Chapter 10

1 In Chapters 8 and 9, we have learnt aboutthe motion of objects and force as the causeof motion. We have learnt that a force isneeded to change the speed or the directionof motion of an object. We always observe thatan object dropped from a height falls towardsthe earth. We know that all the planets goaround the Sun. The moon goes around theearth. In all these cases, there must be someforce acting on the objects, the planets andon the moon. Isaac Newton could grasp thatthe same force is responsible for all force is called the gravitational this Chapter we shall learn aboutgravitation and the universal law ofgravitation. We shall discuss the motion ofobjects under the influence of gravitationalforce on the earth. We shall study how theweight of a body varies from place to shall also discuss the conditions forobjects to float in GravitationWe know that the moon goes around theearth. An object when thrown upwards,reaches a certain height and then fallsdownwards.

2 It is said that when Newton wassitting under a tree, an apple fell on him. Thefall of the apple made Newton start thought that: if the earth can attract anapple, can it not attract the moon? Is the forcethe same in both cases? He conjectured thatthe same type of force is responsible in boththe cases. He argued that at each point of itsorbit, the moon falls towards the earth,instead of going off in a straight line. So, itmust be attracted by the earth. But we donot really see the moon falling towards us try to understand the motion ofthe moon by recalling activity Take a piece of thread. Tie a small stone at one end. Hold theother end of the thread and whirl itround, as shown in Fig. Note the motion of the stone. Release the thread. Again, note the direction of motion ofthe :A stone describing a circular path with avelocity of constant the thread is released, the stonemoves in a circular path with a certain speedand changes direction at every point.

3 Thechange in direction involves change in velocityor acceleration. The force that causes thisacceleration and keeps the body moving alongthe circular path is acting towards the force is called the centripetal (meaning centre-seeking ) force. In the absence of LAW OF GRAVITATIONE very object in the universe attracts everyother object with a force which is proportionalto the product of their masses and inverselyproportional to the square of the distancebetween them. The force is along the linejoining the centres of two , the stone flies off along a straight straight line will be a tangent to thecircular to knowTangent to a circleA straight line that meets the circleat one and only one point is called atangent to the circle. Straight lineABC is a tangent to the circle atpoint motion of the moon around the earthis due to the centripetal force. The centripetalforce is provided by the force of attraction ofthe earth. If there were no such force, themoon would pursue a uniform straight is seen that a falling apple is attractedtowards the earth.

4 Does the apple attract theearth? If so, we do not see the earth movingtowards an apple. Why?According to the third law of motion, theapple does attract the earth. But accordingto the second law of motion, for a given force,acceleration is inversely proportional to themass of an object [Eq. ( )]. The mass of anapple is negligibly small compared to that ofthe earth. So, we do not see the earth movingtowards the apple. Extend the same argumentfor why the earth does not move towards our solar system, all the planets goaround the Sun. By arguing the same way,we can say that there exists a force betweenthe Sun and the planets. From the above factsNewton concluded that not only does theearth attract an apple and the moon, but allobjects in the universe attract each other. Thisforce of attraction between objects is calledthe gravitational =dFig. :The gravitational force between twouniform objects is directed along the linejoining their two objects A and B of masses M andm lie at a distance d from each other as shownin Fig.

5 Let the force of attraction betweentwo objects be F. According to the universallaw of gravitation, the force between twoobjects is directly proportional to the productof their masses. That is,F M m( )And the force between two objects is inverselyproportional to the square of the distancebetween them, that is, 21Fd( )Combining Eqs. ( ) and ( ), we getF 2 M md( )or, G2M mF =d( )where G is the constant of proportionality andis called the universal gravitation multiplying crosswise, Eq. ( ) gives F d 2 = G M m2022-23 GRAVITATION133 Isaac Newton was bornin Woolsthorpe nearGrantham, is generallyregarded as the mostoriginal andinfluential theorist inthe history of was born in a poorfarming family. But hewas not good atfarming. He was sentto study at CambridgeUniversity in 1661. In1665 a plague brokeout in Cambridge and so Newton took a yearoff. It was during this year that the incident ofthe apple falling on him is said to haveoccurred.

6 This incident prompted Newton toexplore the possibility of connecting gravitywith the force that kept the moon in its led him to the universal law ofgravitation. It is remarkable that many greatscientists before him knew of gravity but failedto realise formulated the well-known laws ofmotion. He worked on theories of light andcolour. He designed an astronomical telescopeto carry out astronomical was also a great mathematician. Heinvented a new branch of mathematics, calledcalculus. He used it to prove that for objectsoutside a sphere of uniform density, the spherebehaves as if the whole of its mass isconcentrated at its centre. Newtontransformed the structure of physicalscience with his three laws of motion and theuniversal law of gravitation. As the keystoneof the scientific revolution of the seventeenthcentury, Newton s work combined thecontributions of Copernicus, Kepler, Galileo,and others into a new powerful is remarkable that though thegravitational theory could not be verified atthat time, there was hardly any doubt aboutits correctness.

7 This is because Newton basedhis theory on sound scientific reasoning andbacked it with mathematics. This made thetheory simple and elegant. These qualities arenow recognised as essential requirements of agood scientific Newton(1642 1727)How did Newton guess theinverse-square rule?There has always been a great interestin the motion of planets. By the 16thcentury, a lot of data on the motion ofplanets had been collected by manyastronomers. Based on these dataJohannes Kepler derived three laws,which govern the motion of are called Kepler s laws. These orbit of a planet is an ellipse withthe Sun at one of the foci, as shown inthe figure given below. In this figure Ois the position of the line joining the planet and the Sunsweep equal areas in equal intervalsof time. Thus, if the time of travel fromA to B is the same as that from C to D,then the areas OAB and OCD cube of the mean distance of aplanet from the Sun is proportional tothe square of its orbital period T.

8 Or,r3/T2 = is important to note that Keplercould not give a theory to explainthe motion of planets. It was Newtonwho showed that the cause of theplanetary motion is the gravitationalforce that the Sun exerts on them. Newtonused the third lawof Kepler tocalculate thegravitational forceof attraction. Thegravitational forceof the earth isweakened by distance. A simple argumentgoes like this. We can assume that theplanetary orbits are circular. Suppose theorbital velocity is v and the radius of theorbit is r. Then the force acting on anorbiting planet is given by F v2 T denotes the period, then v = 2 r/T,so that v2 r2 can rewrite this as v2 (1/r) ( r3/T2). Since r3/T2 is constant by Kepler sthird law, we have v2 1/r. Combiningthis with F v2/ r, we get, F 1/ Eq. ( ), the force exerted by theearth on the moon isG2M mF = 10N m kg6 10 kg 10 kg( 10 m) = = 1020 , the force exerted by the earth onthe moon is 1020 the universal law the formula to find themagnitude of the gravitationalforce between the earth and anobject on the surface of the OF THE UNIVERSALLAW OF GRAVITATIONThe universal law of gravitation successfullyexplained several phenomena which werebelieved to be unconnected:(i)the force that binds us to the earth;(ii)the motion of the moon around theearth;(iii)the motion of planets around the Sun;and(iv)the tides due to the moon and the FallLet us try to understand the meaning of freefall by performing this Take a stone.

9 Throw it upwards. It reaches a certain height and then itstarts falling have learnt that the earth attractsobjects towards it. This is due to thegravitational force. Whenever objects falltowards the earth under this force alone, wesay that the objects are in free fall. Is there anyor2G= F dMm( )The SI unit of G can be obtained bysubstituting the units of force, distance andmass in Eq. ( ) as N m2 kg value of G was found out byHenry Cavendish (1731 1810) by using asensitive balance. The accepted value of G 10 11 N m2 kg know that there exists a force ofattraction between any two objects. Computethe value of this force between you and yourfriend sitting closeby. Conclude how you donot experience this force!The law is universal in the sense thatit is applicable to all bodies, whetherthe bodies are big or small, whetherthey are celestial or that F is inverselyproportional to the square of dmeans, for example, that if d getsbigger by a factor of 6, F becomes136times The mass of the earth is6 1024 kg and that of the moon 1022 kg.

10 If the distance between theearth and the moon is 105 km,calculate the force exerted by the earth onthe moon. (Take G = 10 11 N m2 kg-2)Solution:The mass of the earth, M = 6 1024 kgThe mass of the moon,m = 1022 kgThe distance between the earth and themoon,d= 105 km= 105 1000 m= 108 mG = 10 11 N m2 kg 2 More to knowQ2022-23 GRAVITATION135calculations, we can take g to be more or lessconstant on or near the earth. But for objectsfar from the earth, the acceleration due togravitational force of earth is given byEq. ( ). CALCULATE THE VALUE OF gTo calculate the value of g, we should putthe values of G, M and R in Eq. ( ),namely, universal gravitational constant,G = 10 11 N m2 kg-2, mass of the earth,M = 6 1024 kg, and radius of the earth,R = 106 = 10 N m kg6 10 kg=( 10 m) = m s , the value of acceleration due to gravityof the earth, g = m s OF OBJECTS UNDER THEINFLUENCE OF GRAVITATIONALFORCE OF THE EARTHLet us do an activity to understand whetherall objects hollow or solid, big or small, willfall from a height at the same Take a sheet of paper and a stone.