation. This is the law of areas. Gravitation is a central force and hence the law of areas follows. Example 8.1 Let the speed of the planet at the perihelion P in Fig. 8.1(a) be v P and the Sun-planet distance SP be r P . Relate {r P , v P } to the corresponding quantities at the aphelion {r A, v A }. Will the planet take equal times to traverse BAC and CPB ? Answer The magnitude of the angular momentum at P is L p = m p r p v p , since inspection tells us that r p and v p are mutually perpendicular. Similarly, L A = m p r A v A . From angular momentum conservation m p r p v p = m p r A v A or v v p A = r r A p t Since r A > r p , v p > v A . The area SBAC bounded by the ellipse and the radius vectors SB and SC is larger than SBPC in Fig. 8.1. From Kepler’s second law, equal areas are swept in equal times. Hence the planet will take a longer time to traverse BAC than CPB. 8.3 UNIVERSAL LAW OF GRAVITATION Legend has it that observing an apple falling from a tree, Newton was inspired to arrive at an universal law of gravitation that led to an explanation of terrestrial gravitation as well as of Kepler’s laws. Newton’s reasoning was that the moon revolving in an orbit of radius R m was subject to a centripetal acceleration due to earth’s gravity of magnitude 2 2 2 4 m m m R V a R T p = = (8.3) where V is the speed of the moon related to the time period T by the relation 2 / m V R T p = . The time period T is about 27.3 days and R m was already known then to be about 3.84 × 10 8 m. If we substitute these numbers in Eq. (8.3), we get a value of a m much smaller than the value of acceleration due to gravity g on the surface of the earth, arising also due to earth’s gravitational attraction. Johannes Kepler (1571–1630) was a scientist of German origin. He formulated the three laws of planetary motion based on the painstaking observations of Tycho Brahe and coworkers. Kepler himself was an assistant to Brahe and it took him sixteen long years to arrive at the three planetary laws. He is also known as the founder of geometrical optics, being the first to describe what happens to light after it enters a telescope. * Refer to information given in the Box on Page 182 2020-21 186 PHYSICS Central Forces We know the time rate of change of the angular momentum of a single particle about the origin is d dt = × l r F The angular momentum of the particle is conserved, if the torque = × r F t t t t due to the force F on it vanishes. This happens either when F is zero or when F is along r. We are interested in forces which satisfy the latter condition. Central forces satisfy this condition. A ‘central’ force is always directed towards or away from a fixed point, i.e., along the position vector of the point of application of the force with respect to the fixed point. (See Figure below.) Further, the magnitude of a central force F depends on r, the distance of the point of application of the force from the fixed point; F = F(r). In the motion under a central force the angular momentum is always conserved. Two important results follow from this: (1) The motion of a particle under the central force is always confined to a plane. (2) The position vector of the particle with respect to the centre of the force (i.e. the fixed point) has a constant areal velocity. In other words the position vector sweeps out equal areas in equal times as the particle moves under the influence of the central force. Try to prove both these results. You may need to know that the areal velocity is given by : dA/dt = ½ r v sin a. An immediate application of the above discussion can be made to the motion of a planet under the gravitational force of the sun. For convenience the sun may be taken to be so heavy that it is at rest. The gravitational force of the sun on the planet is directed towards the sun. This force also satisfies the requirement F = F(r), since F = G m 1 m 2 /r 2 where m 1 and m 2 are respectively the masses of the planet and the sun and G is the universal constant of gravitation. The two results (1) and (2) described above, therefore, apply to the motion of the planet. In fact, the result (2) is the well-known second law of Kepler. Tr is the trejectory of the particle under the central force. At a position P, the force is directed along OP, O is the centre of the force taken as the origin. In time ?t, the particle moves from P to P', arc PP' = ?s = v ?t. The tangent PQ at P to the trajectory gives the direction of the velocity at P. The area swept in ?t is the area of sector POP' ( ) sin r a ˜ PP'/2 = (r v sin a) ?t/2.) 2020-21 Page 5 CHAPTER EIGHT GRAVITATION 8.1 INTRODUCTION Early in our lives, we become aware of the tendency of all material objects to be attracted towards the earth. Anything thrown up falls down towards the earth, going uphill is lot more tiring than going downhill, raindrops from the clouds above fall towards the earth and there are many other such phenomena. Historically it was the Italian Physicist Galileo (1564-1642) who recognised the fact that all bodies, irrespective of their masses, are accelerated towards the earth with a constant acceleration. It is said that he made a public demonstration of this fact. To find the truth, he certainly did experiments with bodies rolling down inclined planes and arrived at a value of the acceleration due to gravity which is close to the more accurate value obtained later. A seemingly unrelated phenomenon, observation of stars, planets and their motion has been the subject of attention in many countries since the earliest of times. Observations since early times recognised stars which appeared in the sky with positions unchanged year after year. The more interesting objects are the planets which seem to have regular motions against the background of stars. The earliest recorded model for planetary motions proposed by Ptolemy about 2000 years ago was a ‘geocentric’ model in which all celestial objects, stars, the sun and the planets, all revolved around the earth. The only motion that was thought to be possible for celestial objects was motion in a circle. Complicated schemes of motion were put forward by Ptolemy in order to describe the observed motion of the planets. The planets were described as moving in circles with the centre of the circles themselves moving in larger circles. Similar theories were also advanced by Indian astronomers some 400 years later. However a more elegant model in which the Sun was the centre around which the planets revolved – the ‘heliocentric’ model – was already mentioned by Aryabhatta (5 th century A.D.) in his treatise. A thousand years later, a Polish monk named Nicolas 8.1 Introduction 8.2 Kepler’s laws 8.3 Universal law of gravitation 8.4 The gravitational constant 8.5 Acceleration due to gravity of the earth 8.6 Acceleration due to gravity below and above the surface of earth 8.7 Gravitational potential energy 8.8 Escape speed 8.9 Earth satellites 8.10 Energy of an orbiting satellite 8.11 Geostationary and polar satellites 8.12 Weightlessness Summary Points to ponder Exercises Additional exercises 2020-21 184 PHYSICS B A C P S S' 2b 2a Copernicus (1473-1543) proposed a definitive model in which the planets moved in circles around a fixed central sun. His theory was discredited by the church, but notable amongst its supporters was Galileo who had to face prosecution from the state for his beliefs. It was around the same time as Galileo, a nobleman called Tycho Brahe (1546-1601) hailing from Denmark, spent his entire lifetime recording observations of the planets with the naked eye. His compiled data were analysed later by his assistant Johannes Kepler (1571- 1640). He could extract from the data three elegant laws that now go by the name of Kepler’s laws. These laws were known to Newton and enabled him to make a great scientific leap in proposing his universal law of gravitation. 8.2 KEPLER’S LAWS The three laws of Kepler can be stated as follows: 1. Law of orbits : All planets move in elliptical orbits with the Sun situated at one of the foci Fig. 8.1(a) An ellipse traced out by a planet around the sun. The closest point is P and the farthest point is A, P is called the perihelion and A the aphelion. The semimajor axis is half the distance AP. Fig. 8.1(b) Drawing an ellipse. A string has its ends fixed at F 1 and F 2 . The tip of a pencil holds the string taut and is moved around. of the ellipse (Fig. 8.1a). This law was a deviation from the Copernican model which allowed only circular orbits. The ellipse, of which the circle is a special case, is a closed curve which can be drawn very simply as follows. Select two points F 1 and F 2 . Take a length of a string and fix its ends at F 1 and F 2 by pins. With the tip of a pencil stretch the string taut and then draw a curve by moving the pencil keeping the string taut throughout.(Fig. 8.1(b)) The closed curve you get is called an ellipse. Clearly for any point T on the ellipse, the sum of the distances from F 1 and F 2 is a constant. F 1 , F 2 are called the focii. Join the points F 1 and F 2 and extend the line to intersect the ellipse at points P and A as shown in Fig. 8.1(b). The midpoint of the line PA is the centre of the ellipse O and the length PO = AO is called the semi- major axis of the ellipse. For a circle, the two focii merge into one and the semi-major axis becomes the radius of the circle. 2. Law of areas : The line that joins any planet to the sun sweeps equal areas in equal intervals of time (Fig. 8.2). This law comes from the observations that planets appear to move slower when they are farther from the sun than when they are nearer. Fig. 8.2 The planet P moves around the sun in an elliptical orbit. The shaded area is the area ?A swept out in a small interval of time ?t. 3. Law of periods : 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. Table 8.1 gives the approximate time periods of revolution of eight* planets around the sun along with values of their semi-major axes. * Refer to information given in the Box on Page 182 2020-21 GRAVITATION 185 t Table 8.1Data from measurement of planetary motions given below confirm Kepler’s Law of Periods (a = Semi-major axis in units of 10 10 m. T = Time period of revolution of the planet in years(y). Q = The quotient ( T 2 /a 3 ) in units of 10 -34 y 2 m -3 .) Planet a T Q Mercury 5.79 0.24 2.95 Venus 10.8 0.615 3.00 Earth 15.0 1 2.96 Mars 22.8 1.88 2.98 Jupiter 77.8 11.9 3.01 Saturn 143 29.5 2.98 Uranus 287 84 2.98 Neptune 450 165 2.99 Pluto* 590 248 2.99 The law of areas can be understood as a consequence of conservation of angular momentum whch is valid for any central force . A central force is such that the force on the planet is along the vector joining the Sun and the planet. Let the Sun be at the origin and let the position and momentum of the planet be denoted by r and p respectively. Then the area swept out by the planet of mass m in time interval ?t is (Fig. 8.2) ?A given by ?A = ½ (r × v?t) (8.1) Hence ?A /?t =½ (r × p)/m, (since v = p/m) = L / (2 m) (8.2) where v is the velocity, L is the angular momentum equal to ( r × p). For a central force, which is directed along r, L is a constant as the planet goes around. Hence, ? A /?t is a constant according to the last equation. This is the law of areas. Gravitation is a central force and hence the law of areas follows. Example 8.1 Let the speed of the planet at the perihelion P in Fig. 8.1(a) be v P and the Sun-planet distance SP be r P . Relate {r P , v P } to the corresponding quantities at the aphelion {r A, v A }. Will the planet take equal times to traverse BAC and CPB ? Answer The magnitude of the angular momentum at P is L p = m p r p v p , since inspection tells us that r p and v p are mutually perpendicular. Similarly, L A = m p r A v A . From angular momentum conservation m p r p v p = m p r A v A or v v p A = r r A p t Since r A > r p , v p > v A . The area SBAC bounded by the ellipse and the radius vectors SB and SC is larger than SBPC in Fig. 8.1. From Kepler’s second law, equal areas are swept in equal times. Hence the planet will take a longer time to traverse BAC than CPB. 8.3 UNIVERSAL LAW OF GRAVITATION Legend has it that observing an apple falling from a tree, Newton was inspired to arrive at an universal law of gravitation that led to an explanation of terrestrial gravitation as well as of Kepler’s laws. Newton’s reasoning was that the moon revolving in an orbit of radius R m was subject to a centripetal acceleration due to earth’s gravity of magnitude 2 2 2 4 m m m R V a R T p = = (8.3) where V is the speed of the moon related to the time period T by the relation 2 / m V R T p = . The time period T is about 27.3 days and R m was already known then to be about 3.84 × 10 8 m. If we substitute these numbers in Eq. (8.3), we get a value of a m much smaller than the value of acceleration due to gravity g on the surface of the earth, arising also due to earth’s gravitational attraction. Johannes Kepler (1571–1630) was a scientist of German origin. He formulated the three laws of planetary motion based on the painstaking observations of Tycho Brahe and coworkers. Kepler himself was an assistant to Brahe and it took him sixteen long years to arrive at the three planetary laws. He is also known as the founder of geometrical optics, being the first to describe what happens to light after it enters a telescope. * Refer to information given in the Box on Page 182 2020-21 186 PHYSICS Central Forces We know the time rate of change of the angular momentum of a single particle about the origin is d dt = × l r F The angular momentum of the particle is conserved, if the torque = × r F t t t t due to the force F on it vanishes. This happens either when F is zero or when F is along r. We are interested in forces which satisfy the latter condition. Central forces satisfy this condition. A ‘central’ force is always directed towards or away from a fixed point, i.e., along the position vector of the point of application of the force with respect to the fixed point. (See Figure below.) Further, the magnitude of a central force F depends on r, the distance of the point of application of the force from the fixed point; F = F(r). In the motion under a central force the angular momentum is always conserved. Two important results follow from this: (1) The motion of a particle under the central force is always confined to a plane. (2) The position vector of the particle with respect to the centre of the force (i.e. the fixed point) has a constant areal velocity. In other words the position vector sweeps out equal areas in equal times as the particle moves under the influence of the central force. Try to prove both these results. You may need to know that the areal velocity is given by : dA/dt = ½ r v sin a. An immediate application of the above discussion can be made to the motion of a planet under the gravitational force of the sun. For convenience the sun may be taken to be so heavy that it is at rest. The gravitational force of the sun on the planet is directed towards the sun. This force also satisfies the requirement F = F(r), since F = G m 1 m 2 /r 2 where m 1 and m 2 are respectively the masses of the planet and the sun and G is the universal constant of gravitation. The two results (1) and (2) described above, therefore, apply to the motion of the planet. In fact, the result (2) is the well-known second law of Kepler. Tr is the trejectory of the particle under the central force. At a position P, the force is directed along OP, O is the centre of the force taken as the origin. In time ?t, the particle moves from P to P', arc PP' = ?s = v ?t. The tangent PQ at P to the trajectory gives the direction of the velocity at P. The area swept in ?t is the area of sector POP' ( ) sin r a ˜ PP'/2 = (r v sin a) ?t/2.) 2020-21 GRAVITATION 187 t This clearly shows that the force due to earth’s gravity decreases with distance. If one assumes that the gravitational force due to the earth decreases in proportion to the inverse square of the distance from the centre of the earth, we will have a m a 2 m R - ; g a 2 E R - and we get 2 2 m m E R g a R = 3600 (8.4) in agreement with a value of g 9.8 m s -2 and the value of a m from Eq. (8.3). These observations led Newton to propose the following Universal Law of Gravitation : Every body in the universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The quotation is essentially from Newton’s famous treatise called ‘Mathematical Principles of Natural Philosophy’ (Principia for short). Stated Mathematically, Newton’s gravitation law reads : The force F on a point mass m 2 due to another point mass m 1 has the magnitude 1 2 2 | | m m G r = F (8.5) Equation (8.5) can be expressed in vector form as $ ( ) $ 1 2 1 2 2 2 – – m m m m G G r r = = F r r $ 1 2 3 – m m G = r r where G is the universal gravitational constant, $ r is the unit vector from m 1 to m 2 and r = r 2 – r 1 as shown in Fig. 8.3. The gravitational force is attractive, i.e., the force F is along – r. The force on point mass m 1 due to m 2 is of course – F by Newton’s third law. Thus, the gravitational force F 12 on the body 1 due to 2 and F 21 on the body 2 due to 1 are related as F 12 = – F 21 . Before we can apply Eq. (8.5) to objects under consideration, we have to be careful since the law refers to point masses whereas we deal with extended objects which have finite size. If we have a collection of point masses, the force on any one of them is the vector sum of the gravitational forces exerted by the other point masses as shown in Fig 8.4. Fig. 8.4 Gravitational force on point mass m 1 is the vector sum of the gravitational forces exerted by m 2 , m 3 and m 4 . The total force on m 1 is 2 1 1 2 21 Gm m r = F $ 3 1 21 2 31 Gm m r + r $ $ 4 1 31 41 2 41 Gm m r + r r Example 8.2 Three equal masses of m kg each are fixed at the vertices of an equilateral triangle ABC. (a) What is the force acting on a mass 2m placed at the centroid G of the triangle? (b) What is the force if the mass at the vertex A is doubled ? Take AG = BG = CG = 1 m (see Fig. 8.5) Answer (a) The angle between GC and the positive x-axis is 30° and so is the angle between GB and the negative x-axis. The individual forces in vector notation are Fig. 8.3 Gravitational force on m 1 due to m 2 is along r where the vector r is (r 2 – r 1 ). O 2020-21Read More

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