Gravitation Class IX
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we raise a body above the ground and release it, the body alls down towards the earth. Since the body starts moving downwards, a orce must be acting on it. This orce is due to the attraction between the earth and the body and is called the Y? Earth attracts all the objects towards its center. Y? t was who said that every object in the universe attracts every other object with a certain orce called Y? ewton proposed that motion o moon around the earth is caused due to orce o gravity. Y? hen a body, like moon, is moving along a circular path, its velocity changes at every point. Y? This change in the velocity or acceleration, is due to the change in the direction o motion o the body. Y? The orce that produces this acceleration and keeps the body moving along the circular path, acts towards the center and is called the . Y? n the absence o this orce, the body lies o along a tangent to the circular path. Y? 0e reasoned that the moon is kept in its orbit by the centripetal orce provided by the orce o attraction o the earth. Y? Gt each point o its orbit the moon alls towards the earth instead o going straight. Y? ^ut it does not actually all on the earth because when it curves towards the earth, the sur ace o earth being spherical curves away rom under it.
u ± Glso known as ewton¶s law o gravitation. Y? Gccording to the law: ³Every particle in the universe attracts every other particle with a orce which is directly proportional to the product o their masses and inversely proportional to the square o the distance between them.´ Y? The orce is along the line joining the two particles. Y? Thus, i particle 1 attracts particle 2 with a orce F12, then particle 2 pulls particle 1 with a orce F21, o equal magnitude. Y? ^oth the orces are along the line joining the particles. Y? Gccording to the ewton¶s third law o motion, orce F12 and F21 orm a pair o orces (action and reaction), which are equal in magnitude and opposite in direction.
Y? Suppose two particles G and ^ o masses m1 and m2 are lying at a distance r rom
each other. Y? Let the orce o attraction between these two particles be F such that
F O m1m2 and F O 1 r2 So, F O m1 m2 r2 or ëc
here G is a constant known as 2
Y? The S unit o G is m /kg . -11 Y? The value o G was experimentally measured by 0 as 6.67 x 10
m2/kg2. Y? The value o G does not depend on the medium between the two particles or the masses o the bodies or the distance between them. Y? the masses m1 and m2 o the two particles are 1 kg each and the distance r between them is 1m, then F=Gx1x1 12 or F = G Y? Thus, the gravitational constant G is numerically equal to the orce o attraction
which exists between two particles o unit masses kept at a unit distance rom each other. Y? the masses o the objects are small, then the gravitational orce between them is very small. Though various objects on the earth attract one another constantly, they do not cause any motion because orce o attraction between them is very small. Y? , however, at least one o the bodies is large, like sun or earth, then the gravitational orce becomes very large. Y? Since the gravitational orce between two bodies is inversely proportional to the square o the distance between them, there ore, i we double the distance between
Y? Y? Y?
Y? Y? Y?
them, the orce becomes one- ourth and i we halve the distance between them, the orce becomes our times. t is the gravitational orce between the sun and the earth which keeps the earth in uni orm circular motion around the sun. The gravitational orce between the earth and the moon makes the moon revolve at uni orm speed around the earth. The tides in the sea ormed by the rising and alling o water level in the sea, are due to the orce o attraction which the sun and the moon exert on the water sur ace in the sea. The gravitational orce o the earth is responsible or holding the atmosphere above the earth, or the rain alling on the earth, or the low o rivers and or keeping us irmly on the ground. ewton¶s third law o motion also holds good or the orce o gravitation. Gccordingly, when earth exerts a orce o attraction on an object, say a stone, then the stone also exerts an equal orce on the earth in the opposite direction. ^ut we don¶t see earth rising up towards the stone. This is because the mass o a stone is very small due to which the gravitational orce produces a large acceleration in it (a = F/m). ^ut the mass o earth is very large, so the same gravitational orce produces very, very small acceleration in the earth (which cannot be observed). ewton¶s law o gravitation is called universal law o gravitation because it is applicable to all bodies having mass, whether big or small, whether terrestrial or celestial.
± The universal law o gravitation success ully explained several phenomena which were believed to be unconnected: 1.? The orce that binds us to the earth. 2.? The motion o the moon around the earth. 3.? The motion o planets around the Sun. 4.? The tides due to the moon and the Sun.
G ± henever an object alls towards the earth due to its gravitational orce, there is change in the magnitude o its velocity, though there is no change in the direction o motion o the object. Gny change in velocity involves acceleration, so the object accelerates while alling towards the earth. Y? The uni orm acceleration produced in a reely alling body due to gravitational pull o the earth is known as . The direction o this acceleration is towards the center o the earth, i.e., in the vertically downward direction. Y? The alling o a body rom a height towards the earth under the gravitational orce o earth alone is called . Y? Gcceleration due to gravity is denoted by the letter g.
Y? The value o g does not depend upon the mass o the body. This means that all
bodies hollow or solid, big or small all at the same rate. Y? Earlier it was thought that lighter objects all slowly and heavier objects all more
rapidly when dropped rom the same height and at the same time. ^ut it is not so. a eather and a coin are dropped rom the same height at same time, the metal coin reaches the ground irst and eather takes more time to reach the ground. This is because the sur ace area o eather is quite large as compared to its mass, so it experiences much more resistance rom air and its speed is slowed down. The metal coin having small sur ace area & large density, does not get so much resistance rom air and alls to the ground at a aster rate. , however, the eather and the coin are dropped in vacuum, both will reach the ground at the same time. 2 Y? The value o g is 9.8 m/s . 2 Y? hen a body is dropped reely, it alls with an acceleration o 9.8 m/s and when a body is thrown vertically upwards, it undergoes a retardation o 9.8 m/s2. Y? Calculation o the value o g ± a body o mass m is dropped rom a distance rom the center o earth o mass M, then the orce exerted by the earth on the body is given by ewton¶s law o gravitation as: F=GMxm
This orce produces acceleration in the body due to which it moves downwards. From ewton¶s second law o motion: Force = mass x acceleration F = m x a or F = m x g where g is acceleration due to gravity ««« (2)
Comparing equations (1) and (2): mg = GMm r2 c
here G = Gravitational constant M = Mass o the earth
r = Distance o body rom center o the earth e take the distance rom center o the earth because as ar as the gravitational orce at a point outside or on it is concerned, a spherical body, like earth, behaves as i the whole o its mass is concentrated at its center. u? the particle is on or near the sur ace o the earth, then r is taken equal to R, the radius o the earth. u? Thus, or bodies on or near the sur ace o the earth, u?
g = GM R2 Putting the values o G, M and R: g = 6.67 x 10-11 x 6 x 1024 (6.4 x 106)2 g = 9.8 m/s2 u?
The acceleration due to gravity acts in the direction o the line joining the body to the center o the earth.
Î ± Gs G and M are always constant, the value o g is constant as long as radius o earth remains constant. Y? The earth is not a per ect sphere. Due to lattening o earth at poles, all the places on its sur ace are not at the same distance rom its center and so the value o g varies with latitude. Y? Since the radius o earth at poles is minimum, the value o g is maximum at poles. Y? Ggain, the radius o earth is maximum at the equator, so the value o g is minimum at equator. Y? The earth rotates about its north ± south axis. Y? This rotation also a ects the e ective value o g such that it is larger at poles and smaller at equator. Y? Gs we go up rom the sur ace o the earth, the distance rom the center o the earth increases, and hence the value o g decreases. Y? Gs we go deep inside the earth, the value o g decreases. Gt the center o earth it becomes zero. Y? There is also a local variation in g because the earth doesnot have a uni orm density. Gt some places there is water, at others there are coal deposits, metal ores, etc. Measurement o g at di erent places on the sur ace o earth can be used to guess the presence o ores, etc.
± The ormula g = GM/R2 involves only mass o the earth and not o the body on which orce o gravity acts. Since the acceleration due to gravity doesnot depend on the mass o the body, all the bodies, whether heavy or light, all with the same acceleration towards the sur ace o the earth. The di erence in the time taken by them to reach the earth sur ace is due to the resistance o ered by the air according to their sur ace area and density. Y? Since the reely alling bodies all with uni ormly accelerated motion, the three equations o uni ormly accelerated motion o bodies become valid with acceleration µa¶ replaced by µg¶ and distance µs¶ replaced by vertical distance or height µh¶. Y? These equations are: v = u + gt h = ut + ½gt2 v2 = u2 + 2gh Y? hen a body is alling vertically downwards, its velocity is increasing so the
acceleration due to gravity, g, is taken as positive. g = +9.8 m/s2 or a reely alling body. Y? hen a body is thrown vertically upwards, its velocity is decreasing, so the
acceleration due to gravity, g, is taken as negative. g = - 9.8 m/s2 or a body thrown upwards. Y? hen a body is dropped reely rom a height, its initial velocity µu¶ becomes zero. Y? hen a body is thrown vertically upwards, its inal velocity µv¶ becomes zero. Y? The time taken by a body to rise to the highest point is equal to the time taken by
it to all rom the same height. Y? The distance traveled by a reely alling body is directly proportional to the square
o time o all.
± The quantity o matter contained in a body is called its
. ts S.. unit is (kg). Y? G body contains same quantity o matter wherever it be ± whether on earth, moon, or even in outer space. There ore, mass o a body is constant and does not change rom place to place. Y? Mass o a body is a measure o inertia o the body hence it is also known as inertial mass. Greater the mass o a body, greater is the inertia.
Y? The mass o a body cannot be zero. Y? Mass is a quantity and can be measured by a (or beam
balance). Y? hen the earth attracts two objects with equal orce o gravity, then the masses o two objects will be equal.
± The orce with which a body is attracted towards the center o the earth is called its . ts S. unit is (). Y? The relation between mass m o an object and its weight is: = G Mem = m (GMe) = mg Re2
Re2 = mg
Or = m x 9.8 m/s2 Y? Gnother unit o weight called gravitational unit o weight is !
(kg.wt.) Y? One kilogram weight is that gravitational orce which acts on a body o mass 1kg. 1 kg. wt. = 1kg x 9.8 m/s2 = 9.8 x 1 kg x 1 m/s2 ^ut 1 kg x 1 m/s2 = 1 X "# Thus the weight o 1 kilogram mass is 9.8 . Y? eight is a quantity and can be measured by a . Y? Gt a given place, the value o g is constant, so the weight o a body is directly
proportional to its mass. Y? Since the value o g changes rom place to place, there ore, the weight o body also changes rom place to place. Thus, weight o a body is not constant. Y? n the interplanetary space, where g = 0, the weight o a body becomes zero and we eel weightlessness. Y? Gs the value o g is slightly greater at poles than at equator, the weight o a body will also be greater at poles than at equator.
$ ± a body is taken to the sur ace o the moon, then the earth¶s attraction on the body will be very small as compared to that o the moon. Y? The weight o a body on the moon is the orce with which the moon attracts the body. Y? The mass o moon is less than that o the earth. Due to this the moon exerts lesser orce o attraction on objects. Y? Let the mass o an object be m. Let its weight on the moon be m. Let the mass o the moon be Mm and its radius be Rm. The weight o the object on the moon can be given by: m = G Mm x m R2m Y? The gravitational pull o the moon is about one-sixth that o the earth, there ore
the weight o the object on the moon will be about one-sixth o what it is on the earth. m = 1 x e 6 u?
± The orce acting on a body perpendicular to its sur ace is called thrust. Y? The S.. unit o thrust is ewton (). Y? e.g. For ixing a poster on a bulletin board one has to press drawing pins with the thumb. hen pressing a drawing pin, orce is applied on the sur ace area o its head. The orce is directed perpendicular to the sur ace o the board. This orce is called thrust.
± The thrust per unit area is called pressure. Pressure = Thrust Grea 2
Y? The S.. unit o pressure is ewton per square metre (/m ) which is also called
(Pa). Many times a bigger unit o pressure called kilopascal (kPa) is used. Y? The pressure depends on two actors:
3.? Force applied 4.? Grea over which orce acts. Y? The same orce can produce di erent pressures depending on the area over which it acts e.g. when a orce acts over a large area o an object, it produces a small pressure. ^ut i the same orce acts over a small area o the object, it produces a large pressure.
Y? Consider the case o two similar bricks lying on the ground, one in the lying
position and another in the standing position. The two bricks exert the same orce on the ground because they have the same weight. ^ut the two bricks exert di erent pressures on the ground because their areas in contact with the ground are di erent. The brick in the lying position has a large area in contact with the ground. So, the orce o the weight o the brick alls on a large area o the ground and the µ orce per unit area µ or pressure on the ground is less. The brick in the standing position has a small area in contact with the ground. So, the orce o the weight o the brick alls on a smaller area o the ground and the pressure on the ground is more. G school bag has wide straps made o thick cloth so that the weight o bag may all over a large area o the shoulder o the child producing less pressure on the shoulder. Gnd due to less pressure, it is more com ortable to carry the heavy school bag. On the other hand, i the school bag has a strap made o thin string, then the weight o school bag will all over a small area o the shoulder. This will produce a large pressure on the shoulder o the child and it will become very pain ul to carry the heavy school bag. G sharp kni e cuts better than a blunt kni e. G sharp kni e has a very thin edge to its blade. Due to its very thin edge, the orce o our hand alls over a very small area o the object producing a large pressure. Gnd this large pressure cuts the object easily. On the other hand, a blunt kni e does not cut an object easily because due to its thicker edge, the orce o our hand alls over a larger area o the object and produces lesser pressure. This lesser pressure cuts the object with di iculty. The tip o a sewing needle is sharp so that due to its sharp tip, the needle may put the orce on a very small area o the cloth, producing a large pressure su icient to pierce the cloth being stitched. The pressure on ground is more when a man is walking than when he is standing. hen a man is walking, then at one time only his one oot is on the ground. Due to this, the orce o weight o man alls on a smaller area o the ground and produces more pressure on the ground. On the other hand, when the man is standing, then both his eet are on the ground. Due to this the orce o weight o the man alls on a larger area o the ground and produces lesser pressure on the ground. The depression is much more when a man stands on the cushion than when he lies down on it. hen a man stands on a cushion then only his two eet (having small area) are in contact with the cushion. Due to this the weight o man alls on a small area o the cushion producing a large pressure. This large pressure causes a big depression in the cushion. On the other hand, when the same man is lying on the cushion, then his whole body (having large area) is in contact with the cushion. n this case the weight o man alls on a much larger area o the cushion producing much smaller pressure. Gnd this smaller pressure produces a very little depression in the cushion. The tractors have broad tyres so that there is less pressure on the ground and the tyres do not sink into comparatively so t ground in the ields.
Y? G wide steel belt is provided over the wheels o army tanks so that they exert less
pressure on the ground and do not sink into it. Y? ooden sleepers (or concrete sleepers) are kept below the railway line so that
there is less pressure o the train on the ground and railway line may not sink into the ground. Y? The snow shoes have large, lat soles so that there is less pressure on the so t snow and this stops the wearer rom sinking into it. Y? t is easier to walk on so t sand i we have lat shoes rather than shoes with small heels (or pencil heels). This is because a lat shoe has a greater area in contact with the so t sand due to which there is less pressure on the so t ground. Due to this the lat shoes do not sink much in so t sand and it is easy to walk on it. On the other hand, a small heel (or sharp heel) has a small area is contact with the so t sand and so exerts a greater pressure on the so t sand. Due to this greater pressure, the small heels tend to sink deep into so t sand making it di icult or the wearer to walk on so t sand. Y? The oundations o buildings and dams are laid on a larger area o ground so that the weight o the building or dam (to be constructed) produces less pressure on ground and the building or dam may not sink into the ground.
- The pressure at any place due to the atmosphere is called atmospheric pressure. ts value varies rom place to place and also with the time. Gtmospheric pressure at the earth¶s sur ace near the sea level is around 1.01x105 Pa. This value is known as 1atmosphere o pressure (1atmosphere = 760mm o 0g).
± Gll liquids and gases are luids. Y? G solid exerts pressure on a sur ace due to its weight Y? Similarly, luids have weight, and they also exert pressure on the base and walls o the container in which they are enclosed. Y? Pressure exerted in any con ined mass o luid is transmitted undiminished in all directions. Y? The pressure in a liquid is the same at all points at the same horizontal level. Gs we go deeper in the liquid, the pressure increases.
^ ± Y? hen an object is placed in a liquid, the liquid exerts an upward orce on it e.g. hen a piece o cork is held below the sur ace o water and then released the cork immediately rises to the sur ace. Y? t is a common experience that a mug illed with water appears to be heavier when it is li ted above the sur ace o water in a bucket.
Y? n general, whenever an object is immersed in water, it appears to lose some
Y? Y? Y?
weight and eels lighter. The weight o the object in water is called t is less than its true weight. The objects appear to be less heavy when submerged in water because the water exerts an upward orce on them. The upward orce acting on an object immersed in a liquid is called
The buoyant orce is also known as t is due to the buoyant orce exerted by the liquid that the weight o an object appears to be less in the liquid than its actual weight in air. t is due to the buoyant orce exerted by water that we are able to swim in water and ships loat on water. The tendency o a liquid to exert an upward orce on an object placed in it is called Gs more and more volume o the object is immersed in a liquid, the upward buoyant orce acting on it increases. ^ut once the object is completely immersed in a liquid, then lowering it urther in the liquid does not increase the buoyant orce. This means that maximum upward buoyant orce acts on an object when it is completely immersed in the liquid.
± 1.? The buoyant orce exerted by a liquid depends on the $ immersed in the liquid. Y? Gs the volume o the solid object immersed inside the liquid increases, the
upward buoyant orce also increases. Gnd when the object is completely immersed in the liquid, the buoyant orce becomes maximum and remains constant. Y? The magnitude o buoyant orce acting on a solid object does not depend on the nature o the solid object, e.g. i two balls made o di erent metals having di erent weights but equal volumes are ully immersed in a liquid, they will experience an equal loss in weight and thus equal upward buoyant orce. This is because both the balls displace equal weight o the liquid due to their equal volumes. 1.? The buoyant orce exerted by a liquid depends on the % in which the object is immersed. Y? The liquid having higher density exerts more upward buoyant orce on an object
than another liquid having lower density. Thus, as the density o liquid increases, the buoyant orce exerted by it also increases, Y? e.g. sea water has higher density than resh water, there ore, sea-water will exert more buoyant orce on an object immersed in it than the resh water. t is easier to swim in sea water because it exerts a greater buoyant orce on the swimmer.
Y? Similarly, mercury is a liquid having very high density. So, mercury will exert a
very great buoyant orce on an object immersed in it. Even a very heavy material like an iron block loats in mercury because mercury exerts a very high buoyant orce on iron block due to its very high density.
$ % ± G wooden block loats in water whereas a steel rod sinks in it. Thus some objects loat and some sink in water. hen an object is put in a liquid, then two orces act on it: 1.? eight () o the object acting downwards, 2.? ^uoyant orce (^) acting upwards. Gn object will loat or sink in a liquid will depend on the relative magnitude o these two orces acting on the object in opposite directions. Three cases arise: 1.? ^ exerted by the liquid < o the object, the object will sink in the liquid. 2.? ^ = , the object will loat in the liquid. 3.? ^ > , the object will rise in the liquid and then loat. Thus an object will loat in a liquid i the upward buoyant orce it receives rom the liquid is great enough to overcome the downward orce o its weight. For an object to loat, eight o object = ^uoyant orce ^ut, ^uoyant orce = eight o liquid displaced by the object Xeight o object = eight o liquid displaced by the object. Thus an object will loat in a liquid i the weight o object is equal to the weight o liquid displaced by it.
The above relation holds true i the object has a lower density than the liquid. the object has a higher density than the liquid, then the weight o liquid displaced will be less than the weight o object, and the object will sink. u? Gn object will also loat in a liquid i its density is equal to that o the liquid. u? hen we put a piece o iron in water, it sinks immediately because iron is denser than water. ^ut a ship made rom iron and steel loats on water. This is because a ship is a hollow object having a lot o air in it. Gir has low density due to which the average density o ship becomes less than the density o water and the ship loats in water. u? This can be explained in another way. G heavy ship loats in water as it displaces a large weight o water which provides a great buoyant orce to keep it a loat. u? u?
G & ± ³hen an object is wholly or partially immersed in a liquid, it experiences a buoyant orce (or upthrust) which is equal to the weight o liquid displaced by the object´.
^uoyant orce acting = eight o liquid displaced on an object u?
by that object
Grchimedes¶ principle is applicable to objects in luids, i.e. liquids as well as gases. Gases (like air) exert an upward orce (or buoyant orce) on the objects placed in them but in most cases it is so small that we usually ignore it. t is the buoyant orce due to displaced air which makes a balloon rise in air. ^ ^ '
G G & ± 1.? t is used in designing ships and submarines. 2.? t is used in determining the relative density o a substance. 3.? The lactometers used or determining the purity o milk are based on Grchimedes¶ principle. 4.? The hydrometers used or determining the density o liquids are based on Grchimedes¶ principle.
± The density o a substance is de ined as mass o the substance per unit volume. Density = Mass o the substance Volume o the substance
The S unit o density is kilograms per cubic meter (Kg/m3). The density o a substance, under speci ied conditions, is always the same. So, the density o a substance is one o its characteristic properties. u? The density o a given substance can help us to determine its purity. 3 u? Di erent substances have di erent densities e.g. density o water is 1000 Kg/m which means that the mass o 1 cubic metre volume o water is 1000 kg. u? u?
· ± The relative density o a substance is the ratio o its density to that o water. Relative density o a substance = Density o the substance Density o water
Since the relative density is a ratio, it has no units. t is a pure number. The relative density o a substance expresses the heaviness (or density) o the substance in comparison to water e.g. the relative density o iron is 7.8, which means iron is 7.8 times as heavy as an equal volume o water. u? The relative density o water is 1. the relative density o a substance is more than 1, then it will be heavier than water and hence it will sink in water. On the other hand, i the relative density o a substance is less than 1, then it will be lighter than water and hence loat in water. e.g. ce has a density o about 900 kg/m3 and water has a density 1000kg/m3. Thus an ice cube has a relative density o 0.9 so it loats in water. The relative density o iron is7.8, so an iron nail sinks in water. u? u?