C hapter FORCE. Fig. 9.2: (a) A spring expands on application of force; (b) A spherical rubber ball becomes oblong as we apply force on it.

Transcription

1 C haper 9 FORCE AND LAWS OF MOTION In he previous chaper, we described he moion of an objec along a sraigh line in erms of is posiion, velociy and acceleraion. We saw ha such a moion can be uniform or non-uniform. We have no ye discovered wha causes he moion. Why does he speed of an objec change wih ime? Do all moions require a cause? If so, wha is he naure of his cause? In his chaper we shall make an aemp o quench all such curiosiies. For many cenuries, he problem of moion and is causes had puzzled scieniss and philosophers. A ball on he ground, when given a small hi, does no move forever. Such observaions sugges ha res is he naural sae of an objec. This remained he belief unil Galileo Galilei and Isaac Newon developed an enirely differen approach o undersand moion. In our everyday life we observe ha some effor is required o pu a saionary objec ino moion or o sop a moving objec. We ordinarily experience his as a muscular effor and say ha we mus push or hi or pull on an objec o change is sae of moion. The concep of force is based on his push, hi or pull. Le us now ponder abou a force. Wha is i? In fac, no one has seen, ased or fel a force. However, we always see or feel he effec of a force. I can only be explained by describing wha happens when a force is applied o an objec. Pushing, hiing and pulling of objecs are all ways of bringing objecs in moion (Fig. 9.1). They move because we make a force ac on hem. From your sudies in earlier classes, you are also familiar wih he fac ha a force can be used o change he magniude of velociy of an objec (ha is, o make he objec move faser or slower) or o change is direcion of moion. We also know ha a force can change he shape and size of objecs (Fig. 9.2). (a) The rolley moves along he direcion we push i. (b) The drawer is pulled. (a) (b) (c) The hockey sick his he ball forward Fig. 9.1: Pushing, pulling, or hiing objecs change heir sae of moion. Fig. 9.2: (a) A spring expands on applicaion of force; (b) A spherical rubber ball becomes oblong as we apply force on i.

2 9.1 Balanced and Unbalanced Forces Fig. 9.3 shows a wooden block on a horizonal able. Two srings X and Y are ied o he wo opposie faces of he block as shown. If we apply a force by pulling he sring X, he block begins o move o he righ. Similarly, if we pull he sring Y, he block moves o he lef. Bu, if he block is pulled from boh he sides wih equal forces, he block will no move. Such forces are called balanced forces and do no change he sae of res or of moion of an objec. Now, le us consider a siuaion in which wo opposie forces of differen magniudes pull he block. In his case, he block would begin o move in he direcion of he greaer force. Thus, he wo forces are no balanced and he unbalanced force acs in he direcion he block moves. This suggess ha an unbalanced force acing on an objec brings i in moion. Fig. 9.3: Two forces acing on a wooden block Wha happens when some children ry o push a box on a rough floor? If hey push he box wih a small force, he box does no move because of fricion acing in a direcion opposie o he push [Fig. 9.4(a)]. This fricion force arises beween wo surfaces in conac; in his case, beween he boom of he box and floor s rough surface. I balances he pushing force and herefore he box does no move. In Fig. 9.4(b), he children push he box harder bu he box sill does no move. This is because he fricion force sill balances he pushing force. If he children push he box harder sill, he pushing force becomes bigger han he fricion force [Fig. 9.4(c)]. There is an unbalanced force. So he box sars moving. Wha happens when we ride a bicycle? When we sop pedalling, he bicycle begins o slow down. This is again because of he fricion forces acing opposie o he direcion of moion. In order o keep he bicycle moving, we have o sar pedalling again. I hus appears ha an objec mainains is moion under he coninuous applicaion of an unbalanced force. However, i is quie incorrec. An objec moves wih a uniform velociy when he forces (pushing force and fricional force) acing on he objec are balanced and here is no ne exernal force on i. If an unbalanced force is applied on he objec, here will be a change eiher in is speed or in he direcion of is moion. Thus, o accelerae he moion of an objec, an unbalanced force is required. And he change in is speed (or in he direcion of moion) would coninue as long as his unbalanced force is applied. However, if his force is (a) (b) (c) Fig. 9.4 FORCE AND LAWS OF MOTION 115

3 removed compleely, he objec would coninue o move wih he velociy i has acquired ill hen. 9.2 Firs Law of Moion By observing he moion of objecs on an inclined plane Galileo deduced ha objecs move wih a consan speed when no force acs on hem. He observed ha when a marble rolls down an inclined plane, is velociy increases [Fig. 9.5(a)]. In he nex chaper, you will learn ha he marble falls under he unbalanced force of graviy as i rolls down and aains a definie velociy by he ime i reaches he boom. Is velociy decreases when i climbs up as shown in Fig. 9.5(b). Fig. 9.5(c) shows a marble resing on an ideal fricionless plane inclined on boh sides. Galileo argued ha when he marble is released from lef, i would roll down he slope and go up on he opposie side o he same heigh from which i was released. If he inclinaions of he planes on boh sides are equal hen he marble will climb he same disance ha i covered while rolling down. If he angle of inclinaion of he righ-side plane were gradually decreased, hen he marble would ravel furher disances ill i reaches he original heigh. If he righ-side plane were ulimaely made horizonal (ha is, he slope is reduced o zero), he marble would coninue o ravel forever rying o reach he same heigh ha i was released from. The unbalanced forces on he marble in his case are zero. I hus suggess ha an unbalanced (exernal) force is required o change he moion of he marble bu no ne force is needed o susain he uniform moion of he marble. In pracical siuaions i is difficul o achieve a zero unbalanced force. This is because of he presence of he fricional force acing opposie o he direcion of moion. Thus, in pracice he marble sops afer ravelling some disance. The effec of he fricional force may be minimised by using a smooh marble and a smooh plane and providing a lubrican on op of he planes. Fig. 9.5: (a) he downward moion; (b) he upward moion of a marble on an inclined plane; and (c) on a double inclined plane. Newon furher sudied Galileo s ideas on force and moion and presened hree fundamenal laws ha govern he moion of objecs. These hree laws are known as Newon s laws of moion. The firs law of moion is saed as: An objec remains in a sae of res or of uniform moion in a sraigh line unless compelled o change ha sae by an applied force. In oher words, all objecs resis a change in heir sae of moion. In a qualiaive way, he endency of undisurbed objecs o say a res or o keep moving wih he same velociy is called ineria. This is why, he firs law of moion is also known as he law of ineria. Cerain experiences ha we come across while ravelling in a moorcar can be explained on he basis of he law of ineria. We end o remain a res wih respec o he sea unil he drives applies a braking force o sop he moorcar. Wih he applicaion of brakes, he car slows down bu our body ends o coninue in he same sae of moion because of is ineria. A sudden applicaion of brakes may hus cause injury o us by 116 SCIENCE

4 Galileo Galilei was born on 15 February 1564 in Pisa, Ialy. Galileo, righ from his childhood, had ineres in mahemaics and naural philosophy. Bu his faher Vincenzo Galilei waned him o become a medical docor. Accordingly, Galileo enrolled himself for a medical degree a he Galileo Galilei ( ) Universiy of Pisa in 1581 which he never compleed because of his real ineres in mahemaics. In 1586, he wroe his firs scienific book The Lile Balance [La Balancia], in which he described Archimedes mehod of finding he relaive densiies (or specific graviies) of subsances using a balance. In 1589, in his series of essays De Mou, he presened his heories abou falling objecs using an inclined plane o slow down he rae of descen. In 1592, he was appoined professor of mahemaics a he Universiy of Padua in he Republic of Venice. Here he coninued his observaions on he heory of moion and hrough his sudy of inclined planes and he pendulum, formulaed he correc law for uniformly acceleraed objecs ha he disance he objec moves is proporional o he square of he ime aken. Galileo was also a remarkable crafsman. He developed a series of elescopes whose opical performance was much beer han ha of oher elescopes available during hose days. Around 1640, he designed he firs pendulum clock. In his book Sarry Messenger on his asronomical discoveries, Galileo claimed o have seen mounains on he moon, he milky way made up of iny sars, and four small bodies orbiing Jupier. In his books Discourse on Floaing Bodies and Leers on he Sunspos, he disclosed his observaions of sunspos. Using his own elescopes and hrough his observaions on Saurn and Venus, Galileo argued ha all he planes mus orbi he Sun and no he earh, conrary o wha was believed a ha ime. impac or collision wih he panels in fron. Safey bels are worn o preven such accidens. Safey bels exer a force on our body o make he forward moion slower. An opposie experience is encounered when we are sanding in a bus and he bus begins o move suddenly. Now we end o fall backwards. This is because he sudden sar of he bus brings moion o he bus as well as o our fee in conac wih he floor of he bus. Bu he res of our body opposes his moion because of is ineria. When a moorcar makes a sharp urn a a high speed, we end o ge hrown o one side. This can again be explained on he basis of he law of ineria. We end o coninue in our sraigh-line moion. When an unbalanced force is applied by he engine o change he direcion of moion of he moorcar, we slip o one side of he sea due o he ineria of our body. The fac ha a body will remain a res unless aced upon by an unbalanced force can be illusraed hrough he following aciviies: Aciviy 9.1 Make a pile of similar carom coins on a able, as shown in Fig Aemp a sharp horizonal hi a he boom of he pile using anoher carom coin or he sriker. If he hi is srong enough, he boom coin moves ou quickly. Once he lowes coin is removed, he ineria of he oher coins makes hem fall verically on he able. Fig. 9.6: Only he carom coin a he boom of a pile is removed when a fas moving carom coin (or sriker) his i. FORCE AND LAWS OF MOTION 117

5 Aciviy 9.2 Se a five-rupee coin on a siff playing card covering an empy glass umbler sanding on a able as shown in Fig Give he card a sharp horizonal flick wih a finger. If we do i fas hen he card shoos away, allowing he coin o fall verically ino he glass umbler due o is ineria. The ineria of he coin ries o mainain is sae of res even when he card flows off. Fig. 9.7: When he playing card is flicked wih he finger he coin placed over i falls in he umbler. Aciviy 9.3 Place a waer-filled umbler on a ray. Hold he ray and urn around as fas as you can. We observe ha he waer spills. Why? Observe ha a groove is provided in a saucer for placing he ea cup. I prevens he cup from oppling over in case of sudden jerks. 9.3 Ineria and Mass All he examples and aciviies given so far illusrae ha here is a resisance offered by an objec o change is sae of moion. If i is a res i ends o remain a res; if i is moving i ends o keep moving. This propery of an objec is called is ineria. Do all bodies have he same ineria? We know ha i is easier o push an empy box han a box full of books. Similarly, if we kick a fooball i flies away. Bu if we kick a sone of he same size wih equal force, i hardly moves. We may, in fac, ge an injury in our foo while doing so! Similarly, in aciviy 9.2, insead of a five-rupees coin if we use a one-rupee coin, we find ha a lesser force is required o perform he aciviy. A force ha is jus enough o cause a small car o pick up a large velociy will produce a negligible change in he moion of a rain. This is because, in comparison o he car he rain has a much lesser endency o change is sae of moion. Accordingly, we say ha he rain has more ineria han he car. Clearly, heavier or more massive objecs offer larger ineria. Quaniaively, he ineria of an objec is measured by is mass. We may hus relae ineria and mass as follows: Ineria is he naural endency of an objec o resis a change in is sae of moion or of res. The mass of an objec is a measure of is ineria. Quesions 1. Which of he following has more ineria: (a) a rubber ball and a sone of he same size? (b) a bicycle and a rain? (c) a fiverupees coin and a one-rupee coin? 2. In he following example, ry o idenify he number of imes he velociy of he ball changes: A fooball player kicks a fooball o anoher player of his eam who kicks he fooball owards he goal. The goalkeeper of he opposie eam collecs he fooball and kicks i owards a player of his own eam. Also idenify he agen supplying he force in each case. 3. Explain why some of he leaves may ge deached from a ree if we vigorously shake is branch. 4. Why do you fall in he forward direcion when a moving bus brakes o a sop and fall backwards when i acceleraes from res? 9.4 Second Law of Moion The firs law of moion indicaes ha when an unbalanced exernal force acs on an 118 SCIENCE

6 objec, is velociy changes, ha is, he objec ges an acceleraion. We would now like o sudy how he acceleraion of an objec depends on he force applied o i and how we measure a force. Le us recoun some observaions from our everyday life. During he game of able ennis if he ball his a player i does no hur him. On he oher hand, when a fas moving cricke ball his a specaor, i may hur him. A ruck a res does no require any aenion when parked along a roadside. Bu a moving ruck, even a speeds as low as 5 m s 1, may kill a person sanding in is pah. A small mass, such as a bulle may kill a person when fired from a gun. These observaions sugges ha he impac produced by he objecs depends on heir mass and velociy. Similarly, if an objec is o be acceleraed, we know ha a greaer force is required o give a greaer velociy. In oher words, here appears o exis some quaniy of imporance ha combines he objec s mass and is velociy. One such propery called momenum was inroduced by Newon. The momenum, p of an objec is defined as he produc of is mass, m and velociy, v. Tha is, p = mv (9.1) Momenum has boh direcion and magniude. Is direcion is he same as ha of velociy, v. The SI uni of momenum is kilogram-mere per second (kg m s -1 ). Since he applicaion of an unbalanced force brings a change in he velociy of he objec, i is herefore clear ha a force also produces a change of momenum. Le us consider a siuaion in which a car wih a dead baery is o be pushed along a sraigh road o give i a speed of 1 m s -1, which is sufficien o sar is engine. If one or wo persons give a sudden push (unbalanced force) o i, i hardly sars. Bu a coninuous push over some ime resuls in a gradual acceleraion of he car o his speed. I means ha he change of momenum of he car is no only deermined by he magniude of he force bu also by he ime during which he force is exered. I may hen also be concluded ha he force necessary o change he momenum of an objec depends on he ime rae a which he momenum is changed. The second law of moion saes ha he rae of change of momenum of an objec is proporional o he applied unbalanced force in he direcion of force MATHEMATICAL FORMULATION OF SECOND LAW OF MOTION Suppose an objec of mass, m is moving along a sraigh line wih an iniial velociy, u. I is uniformly acceleraed o velociy, v in ime, by he applicaion of a consan force, F hroughou he ime,. The iniial and final momenum of he objec will be, p 1 = mu and p 2 = mv respecively. The change in momenum p 2 p 1 mv mu m (v u). The rae of change of momenum Or, he applied force, m ( v u) F m ( v u) ( ) F = km v u (9.2) = k m a (9.3) Here a [ = (v u)/ ] is he acceleraion, which is he rae of change of velociy. The quaniy, k is a consan of proporionaliy. The SI unis of mass and acceleraion are kg and m s -2 respecively. The uni of force is so chosen ha he value of he consan, k becomes one. For his, one uni of force is defined as he amoun ha produces an acceleraion of 1 m s -2 in an objec of 1 kg mass. Tha is, 1 uni of force = k (1 kg) (1 m s -2 ). Thus, he value of k becomes 1. From Eq. (9.3) F = ma (9.4) The uni of force is kg m s -2 or newon, which has he symbol N. The second law of FORCE AND LAWS OF MOTION 119

7 moion gives us a mehod o measure he force acing on an objec as a produc of is mass and acceleraion. The second law of moion is ofen seen in acion in our everyday life. Have you noiced ha while caching a fas moving cricke ball, a fielder in he ground gradually pulls his hands backwards wih he moving ball? In doing so, he fielder increases he ime during which he high velociy of he moving ball decreases o zero. Thus, he acceleraion of he ball is decreased and herefore he impac of caching he fas moving ball (Fig. 9.8) is also reduced. If he ball is sopped suddenly hen is high velociy decreases o zero in a very shor inerval of ime. Thus, he rae of change of momenum of he ball will be large. Therefore, a large force would have o be applied for holding he cach ha may hur he palm of he fielder. In a high jump ahleic even, he ahlees are made o fall eiher on a cushioned bed or on a sand bed. This is o increase he ime of he ahlee s fall o sop afer making he jump. This decreases he rae of change of momenum and hence he force. Try o ponder how a karae player breaks a slab of ice wih a single blow. Fig. 9.8: A fielder pulls his hands gradually wih he moving ball while holding a cach. The firs law of moion can be mahemaically saed from he mahemaical expression for he second law of moion. Eq. (9.4) is or or F = ma F ( ) = m v u F = mv mu (9.5) Tha is, when F = 0, v = u for whaever ime, is aken. This means ha he objec will coninue moving wih uniform velociy, u hroughou he ime,. If u is zero hen v will also be zero. Tha is, he objec will remain a res. Example 9.1 A consan force acs on an objec of mass 5 kg for a duraion of 2 s. I increases he objec s velociy from 3 m s 1 o 7 m s -1. Find he magniude of he applied force. Now, if he force was applied for a duraion of 5 s, wha would be he final velociy of he objec? Soluion: We have been given ha u = 3 m s 1 and v = 7 m s -1, = 2 s and m = 5 kg. From Eq. (9.5) we have, ( ) F = m v u Subsiuion of values in his relaion gives F = 5 kg (7 m s -1 3 m s -1 )/2 s = 10 N. Now, if his force is applied for a duraion of 5 s ( = 5 s), hen he final velociy can be calculaed by rewriing Eq. (9.5) as v = u + F m On subsiuing he values of u, F, m and, we ge he final velociy, v = 13 m s SCIENCE

8 Example 9.2 Which would require a greaer force acceleraing a 2 kg mass a 5 m s 2 or a 4 kg mass a 2 m s -2? Soluion: From Eq. (9.4), we have F = ma. Here we have m 1 = 2 kg; a 1 = 5 m s -2 and m 2 = 4 kg; a 2 = 2 m s -2. Thus, F 1 = m 1 a 1 = 2 kg 5 m s -2 = 10 N; and F 2 = m 2 a 2 = 4 kg 2 m s -2 = 8 N. F 1 > F 2. Thus, acceleraing a 2 kg mass a 5 m s -2 would require a greaer force. Example 9.3 A moorcar is moving wih a velociy of 108 km/h and i akes 4 s o sop afer he brakes are applied. Calculae he force exered by he brakes on he moorcar if is mass along wih he passengers is 1000 kg. Soluion: The iniial velociy of he moorcar u = 108 km/h = m/(60 60 s) = 30 m s -1 and he final velociy of he moorcar v = 0 m s -1. The oal mass of he moorcar along wih is passengers = 1000 kg and he ime aken o sop he moorcar, = 4 s. From Eq. (9.5) we have he magniude of he force applied by he brakes F as m(v u)/. On subsiuing he values, we ge F = 1000 kg (0 30) m s -1 /4 s = 7500 kg m s -2 or 7500 N. The negaive sign ells us ha he force exered by he brakes is opposie o he direcion of moion of he moorcar. Example 9.4 A force of 5 N gives a mass m 1, an acceleraion of 10 m s 2 and a mass m 2, an acceleraion of 20 m s -2. Wha acceleraion would i give if boh he masses were ied ogeher? Soluion: From Eq. (9.4) we have m 1 = F/a 1 ; and m 2 = F/a 2. Here, a 1 = 10 m s -2 ; a 2 = 20 m s -2 and F = 5 N. Thus, m 1 = 5 N/10 m s -2 = 0.50 kg; and m 2 = 5 N/20 m s -2 = 0.25 kg. If he wo masses were ied ogeher, he oal mass, m would be m = 0.50 kg kg = 0.75 kg. The acceleraion, a produced in he combined mass by he 5 N force would be, a = F/m = 5 N/0.75 kg = 6.67 m s -2. Example 9.5 The velociy-ime graph of a ball of mass 20 g moving along a sraigh line on a long able is given in Fig Fig. 9.9 How much force does he able exer on he ball o bring i o res? Soluion: The iniial velociy of he ball is 20 cm s -1. Due o he fricion force exered by he able, he velociy of he ball decreases down o zero in 10 s. Thus, u = 20 cm s 1 ; v = 0 cm s -1 and = 10 s. Since he velociy-ime graph is a sraigh line, i is clear ha he ball moves wih a consan acceleraion. The acceleraion a is, a = v u = (0 cm s cm s -1 )/10 s = 2 cm s -2 = 0.02 m s -2. FORCE AND LAWS OF MOTION 121

9 The force exered on he ball F is, F = ma = (20/1000) kg ( 0.02 m s-2 ) = N. The negaive sign implies ha he fricional force exered by he able is opposie o he direcion of moion of he ball. 9.5 Third Law of Moion The firs wo laws of moion ell us how an applied force changes he moion and provide us wih a mehod of deermining he force. The hird law of moion saes ha when one objec exers a force on anoher objec, he second objec insananeously exers a force back on he firs. These wo forces are always equal in magniude bu opposie in direcion. These forces ac on differen objecs and never on he same objec. In he game of fooball someimes we, while looking a he fooball and rying o kick i wih a greaer force, collide wih a player of he opposie eam. Boh feel hur because each applies a force o he oher. In oher words, here is a pair of forces and no jus one force. The wo opposing forces are also known as acion and reacion forces. Le us consider wo spring balances conneced ogeher as shown in Fig The fixed end of balance B is aached wih a rigid suppor, like a wall. When a force is applied hrough he free end of spring balance A, i is observed ha boh he spring balances show he same readings on heir scales. I means ha he force exered by spring balance A on balance B is equal bu opposie in direcion o he force exered by he balance B on balance A. The force which balance A exers on balance B is called he acion and he force of balance B on balance A is called he reacion. This gives us an alernaive saemen of he hird law of moion i.e., o every acion here is an equal and opposie reacion. However, i mus be remembered ha he acion and reacion always ac on wo differen objecs. Fig. 9.10: Acion and reacion forces are equal and opposie. Suppose you are sanding a res and inend o sar walking on a road. You mus accelerae, and his requires a force in accordance wih he second law of moion. Which is his force? Is i he muscular effor you exer on he road? Is i in he direcion we inend o move? No, you push he road below backwards. The road exers an equal and opposie reacion force on your fee o make you move forward. I is imporan o noe ha even hough he acion and reacion forces are always equal in magniude, hese forces may no produce acceleraions of equal magniudes. This is because each force acs on a differen objec ha may have a differen mass. When a gun is fired, i exers a forward force on he bulle. The bulle exers an equal and opposie reacion force on he gun. This resuls in he recoil of he gun (Fig. 9.11). Since he gun has a much greaer mass han he bulle, he acceleraion of he gun is much less han he acceleraion of he bulle. The hird law of moion can also be illusraed when a sailor jumps ou of a rowing boa. As he sailor jumps forward, he force on he boa moves i backwards (Fig. 9.12). Fig. 9.11: A forward force on he bulle and recoil of he gun. 122 SCIENCE

10 The car shown in his aciviy can be consruced by using a 12 mm or 18 mm hick plywood board of abou 50 cm 100 cm wih wo pairs of hard ball-bearing wheels (skae wheels are good o use). Skaeboards are no as effecive because i is difficul o mainain sraigh-line moion. Fig. 9.12: As he sailor jumps in forward direcion, he boa moves backwards. Aciviy 9.4 Reques wo children o sand on wo separae cars as shown in Fig Give hem a bag full of sand or some oher heavy objec. Ask hem o play a game of cach wih he bag. Does each of hem receive an insananeous reacion as a resul of hrowing he sand bag (acion)? You can pain a whie line on carwheels o observe he moion of he wo cars when he children hrow he bag owards each oher. 9.6 Conservaion of Momenum Suppose wo objecs (wo balls A and B, say) of masses m A and m B are ravelling in he same direcion along a sraigh line a differen velociies u A and u B, respecively [Fig. 9.14(a)]. And here are no oher exernal unbalanced forces acing on hem. Le u A > u B and he wo balls collide wih each oher as shown in Fig. 9.14(b). During collision which lass for a ime, he ball A exers a force F AB on ball B and he ball B exers a force F BA on ball A. Suppose v A and v B are he velociies of he wo balls A and B afer he collision, respecively [Fig. 9.14(c)]. Fig. 9.14: Conservaion of momenum in collision of wo balls. Fig Now, place wo children on one car and one on anoher car. The second law of moion can be seen, as his arrangemen would show differen acceleraions for he same force. From Eq. (9.1), he momena (plural of momenum) of ball A before and afer he collision are m A u A and m A v A, respecively. The rae of change of is momenum (or F AB, acion) ( va ua ) during he collision will be m A. Similarly, he rae of change of momenum of ball B (= F BA or reacion) during he collision ( vb ub) will be m B. According o he hird law of moion, he force F AB exered by ball A on ball B (acion) FORCE AND LAWS OF MOTION 123

11 and he force F BA exered by he ball B on ball A (reacion) mus be equal and opposie o each oher. Therefore, F AB = F BA (9.6) ( va ua ) ( vb ub) or m A = m B. This gives, m A u A + m B u B = m A v A + m B v B (9.7) Since (m A u A + m B u B ) is he oal momenum of he wo balls A and B before he collision and (m A v A + m B v B ) is heir oal momenum afer he collision, from Eq. (9.7) we observe ha he oal momenum of he wo balls remains unchanged or conserved provided no oher exernal force acs. As a resul of his ideal collision experimen, we say ha he sum of momena of he wo objecs before collision is equal o he sum of momena afer he collision provided here is no exernal unbalanced force acing on hem. This is known as he law of conservaion of momenum. This saemen can alernaively be given as he oal momenum of he wo objecs is unchanged or conserved by he collision. Aciviy 9.5 Take a big rubber balloon and inflae i fully. Tie is neck using a hread. Also using adhesive ape, fix a sraw on he surface of his balloon. Pass a hread hrough he sraw and hold one end of he hread in your hand or fix i on he wall. Ask your friend o hold he oher end of he hread or fix i on a wall a some disance. This arrangemen is shown in Fig Now remove he hread ied on he neck of balloon. Le he air escape from he mouh of he balloon. Observe he direcion in which he sraw moves. Fig Aciviy 9.6 Take a es ube of good qualiy glass maerial and pu a small amoun of waer in i. Place a sop cork a he mouh of i. Now suspend he es ube horizonally by wo srings or wires as shown in Fig Hea he es ube wih a burner unil waer vaporises and he cork blows ou. Observe ha he es ube recoils in he direcion opposie o he direcion of he cork. Fig Also, observe he difference in he velociy he cork appears o have and ha of he recoiling es ube. Example 9.6 A bulle of mass 20 g is horizonally fired wih a velociy 150 m s -1 from a pisol of mass 2 kg. Wha is he recoil velociy of he pisol? Soluion: We have he mass of bulle, m 1 = 20 g (= 0.02 kg) and he mass of he pisol, m 2 = 2 kg; iniial velociies of he bulle (u 1 ) and pisol (u 2 ) = 0, respecively. The final velociy of he bulle, v 1 = m s -1. The direcion of bulle is aken from lef o righ (posiive, by convenion, Fig. 9.17). Le v be he recoil velociy of he pisol. 124 SCIENCE

12 Toal momena of he pisol and bulle before he fire, when he gun is a res = ( ) kg 0 m s 1 = 0 kg m s 1 Toal momena of he pisol and bulle afer i is fired = 0.02 kg (+ 150 m s 1 ) + 2 kg v m s 1 = (3 + 2v) kg m s 1 According o he law of conservaion of momenum Toal momena afer he fire = Toal momena before he fire 3 + 2v = 0 v = 1.5 m s 1. Negaive sign indicaes ha he direcion in which he pisol would recoil is opposie o ha of bulle, ha is, righ o lef. Fig. 9.17: Recoil of a pisol Example 9.7 A girl of mass 40 kg jumps wih a horizonal velociy of 5 m s -1 ono a saionary car wih fricionless wheels. The mass of he car is 3 kg. Wha is her velociy as he car sars moving? Assume ha here is no exernal unbalanced force working in he horizonal direcion. Soluion: Le v be he velociy of he girl on he car as he car sars moving. The oal momena of he girl and car before he ineracion = 40 kg 5 m s kg 0 m s 1 = 200 kg m s 1. Toal momena afer he ineracion = (40 + 3) kg v m s 1 = 43 v kg m s 1. According o he law of conservaion of momenum, he oal momenum is conserved during he ineracion. Tha is, 43 v = 200 v = 200/43 = m s 1. The girl on car would move wih a velociy of 4.65 m s 1 in he direcion in which he girl jumped (Fig. 9.18). (a) (b) Fig. 9.18: The girl jumps ono he car. FORCE AND LAWS OF MOTION 125

13 Example 9.8 Two hockey players of opposie eams, while rying o hi a hockey ball on he ground collide and immediaely become enangled. One has a mass of 60 kg and was moving wih a velociy 5.0 m s 1 while he oher has a mass of 55 kg and was moving faser wih a velociy 6.0 m s 1 owards he firs player. In which direcion and wih wha velociy will hey move afer hey become enangled? Assume ha he fricional force acing beween he fee of he wo players and ground is negligible. Soluion: If v is he velociy of he wo enangled players afer he collision, he oal momenum hen = (m 1 + m 2 ) v = ( ) kg v m s 1 = 115 v kg m s 1. Equaing he momena of he sysem before and afer collision, in accordance wih he law of conservaion of momenum, we ge v = 30/115 = 0.26 m s 1. Thus, he wo enangled players would move wih velociy 0.26 m s 1 from righ o lef, ha is, in he direcion he second player was moving before he collision. Fig. 9.19: A collision of wo hockey players: (a) before collision and (b) afer collision. Le he firs player be moving from lef o righ. By convenion lef o righ is aken as he posiive direcion and hus righ o lef is he negaive direcion (Fig. 9.19). If symbols m and u represen he mass and iniial velociy of he wo players, respecively. Subscrips 1 and 2 in hese physical quaniies refer o he wo hockey players. Thus, m 1 = 60 kg; u 1 = + 5 m s -1 ; and m 2 = 55 kg; u 2 = 6 m s -1. The oal momenum of he wo players before he collision = 60 kg (+ 5 m s -1 ) + 55 kg ( 6 m s -1 ) = 30 kg m s -1 Quesions 1. If acion is always equal o he reacion, explain how a horse can pull a car. 2. Explain, why is i difficul for a fireman o hold a hose, which ejecs large amouns of waer a a high velociy. 3. From a rifle of mass 4 kg, a bulle of mass 50 g is fired wih an iniial velociy of 35 m s 1. Calculae he iniial recoil velociy of he rifle. 126 SCIENCE

14 4. Two objecs of masses 100 g and 200 g are moving along he same line and direcion wih velociies of 2 m s 1 and 1 m s 1, respecively. They collide and afer he collision, he firs objec moves a a velociy of 1.67 m s 1. Deermine he velociy of he second objec. Wha you have learn CONSERVATION LAWS All conservaion laws such as conservaion of momenum, energy, angular momenum, charge ec. are considered o be fundamenal laws in physics. These are based on observaions and experimens. I is imporan o remember ha a conservaion law canno be proved. I can be verified, or disproved, by experimens. An experimen whose resul is in conformiy wih he law verifies or subsaniaes he law; i does no prove he law. On he oher hand, a single experimen whose resul goes agains he law is enough o disprove i. The law of conservaion of momenum has been deduced from large number of observaions and experimens. This law was formulaed nearly hree cenuries ago. I is ineresing o noe ha no a single siuaion has been realised so far, which conradics his law. Several experiences of every-day life can be explained on he basis of he law of conservaion of momenum. Firs law of moion: An objec coninues o be in a sae of res or of uniform moion along a sraigh line unless aced upon by an unbalanced force. The naural endency of objecs o resis a change in heir sae of res or of uniform moion is called ineria. The mass of an objec is a measure of is ineria. Is SI uni is kilogram (kg). Force of fricion always opposes moion of objecs. Second law of moion: The rae of change of momenum of an objec is proporional o he applied unbalanced force in he direcion of he force. The SI uni of force is kg m s 2. This is also known as newon and represened by he symbol N. A force of one newon produces an acceleraion of 1 m s 2 on an objec of mass 1 kg. The momenum of an objec is he produc of is mass and velociy and has he same direcion as ha of he velociy. Is SI uni is kg m s 1. Third law of moion: To every acion, here is an equal and opposie reacion and hey ac on wo differen bodies. In an isolaed sysem, he oal momenum remains conserved. FORCE AND LAWS OF MOTION 127

15 Exercises 1. An objec experiences a ne zero exernal unbalanced force. Is i possible for he objec o be ravelling wih a non-zero velociy? If yes, sae he condiions ha mus be placed on he magniude and direcion of he velociy. If no, provide a reason. 2. When a carpe is beaen wih a sick, dus comes ou of i. Explain. 3. Why is i advised o ie any luggage kep on he roof of a bus wih a rope? 4. A basman his a cricke ball which hen rolls on a level ground. Afer covering a shor disance, he ball comes o res. The ball slows o a sop because (a) he basman did no hi he ball hard enough. (b) velociy is proporional o he force exered on he ball. (c) here is a force on he ball opposing he moion. (d) here is no unbalanced force on he ball, so he ball would wan o come o res. 5. A ruck sars from res and rolls down a hill wih a consan acceleraion. I ravels a disance of 400 m in 20 s. Find is acceleraion. Find he force acing on i if is mass is 7 meric onnes (Hin: 1 meric onne = 1000 kg.) 6. A sone of 1 kg is hrown wih a velociy of 20 m s 1 across he frozen surface of a lake and comes o res afer ravelling a disance of 50 m. Wha is he force of fricion beween he sone and he ice? 7. A 8000 kg engine pulls a rain of 5 wagons, each of 2000 kg, along a horizonal rack. If he engine exers a force of N and he rack offers a fricion force of 5000 N, hen calculae: (a) he ne acceleraing force; (b) he acceleraion of he rain; and (c) he force of wagon 1 on wagon An auomobile vehicle has a mass of 1500 kg. Wha mus be he force beween he vehicle and road if he vehicle is o be sopped wih a negaive acceleraion of 1.7 m s 2? 9. Wha is he momenum of an objec of mass m, moving wih a velociy v? (a) (mv) 2 (b) mv 2 (c) ½ mv 2 (d) mv 10. Using a horizonal force of 200 N, we inend o move a wooden cabine across a floor a a consan velociy. Wha is he fricion force ha will be exered on he cabine? 11. Two objecs, each of mass 1.5 kg, are moving in he same sraigh line bu in opposie direcions. The velociy of each 128 SCIENCE

16 objec is 2.5 m s -1 before he collision during which hey sick ogeher. Wha will be he velociy of he combined objec afer collision? 12. According o he hird law of moion when we push on an objec, he objec pushes back on us wih an equal and opposie force. If he objec is a massive ruck parked along he roadside, i will probably no move. A suden jusifies his by answering ha he wo opposie and equal forces cancel each oher. Commen on his logic and explain why he ruck does no move. 13. A hockey ball of mass 200 g ravelling a 10 m s 1 is sruck by a hockey sick so as o reurn i along is original pah wih a velociy a 5 m s 1. Calculae he change of momenum occurred in he moion of he hockey ball by he force applied by he hockey sick. 14. A bulle of mass 10 g ravelling horizonally wih a velociy of 150 m s 1 srikes a saionary wooden block and comes o res in 0.03 s. Calculae he disance of peneraion of he bulle ino he block. Also calculae he magniude of he force exered by he wooden block on he bulle. 15. An objec of mass 1 kg ravelling in a sraigh line wih a velociy of 10 m s 1 collides wih, and sicks o, a saionary wooden block of mass 5 kg. Then hey boh move off ogeher in he same sraigh line. Calculae he oal momenum jus before he impac and jus afer he impac. Also, calculae he velociy of he combined objec. 16. An objec of mass 100 kg is acceleraed uniformly from a velociy of 5 m s 1 o 8 m s 1 in 6 s. Calculae he iniial and final momenum of he objec. Also, find he magniude of he force exered on he objec. 17. Akhar, Kiran and Rahul were riding in a moorcar ha was moving wih a high velociy on an expressway when an insec hi he windshield and go suck on he windscreen. Akhar and Kiran sared pondering over he siuaion. Kiran suggesed ha he insec suffered a greaer change in momenum as compared o he change in momenum of he moorcar (because he change in he velociy of he insec was much more han ha of he moorcar). Akhar said ha since he moorcar was moving wih a larger velociy, i exered a larger force on he insec. And as a resul he insec died. Rahul while puing an enirely new explanaion said ha boh he moorcar and he insec experienced he same force and a change in heir momenum. Commen on hese suggesions. 18. How much momenum will a dumb-bell of mass 10 kg ransfer o he floor if i falls from a heigh of 80 cm? Take is downward acceleraion o be 10 m s 2. FORCE AND LAWS OF MOTION 129

17 Addiional Exercises A1. The following is he disance-ime able of an objec in moion: Time in seconds Disance in meres (a) Wha conclusion can you draw abou he acceleraion? Is i consan, increasing, decreasing, or zero? (b) Wha do you infer abou he forces acing on he objec? A2. Two persons manage o push a moorcar of mass 1200 kg a a uniform velociy along a level road. The same moorcar can be pushed by hree persons o produce an acceleraion of 0.2 m s -2. Wih wha force does each person push he moorcar? (Assume ha all persons push he moorcar wih he same muscular effor.) A3. A hammer of mass 500 g, moving a 50 m s -1, srikes a nail. The nail sops he hammer in a very shor ime of 0.01 s. Wha is he force of he nail on he hammer? A4. A moorcar of mass 1200 kg is moving along a sraigh line wih a uniform velociy of 90 km/h. Is velociy is slowed down o 18 km/h in 4 s by an unbalanced exernal force. Calculae he acceleraion and change in momenum. Also calculae he magniude of he force required. A5. A large ruck and a car, boh moving wih a velociy of magniude v, have a head-on collision and boh of hem come o a hal afer ha. If he collision lass for 1 s: (a) Which vehicle experiences he greaer force of impac? (b) Which vehicle experiences he greaer change in momenum? (c) Which vehicle experiences he greaer acceleraion? (d) Why is he car likely o suffer more damage han he ruck? 130 SCIENCE