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1 WORK POWER AND ENERGY Consevaive foce a) A foce is said o be consevaive if he wok done by i is independen of pah followed by he body b) Wok done by a consevaive foce fo a closed pah is zeo c) Wok done by a consevaive foce depends only on he iniial and final posiions of he body d) Wok done by a consevaive foce is he poduc of Foce and displacemen e) Duing a ound ip he body aains he same iniial KE Ex Gaviaional foce, Elecosaic foce ec Non  Consevaive foce a) A foce is said o be non consevaive if he wok done by i depends on he pah followed by he body b) Wok done by a nonconsevaive foce fo a closed pah is no zeo c) Duing a ound ip he body aains a diffeen KE as ha of iniial d) Wok done by a nonconsevaive foce is he poduc of Foce and disance e) Due o a nonconsevaive hee may be a loss of mechanical enegy bu he oal enegy is consan Ex Ficional foce 3 Wok: a Wok is said o be done when he poin of applicaion of foce has some displacemen in he diecion of he foce b) The amoun of wok done is given by he do poduc of foce and displacemen Fs W = = Fscosθ c) Wok is independen of he ime aken and is a scala d) If he foce and displacemen ae pependicula o each ohe, hen he wok done is zeo e) A peson owing a boa upseam is a es wih espec o an obseve on he shoe Accoding o he obseve he peson does no pefom any wok Howeve, he peson pefoms wok agains he flow of wae If he sops owing he boa, he boa moves in he diecion of flow of wae and wok is pefomed by he foce due o flow, as hee is displacemen in he diecion of flow f) If he wok is done by a unifomly vaying foce such as esoing foce in a sping, hen he wok done is equal o he poduc of aveage foce and displacemen g) If he foce is vaying non unifomly, hen he wok done = F ds = F ds cos θ h) The aea of F s gaph gives he wok done i) SI uni of wok is joule j) Joule is he wok done when a foce of one Newon displaces a body hough one mee in he diecion of foce k) CGS uni of wok is eg; J = 0 7 egs l) If he foce o is componen is in a diecion opposie o he displacemen, he wok is negaive Ex When a body is lifed veically upwads, he wok done by he gaviaional foce is negaive, as he displacemen is upwad wheeas he gaviaional foce is acing downwads m) The wok done in lifing an objec of mass m hough a heigh h is equal o mgh n) When a body of mass m is aised fom a heigh h o heigh h, hen he wok done = mg(h h ) wwwsakshieducaioncom
2 o) Le a body be lifed hough a heigh 'h' veically upwads by a foce 'F' acing upwads Then, he wok done by he esulan foce is W = (F mg)h p) The wok done on a sping in seching o compessing i hough a disance x is given W = kx whee k is he foce consan o sping consan q) Wok done in changing he elongaion of a sping fom x o x is W = k ( x x ) ) The wok done in pulling he bob of a simple pendulum of lengh L hough an angle W = mgl( cos θ ) s) The wok done in lifing a homogeneous meal od lying on he gound such ha i makes an angle 'θ' wih he hoizonal, is W = mglsin θ ) The wok done in oaing a od o ba of mass m hough an angle θ abou a poin of suspension is mgl W = ( cos θ ) = mglsin (θ/) whee L is he disance of he cene of gaviy fom he poin of suspension u) The wok done in lifing a body of mass 'm' and densiy 'd s ' in a liquid of densiy 'd l ' hough a heigh 'h' unde gaviy is W = m g h dl h ds v) Wok done in pulling back a pa of lengh of a chain hanging fom he edge ono a smooh n mgl hoizonal able compleely is W = n w) Inclined plane : i Wok done in moving a block of mass 'm' up a smooh inclined plane of inclinaion 'θ' hough a disance 's' is W = Fs = (mg sinθ) s ii If he plane is ough, hen W = mg (sinθ+µ k cosθ)s x) Wok done by a posiion dependen foce If he posiion of a body changes fom x o x hen wok done is given by W x = x F dx = aea unde F S cuve y) Wok done by a ime dependen foce F a = m Now dv a = v = dv = a And W = m( v u ) z) Wok done when posiion depends on ime ds v = When = v= v and when = v= v,hen wwwsakshieducaioncom
3 W = m( v v ) 4 Spings: a) The esoing foce on he sping pe uni elongaion is called foce consan o sping consan F =  Kx (Negaive sign indicaes ha he foce is opposie o elongaion) b) The wok done in seching o compessing a sing hough x is given by W = Kx F c) W = x x = Fx F Q K = x F d) W = F F F = Q x = K k k W = K x x Also, W = ( F x F x ) e) If he sping is seched fom x and x hen he wok don is given by ( ) f) If a body is dopped fom a heigh h on o a sping of consan k, and if x is he compession in he sping, hen mg ( h + x) = kx g) If ai ficion is consideed, ( mg f )( h + x) = kx h) If a body of mass on moving wih a speed v collides a sping in is pih and compesses he sping hough x hen, mv = kx If ficion is consideed kx = mv f s = mv µ mgs i) If a sping of sping consan k is cu ino n equal pas he sping consan of each pa is nk k l = k l = k l = = k l + l + l j) If a sping of consan k is cu ino unequal pas, hen ( ) 5 Powe: a Rae of doing wok is called powe wok Powe = = Foce x velociy ime wwwsakshieducaioncom b SI uni of powe is wa and CGS uni is eg/second hose powe = 746 wa c If a vehicle avels wih a speed of v ovecoming a oal esisance of F, hen he powe of he engine is given by P = F v dθ d If a body is oaed in cicula pah, he powe exeed is given by P = τ = τω e If a block of mass 'm' is pulled along he smooh inclined plane of angle 'θ', wih consan velociy 'v', hen he powe exeed is, p = (mg sin θ)v f If he block is pulled up a ough inclined plane hen he powe is P = mg (sinθ + µ k cosθ)v
4 g If he block is pulled down a ough inclined plane hen he powe is P = mg (sinθ  µ k cosθ)v h When wae is coming ou fom a hose pipe of aea of coss secion 'A' wih a velociy 'v' and his a wall nomally and i) sops dead, hen foce exeed by he wae on he wall is Av ρ And he powe exeed by wae is P = A v 3 ρ (ρ = densiy of wae) ii) If wae ebounds wih same velociy (v) afe siking he wall, P = Av 3 ρ dm i When sand dops fom a saionay doppe a a ae of on o a conveye bel moving wih a consan velociy, hen he exa foce equied o keep he bel moving wih a consan speed V is dm given by F = v and he powe equied = P = dm v j If a pump lifs he wae fom a well of deph 'h' and impas some velociy 'v' o he wae, hen he powe of pump P = mgh + mv k Powe exeed by a machine gun which fies 'n' bulles in ime '' is P = l If a pump delives V lies of wae ove a heigh of h mees in one minue, hen he powe of he engine (P) = Vgh 60 m A moo sends a liquid wih a velociy 'V' in a ube of coss secion 'A' and 'd' is he densiy of he liquid, hen he powe of he moo is P = AdV 3 n A body of mass M iniially a es on a smooh hoizonal suface acceleaes unifomly and acquies velociy V in ime The wok done on he body in ime is mv i) Wok done = 6 ENERGY: ii) Aveage powe = W = mv mv iii) Insananeous powe = Fv = wwwsakshieducaioncom mnv a) The enegy possessed by a body by viue of is moion is called kineic enegy Ex A fied bulle, blowing wind, ec Fo a body of mass m is moving wih a velociy v, kineic enegy = mv b) The enegy possessed by a body by viue of is moion is called kineic enegy Ex A ben bow, Wae soed in a esevoi ec c) A flying bid possesses boh KE and PE d) The wok done in lifing an objec of mass m hough a heigh h is soed as poenial enegy in he body and i is equal o mgh e) Wok enegy heoem : The wok done by he consan esulan foce acing on a body is equal o he change in is kineic enegy W = mv mu
5 P f) If he kineic enegy of a body of mass m is E and is momenum is P, hen E = m g) If he momenum of he body inceased by n imes, KE incease by n imes h) If he KE of he body inceases by n imes, he momenum inceases by n imes P i) If he momenum of he body inceases by P%, % incease in KE= + P% 00 j) If he momenum of he body deceases by p%, % decease in KE= P p% 00 k) If he KE of he body inceases by E%, % incease in momenum= + E 00% 00 E 00 l) If he KE of he body deceases by E%, % decease in momenum= + 00% m) If wo bodies, one heavie and he ohe lighe ae moving wih he same momenum, hen he lighe body possesses geae kineic enegy n) If wo bodies, one heavie and he ohe lighe have he same KE hen he heavie body possesses geae momenum o) Two bodies, one is heavie and he ohe is lighe ae moving wih he same momenum If hey ae sopped by he same eading foce, hen i) The disance avelled by he lighe body is geae (s m ) ii) They will come o es wihin he same ime ineval p) Two bodies, one is heavie and he ohe is lighe ae moving wih same kineic enegy If hey ae sopped by he same eading foce, hen i) The disance avelled by boh he bodies ae same ii) The ime aken by he heavie body will be moe ( m ) wwwsakshieducaioncom
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