PHY 143 Basic Physics For Engineers I. E-Notes

Transcription

1 PHY 43 Basic Physics Fo Enginees I E-Noes Pepaed by Mohd Noo Mohd Ali Physics Lecue Applied Science Depamen Univesiy Teknologi MARA Pulau Pinang Offeed Since July 007

2 Physics and measuemens ( hs)... 4 Unis... 4 Fundamenal Quaniies... 4 Deived Quaniies... 4 Pefixes... 4 Significan figue... 5 Convesion of unis... 5 Scala s and vecos (3 hs)... 7 Definiion... 7 Veco Noaion... 7 Addiion (& subacion) of veco componens - Geomeical Mehod... 7 Addiion (& subacion) of veco componens - Uni veco mehod... 8 Resoluion of vecos ino x and y componens... 8 Kinemaics (3 hs)... 0 Posiion, displacemen and velociy... 0 Insananeous velociy is defined as he velociy a a paicula insance in ime Acceleaion, dv/d... Gaph of velociy vesus ime... Consan acceleaion in linea moion... 3 Moion in wo dimension... 3 Fee Fall moion... 4 Newon s Laws of Moion, Linea momenum and collisions (5hs)... 5 Definiion... 5 The Foce Law (Newon s Second Law)... 5 Types of foces (Gaviaional, Nomal & Ficion)... 5 Fee body diagam... 6 Linea momenum and Collisions... 6 Collecing he componens... 7 Collision beween wo bodies... 7 One moving, one saionay iniially... 7 Boh moving wih diffeen velociy in he same diecion iniially... 7 Boh objecs moving owads each ohe iniially Boh objecs move wih diffeen velociies afe collision... 7 One objec saionay, one moving afe collision Boh objecs moves ogehe wih he same velociy afe collision... 7 Wok, enegy and powe (3 hs)... 7 Wok, enegy & powe... 7 Consevaion of mechanical Enegy... 8 Wok-Enegy heoem... 8 Roaional moion (3 hs)... 9 Moion of a body oaing/ spinning abou an axis When a body is oaing/spinning abou an axis hen each poins on he body ae moving in cicula pahs ceneed a he axis of oaion... 9 Angula posiion, angula displacemen and angula velociy... 9 Noe: If moving CCW, angula velociy is posiive, moving CW, angula velociy is negaive... 9 Angula acceleaion... 9 Consan angula acceleaion in oaional moion... 9 Relaion Beween Linea and Roaional Quaniies... 0 Saic equilibium (3 hs)...

3 Equilibium of a paicle... Definiion of Toque... Equilibium of igid body... Oscillaoy moion (3 hs)... Peiodic moion... Simple hamonic moion (SHM)... Simple Pendulum... 3 Consevaion of Enegy in SHM... 4 Mechanics of solids ( hs)... 5 Elasiciy... 5 Young s Modulus... 5 Shea modulus... 5 Bulk modulus... 6 Mechanics of fluids ( hs)... 7 Densiy and elaive densiy... 7 Pessue... 7 Buoyancy... 8 Vibaions and waves ( hs) Inoducion Type of waves Chaaceisic of wave moion The popagaion of Waves... 3 Wave equaions... 3 Tempeaue, Themal expansion and he Ideal Gas Law (4 hs) Tempeaue and hemomees Themal expansion The gas laws and Absolue empeaue The ideal gas law Hea and The Fis Law of Themodynamics (4 hs) Hea as enegy ansfe Specific hea Laen hea of fusion and evapoaion The fis law of hemodynamics wok done by a gas

4 Physics and measuemens ( hs) Physics is an empiical science, which means ha measuemens ae made. To communicae he measuemens, sysem of unis is used. Compaison beween he measuemens can be made if a common uni of measue is used, o a leas a known convesion is know, Unis A common sysem of unis used in physics is he SI (Syseme Inenaional d Unies), which is a evised meic sysem. This sysem uses mee fo lengh, kilogam fo mass and secaon fo ime. Fundamenal Quaniies Fundamenal quaniies ae he basic quaniy of measue. The fundamenal quanies of he SI unis, is coesponding symbol, he uni of measue and symbol fo uni of measue is given in he able below. Quaniy of measue Uni of measue Symbol fo Uni Lengh, l mee m Mass, m kilogam kg Time, second s elecical cuen, I ampee A Tempeaue, T kelvin K amoun of subsance mole mol luminous inensiy. candela cd Deived Quaniies Deived quaniies ae quaniies which can be descibed using he fundamenal quaniies. Some examples ae given below. Deived quaniies Unis Symbol of uni momenum kilogam mee pe second kg m s - Foce Newon, kilogam mee N, kg m s - pe second pe second Enegy Joule, Newon.mee J, kg m s - Deived quaniies ae obained when physical quaniies ae muliplied o divided wih one anohe. Subaion o addiion will no poduce deived quaniies. E.g..3 m + 3. m 3.33 m same quaniy.3 m x 3. m 3.95 m aea, a deived quaniy Pefixes Pefixes ae used wih unis o indicae powe of 0 facos. Pefix Powe of 0 pea (P) 0 5 4

5 ea (T) 0 giga (G) 0 9 mega (M) 0 6 kilo (k) 0 3 deci (d) 0 - ceni 0 - milli (m) 0-3 mico (µ) 0-6 nano (n) 0-9 pico (p) 0 - fempo (f) 0-5 Significan figue When measuemens ae made, he accuacy of he measuing insumens deemine he numbe of significan figues (digis) ha can be aken fo he measuemen. This is he numbe of significan figues due o he pecision of measuing insumen. Example Mee ule smalles division is mm, Theefoe eadings can be made o he neaes mm, e.g m gives 3 significan figues. Venie Calipe eadings can be made o he neaes /0 of mm, eg 5.4 cm 3 sig. fig Micomee scew gauge eadings can be made o /00 of mm, eg mm 4 sig fig When quaniies undego mahemaical opeaion, he numbe of significan figue will usually deceases. The esul of a mahemaical opeaion will usually have he smalle numbe of significan figue fom he significan figues of he quaniies ha undego he mahemaical opeaion. Mahemaical opeaion does no incease he pecision of he esul. Ex m x 4.56 mm x 0-3 m 8.5 x 0-3 m o 3 significan figues m mm m m o he decimal place of he coase insumen (mee ule) Convesion of unis Convesion of unis is done by eaing he pefix value faco and unis as odinay vaiables Convesion due o pefix 3.4 m ino cm 00 cm m 3.4 m x 00 cm / m 34 cm 5

6 O cm x 0 - m 3.4 x cm / x 0 - m 34 cm Convesion due o uni s sysem 5 fee 5 inch ino m fee inch inch.54 cm 5 fee 5 inch 65 inch x.54 cm / inch 65 cm x m / 00 cm.65 m 6

7 Scala s and vecos (3 hs) Definiion Scalas ae physical quaniies which have magniude only. Vecos ae physical quaniies which have magniude and diecion. Example of scalas: disance avelled, ime, enegy Example of vecos: foce, momenum, velociy, acceleaion Veco Noaion A veco is epesened gaphically by an aow. The size (lengh) of he aow epesens he magniude of he veco, he diecion of he aow is he diecion of he veco. In wien fom a veco symbol is wien in bold ypeface, A, o wih an aow A ove he symbol. The diecion has o be saed expliciely Addiion (& subacion) of veco componens - Geomeical Mehod When wo vecos ae added ogehe, he ail of he second veco is conneced o he ip/head of he fis veco. The esul of he addiion is called he esulan veco. Noe : A + B B + A Veco subacion is he addiion of a veco wih a negaive veco. A negaive veco is a veco having he same magniude as he oiginal veco bu opposie in diecion. Example A - B A + -B Noice: B is in opposie diecion of B 7

8 Noe : A + - B - (B + -A) Addiion (& subacion) of veco componens - Uni veco mehod A veco A can be wien in he fom, A A x î + A y ĵ, whee î and ĵ ae uni vecos ( magniude, bu no associaed uni) and poin in he +x and +y diecion especively, and A x and A y ae he x and y componens of he vecos especively. A + B (A x î + A y ĵ ) +( B x î + B y ĵ ) (A x + B x ) î + (A y + B y ) ĵ Resoluion of vecos ino x and y componens. We use igonomey o assis us in woking wih vecos in numbe. A veco can be esolve ino componens along/paallel o a seleced se of axis. Using x-y axes as efeence axes. The veco is esolved ino componens along he x and he y axis. A x A cos θ i A y A sin θ j Noice θ is he angle beween he veco and he posiive x axis B x B cos θ i B y B sin θ j Thus A + B (A x + B x ) î + (A y + B y ) ĵ (A cos θ + B cos θ ) î + (A sin θ + B sin θ ) ĵ 8

9 The magniude of he esulan veco C A + B SQRT[(A cos θ + B cos θ ) + (A sin θ + B sin θ ) ] 9

10 Kinemaics (3 hs) Posiion, displacemen and velociy Posiion he locaion of a poin / paicle / objec, i.e. he labels o coodinae wih espec o a sysem of axes (i.e. x-y-z axes) Posiion Veco is he veco fom he oigin o he posiion / coodinae poin. Disance aveled is he lengh of he pah of he objec Displacemen - The lengh veco along a saigh line beween wo posiions. Nomally he efeence poin is he oigin (0, 0) in he x-y plane (-dim), o x 0 on he x- axis. The posiion veco is epesened by x î + y ĵ (-dim) o x î ( -dim) Speed - The ae of change of disance aveled. Speed is a scala quaniy. Aveage Speed disance aveled divided by ime o cove he disance Insananeous speed is defined as he speed a a paicula insance in ime. Example: A suden walks a disance of 00 mees in 0 seconds. Find he aveage speed of he suden. Aveage speed 00 mees/0 seconds 5 mee/second 5 m/s 5 ms- Velociy - The ae of change of displacemen. Velociy is a veco. Aveage velociy he displacemen divided by ime o cove he disance x x0 x vavg iˆ iˆ 0 Insananeous velociy is defined as he velociy a a paicula insance in ime. v lim 0 x 0

11 The posiion vesus ime gaph is a plo of he paicle posiion wih espec o ime. I shows he paicle posiion a some insances in ime. Fom he definiion of velociy, i x i x x vavg ˆ ˆ 0 0, he aveage velociy beween wo posiion / ime ineval can be obained fom he gaph. The aveage velociy beween 0-5 sec is he slope of he gaph beween 0-5 sec. The aveage velociy beween 5-0 sec, 0-5 sec, 5 0 sec is he slope of he gaph beween 5-0 sec, 0-5 sec, 5 0 sec especively. The aveage velociy beween 0-0 secs is howeve is no he slope of he gaph, and has o be compued. Acceleaion, dv/d Acceleaion - The ae of change of velociy. The acceleaion is a veco. Aveage acceleaion he change in velociy divided by he ime he change occu. 0 0 v v a Along he x axis, i v v a ˆ 0 0 The equaions of moion. i x x vavg ˆ v v a

12 vavg x iˆ vavg s iˆ v u a v u aiˆ iˆ siˆ v avg viˆ uiˆ + aiˆ siˆ v i avg ˆ v u + a s v avg s v avg subsiue vavg v + u v + u s u + a + u s eplace v u + a u + a s s u + Rewiing Then, a s v u a, and v + u s ( v u)( v + u) ( a)( ) v + u as v u + as Gaph of velociy vesus ime

13 The velociy vesus ime gaph gives us he he velociy a some insances in ime. In he above example he paicle has a consan velociy of 4 m s - beween 0 0 sec. Is velociy deceases o m s - in he nex 5 sec (5 0 sec ineval), hen inceases unil i eaches a velociy of 6 m s - a 0 s. The paicle has zeo acceleaion in he ime ineval 0 5 sec, undegoes deceleaion (acceleaion is negaive) beween 5 0 sec, and acceleaion duing he ime ineval 0-0 sec. Consan acceleaion in linea moion A paicle o body moving in a saigh line, fo example along he x- axis is said o be moving in linea moion. The pah of he paicle can be epesened by a saigh line. To simplify analysis, we will only conside he paicle o be moving a consan acceleaion ove a ceain ime. Howeve, he acceleaion of he paicle can be of a diffeen value ove anohe ime ange. A gaph of velociy vesus ime will show saigh line segmens, simila o he velociy vesus ime gaph above. The kinemaics equaions can be use ove he ime when he acceleaion is consan. Example: A paicle moves in a saigh line duing is moion. I moves fom es (velociy 0), wih an acceleaion of 0.5 m s - fo 0 seconds. Is acceleaion hen becomes zeo fo he nex five seconds. I hen deceleaes and sops moving in anohe 5 seconds. Find a) The velociy of he paicle a seconds. b) The disance i moves duing is moion. c) The deceleaion of he paicle. Moion in wo dimension Paicles o bodies which move in a plane, fo example on he x-y plane is said o undego moion in wo- dimension. The velociy of he paicles / bodies can be esolved along wo pependicula diecions (x-y axes). Example: 3

14 An aiplane is moving wih a velociy of 300 km h - in he diecion 030 o. I flew fo 45 minues befoe i changes diecion o 0 o. I hen flew fo anohe 30 minues wih a speed of 350 km h -. Wha is he disance i aveled? Paicles and bodies in pojecile moion also move in wo dimensions he hoizonal and he veical diecion. Example: An aow is sho a angle of 45 0 wih espec o he gound. The aow leaves he bow a 50 ms -. Wha is he hoizonal disance i aveled when i eun o is iniial heigh fom he gound? (Assume hee is no ai esisance and he acceleaion due o gaviy is 9.8 ms - ) Fee Fall moion When an objec is elease fom a heigh above gound, i will fall o he gound wih acceleaion due o gaviy, g which is 9.8 ms-. The acceleaion due o gaviy is always poining owads he gound. If an objec is hown veically upwads wih a ceain speed, i will also fall o he gound wih he acceleaion due o gaviy. Example:. An objec is eleased fom es fom a heigh of 0 m above he gound. Calculae is velociy as i his he gound.. A ball is hown upwads a a speed of 30 ms -. How high does he ball avel, befoe i euns o he gound? Calculae he oal ime i akes duing fligh. 4

15 Newon s Laws of Moion, Linea momenum and collisions (5hs) Lecue Noes PHY43 Definiion Newon s Fis Law saes ha an objec a es will emain a es and an objec in moion will emain in moion unless ac upon by a foce. An objec s velociy does no change if and only if he ne foce acing on he objec is zeo. Newon s Fis Law is also called he law of ineia. Ineia is he esisance o changes in velociy. Ineia is a popey of mae. Ineia is popoional o he mass of he objec. An objec wih a lage mass has a lage ineia han a less massive objec. Newon s Thid Law saes ha fo evey acion hee is an equal and opposie eacion. Foces acs in pais. Evey foce is pa of he ineacion beween wo objecs, and each of he objecs exes a foce on he ohe. Example: When we si on a chai, we exe a foce on he chai. Convesely, he chai exes an equal bu opposie foce on us. The Foce Law (Newon s Second Law) Newon s Second Law saes ha he acceleaion of an objec is popoional o he ne foce on he objec and invesely popoional o he mass of he objec. In mahemaical epesenaion F ne ΣF ma, ΣF is he sum all foces acing on he body of mass m, and a he acceleaion. Example: A consan foce of 0 N is applied o a body of mass N which is iniially a es. Wha is he acceleaion on he body? Calculae is velociy afe 5 seconds. Types of foces (Gaviaional, Nomal & Ficion) A foce is a push o pull. Foces can be caegoized in seveal foms, by how i acs on anohe objec and by he souce of he foce. Conac foces ae foces acing on objecs because he objecs ae in ouch wih one anohe. Fo example he foce fom a chai holding up a siing peson is a conac foce. Long ange foces ae foces which do no equie he objecs o be in ouch. Fo example he foce he sun exes on he eah o keep i oaing aound he sun. The foce beween wo magnes is also an example of a long ange foce. The Gaviaional foce is an example of long ange foce. GMm The gaviaional foce is given as F, whee G is he univesal gaviaional consan, G x 0 - Nm kg -, M and m ae he masses of he objecs in ineacion, and he disance beween he wo objecs. Nea he suface of he eah he gaviaional foce is given as F mg, whee g 9.8 N kg - o ms - is he acceleaion due o gaviy. 5

16 The nomal foce is he eacion foce exeed by a suface in he diecion pependicula (nomal) o he suface. The foce he sea of a chai exes on a peson siing on i is a n example of nomal foce. Ficion o ficional foce is a foce which acs beween wo sliding sufaces and opposes he diecion of moion of he sliding sufaces. Ficion exiss because sufaces ae no micoscopically smooh. The iegulaiies beween he sufaces poduce ficion. Ficion can be educed beween sliding sufaces by polishing he wo sufaces smooh, applying lubican beween he sufaces o inseing balls o olles (beaing). The magniude of he ficional foce is popoional o he nomal foce, f un. The diecion is always opposie he diecion of moion. µ is he coeficien of ficion (saic/dynamic) Fee body diagam A fee body diagam is a diagam of an objec which shows all foces acing on he objec. In a many body poblem, fee body diagams ae dawn fo each body in he poblems. All he foces acing on ha paicula objec ae dawn on he objec. A fee body diagam helps in analyzing a kinemaics poblem. Linea momenum and Collisions Definiion of momenum The linea momenum p of an objec is he poduc of he objec s mass, m and velociy, v p mv Uni : kg ms -, a veco The Pinciple of Consevaion of momenum 6

17 i p i p befoe l l afe Lecue Noes PHY43 The oal linea momenum of an isolaed sysem emains consan (is conseved). An isolaed sysem is one fo which he veco sum of he exenal foces acing on he sysem is zeo. A simple sysem conains wo objec which can ineac wih each ohe. The masses of he objecs emain unchanged, bu hei velociies migh change duing he ineacion. By The Pinciple of Consevaion of Momenum, he oal momenum, (ie he sum of he momenum of he wo objecs) befoe he ineacion is equal o he oal momenum, (ie he sum of he momenum of he wo objecs) afe he ineacion. Sum of momenum befoe ineacion sum of momenum afe ineacion. i m u + m u i i l m v mu mv mv l + l The ineacion beween he objecs is nomally called collision. Soluion Fom, m u + mu mv mv + m ( u ˆ ˆ) ( ˆ ˆ) ( ˆ ˆ) ( ˆ x i + u y j + m uxi + u y j m v xi + v y j + m vxi + v y j Collecing he componens m ( ux iˆ) + m( uxiˆ) m ( vxiˆ) + m( v u + m u m v m v xi m x x x + x m ( u y ˆ) j + m( u y ˆ) j m ( v y ˆ) j + m( v y j m u y + mu y mv y + mv y Collision beween wo bodies One moving, one saionay iniially Boh moving wih diffeen velociy in he same diecion iniially. Boh objecs moving owads each ohe iniially. Boh objecs move wih diffeen velociies afe collision. One objec saionay, one moving afe collision. Boh objecs moves ogehe wih he same velociy afe collision Wok, enegy and powe (3 hs) Wok, enegy & powe Wok is he poduc of he displacemen due o he foce wih he componen of he foce along he diecion of he displacemen. W (F cos θ) s, whee F is he foce, s is he displacemen and θ is he angle beween he foce and displacemen. ˆ) ˆ) ˆ) 7

18 Kineic enegy is he enegy associaed wih moion. The kineic enegy of an objec wih mass m and speed v is given by KE mv Poenial enegy is he enegy associaed wih an objec s posiion o chemical composiion. Poenial enegy in a sping The poenial enegy soed in a sping is given by PE kx Whee k is he sping consan and x, he exension of he sping Gaviaional poenial enegy The enegy of an objec due o is posiion elaive o he suface of he eah. The gaviaional poenial enegy of an objec of mass, m and heigh h, elaive o an abiayly se zeo is PE mgh Uni of Enegy The uni of enegy is joule (J). joule N m joule kg ms- Powe is he ae of doing wok Powe is he ae of enegy used Powe is he ae of enegy supplied P Powe W wok, E Enegy used/supplied ime aken fo wok done Consevaion of mechanical Enegy Mechanical enegy descibes he poenial enegy and kineic enegy pesen in he componens of a mechanical sysem. The mechanical enegy of he sysem is conseved. The enegy changes fom fom kineic enegy o poenial enegy and vice vesa. Noe: Enegy is always conseved, bu in a mechanical sysem he enegy is assumed no o change ino chemical, nuclea o elecical. Wok-Enegy heoem When a ne exenal foce does wok, W on an objec, he kineic enegy of he objec changes fom is iniial value KE o o a final value KE f, he diffeence beween he wo being he wok. W KE f KE o / mv f / mv 0 Enegy appoach in solving kinemaics poblems : Involves kineic and gaviaional poenial enegy, o kineic and sping poenial enegy. 8

19 Roaional moion (3 hs) Moion of a body oaing/ spinning abou an axis. When a body is oaing/spinning abou an axis hen each poins on he body ae moving in cicula pahs ceneed a he axis of oaion If a line is dawn fom each poins o he axis of oaion, he adial lines will have all have he same angula speed abou he axis of oaion. The adial lines moves a he same angula speed. The moion of he adial lines ae common o he body. Thus he oaing body can be descibed using he moion of a adial line on he body. Angula posiion, angula displacemen and angula velociy Angula posiion is defined by θ, whee is he disance fom he axis of oaion and θ he angle he adial line makes wih he posiive x- axis Angula displacemen he change in angula posiion. θ θ f - θ I Angula speed θ θ θ0 Aveage angula speed ω θ dθ Insananeous angula speedω lim 0 d 0 Noe: If moving CCW, angula velociy is posiive, moving CW, angula velociy is negaive Angula acceleaion ω ω ω0 Aveage Angula acceleaion α 0 ω dω Insananeous Angula acceleaion α lim 0 d Consan angula acceleaion in oaional moion When a body is moving wih a consan angula acceleaion in oaional moion, a se of equaions o descibe he moion is obained. 9

20 0 θ α ω ω α ω θ ω ω θ ω ω ω α θ θ θ ϖ ) ( The equaions of moion fo oaional moion ae simila o ha of linea moion. Relaion Beween Linea and Roaional Quaniies The disance along he cicula pah is give as ω θ θ θ d d d d d ds s ) ( Bu T v d ds v T ω Diffeeniaing he velociy wih espec o ime gives he angenial acceleaion T a α ω ω d d d d d dv ) ( a T α The linea velociy and linea acceleaion ae always angen o he cicula pah.

21 Saic equilibium (3 hs) Equilibium of a paicle An paicle is in equilibium if he sum of foces acing on i is zeo. F ma 0 Definiion of Toque Toque/Momen is a veco. I is a esul of a veco muliplicaion called he coss poduc. τ F τ F sinθ ( veco definiion of oque / magniude of oque by veco esoluion and diecon by obsevaion) If he oque/momen causes a oaion in he clockwise diecion, i s diecion is said o be negaive. If he oque/momen causes a oaion in he coune clockwise diecion, i s diecion is said o be posiive. Equilibium of igid body An igid body is in equilibium if he sum of foces acing on i is zeo, and he sum of momens on he body is zeo. Physically, a body in equilibium has zeo anslaional / linea acceleaion and zeo oaional acceleaion. Tanslaional Equilibium F ma 0 Acceleaion is zeo, velociy is consan o zeo. Roaional Equilibium Sum of momen is zeo τ F 0

22 Oscillaoy moion (3 hs) The pendulum and mass sping sysem ae oscillaion sysem whee he plum bob o mass oscillaes abou an equilibium poin. In boh cases he oscillaion follow he simple hamonic moion. Thus he wo sysem ae efeed o as simple hamonic oscillaos. Peiodic moion A moion is peiodic, if i epeas iself a sandad inevals in a specific manne. Simple hamonic moion (SHM) Definiion of Simple Hamonic Moion An objec is said o be moving in a simple hamonic moion if is moion is esiced o a pah, passing hough an equilibium poin, wih is acceleaion popoional o he displacemen fom he equilibium poin and always poining owads he equilibium poin. Using Newon s nd Law, he pevious definiion of S.H.M. imply ha he foce acing on he moving mass is popoional o he displacemen fom he equilibium poin and is poining owads he equilibium poin. F x dv d x bu.. F ma m m d d d x hen... m x d d x o... m kx d F is opposie in diecion o he displacemen, x. Boh ae vecos. k is a consan of popoionaliy The diffeenial equaion has a soluion of he fom d x kx d x( ) Acosω + φ o x( ) Asinω + φ which has sinusoidal fom. x() : displacemen fom he equilibium A : Ampliude max displacemen ω : angula fequency φ : iniial phase T : peiod ime fo one complee cycle f : fequency- no of cycle pe sec /T ω πf π/t

23 The SHM Equaion & Soluion x() A cos ω+φ : displacemen fom equilibium dx()/d -ω A sin ω+φ : velociy d x()/d -ω A cos ω+φ : acceleaion d x()/d - kx() - ω A cos ω+φ - ka cos ω+φ ω k Simple Pendulum The weigh has a componen along he diecion of displacemen bu opposie in diecion. This acs as he foce in he SHM. ma mg sinθ ds m mg sinθ d s lθ ds dθ l d d d s d θ l d d d θ ml mg sinθ d d θ l g sinθ d d θ g sinθ d l d θ g θ d l The final equaion has he SHM fom. Is soluion is fo he angula displacemen θ, whee θ() A sin ω +φ Whee, g ω l ω πf g l g l 3

24 Consevaion of Enegy in SHM Fee vibaion : The SHM sysem is lef o oscillae on is own. The oscillaing mass changes kineic enegy ino poenial enegy and vice vesa duing is moion. If no enegy is los due o ficion o ai esisance, hen he ampliude of he oscillaion emains a consan. Toal enegy emains a consan. Toal enegy ½ mv + ½ kx (sping in SHM) E ½ m(- ω A cos (ω + φ)) + ½ k(a sin (ω + φ)) E ½ m (k/m) A cos (ω + φ) + ½ k A sin (ω + φ) E ½ k A In fee vibaion, he sysem vibaes /oscillaes wih he naual fequency of he sysem. The fequency of SHM / SH oscillaos ae he naual fequency. Enegy is conseved in fee vibaion. In foced vibaion, he sysem is foced o oscillaes / vibaes a a fequency which is no he naual fequency of he sysem i.e.. A souce is needed in foced vibaion, wheeas in fee oscillaion a souce of vibaion is no needed. Enegy is no conseved in foced vibaion. 4

25 Mechanics of solids ( hs) Elasiciy An objec is elasic if i euns o is oiginal dimension when he sess acing upon i is emoved. The aio of sess o sain is linea o a ceain limi, called he popoionaliy limi. Beyond he popoionaliy limi, he elaion is non linea. The objec will eun o is oiginal dimension unil he elasic limi is eached. Beyond he elasic limi he objec will no eun o is oiginal dimension. Is behavio is said o be plasic lage defomaion fo a small incease in applied foce. The aio of ensile sess o sain is he elasic modules. The linea popoionaliy of sess and sain is called Hooke s Law. Young s Modulus Tensile sess is defined as he aio of he applied foce pependicula o he suface o he coss secion aea A; F Tensile sess A Scala, SI Uni : Pascal o Nm- l lo l Tensile sain l l o o Young s Modulus Tensile sess / Tensile sain F l / A l o Shea modulus x Shea sain l o 5

26 The coesponding elasic modulus is called he shea modulus. F// The shea modulus S A x l o Bulk modulus When he sess is a unifom pessue on all sides, and he esuling defomaion is a change in volume, he sess is called bulk (volume) sess and he elaive change in volume, bulk (volume sain). V Bulk (volume) sain V o The aio of sess o sain is called he Bulk Modulus. p Bulk Modulus, B V / V o whee he change in pessue is posiive, while he change in volume is negaive. This gives Bulk Modulus a posiive value. The uni of bulk modulus is Pa. The ecipocal of he bulk modulus is he compessibiliy and is denoed as V / Vo k B p k, whee The unis of compessibiliy is Pa - 6

27 Mechanics of fluids ( hs) Densiy and elaive densiy Densiy and specific weigh The densiy of a fluid is defined as is mass divided by is volume. ρ m / V The SI uni of densiy is kg/m 3. The densiy of wae is 000 kg/m 3 ( g/cm 3 ). The specific weigh of a fluid is is densiy divided by he densiy of wae. The specific weigh has no uni. Pessue Saic pessue The saic pessue is he pessue in a non moving fluids. By definiion Pessue Foce / Aea P F /A, The SI uni of pessue is N/m o Pascal (Pa) Theefoe he pessue due o a saic fluid is he foce exeed by he fluid divided by he aea on which he foce acs. Suppose a column of fluid has a coss secional aea A and heigh h. Is volume is hen given as V A x h. The mass of he fluid, m is given as he poduc of is densiy, ρ and is volume, V. Is weigh hen is W F mg ρvg The pessue on op of he fluid column is P. The pessue exeed by he column is Pl F/A ρvg /A ρahg /A ρhg Then he pessue a he boom of he fluid column is P P + ρhg 7

28 The pessue inside a fluid is P P 0 + ρhg, whee P 0 is he amospheic pessue, ρ he fluid s densiy, g acceleaion due o gaviy and h he deph inside a fluid. Fluid saic Pascal s Law Pascal s Law saes ha he pessue applied o an enclosed fluid is ansmied undiminished o evey poion of he fluids and he walls of he conaining vessel. This is he pinciple used in a hydaulic lif. The pessue on boh pison ae he same. The pessue on he smalle pison is P F /A. The pessue on he lage pison is P F /A. Thus P F /A F /A o F /F A /A o F F A /A Thus a smalle foce can be amplified Buoyancy. When a body is compleely o paially immesed in a fluid, he fluid exes an upwad foce on he body equal o he weigh of he fluid displaced by he body. 8

29 The upwad foce acing on he body is F F A(P am +ρgh (P am +ρgh)) A(ρgh ρgh) Aρg (h - h) ρg A (h - h) ρg V, whee V A (h - h ) is he volume of he fluid displaced by body This is he buoyan foce acing on he body. Fo a body which floas i weigh W mg buoyan foce. (Saic equilibium) Fo a body which sinks W > buoyan foce. 9

30 Vibaions and waves ( hs) Inoducion Waves ae ansmission of disubances hough a medium, ouwads fom a souce poducing he disubances. Waves ae poduced by vibaion of a souce. Type of waves Tansvese wave In a ansvese wave, he disubances causes he paicle of he medium o oscillaes in a diecion pependicula o he diecion of avel of he wave. Souce Longiudinal waves In a longiudinal wave, he disubances causes he paicles of he medium o oscillaes along he diecion of avel of he wave. (In boh he above examples he disubances ae poduced a he souce and changes sinusoidal, poducing he chaaceisic sine wave fom.) Chaaceisic of wave moion Wave paamees The classic wave (ansvese and longiudinal) is poduced by a souce which oscillaes in simple hamonic moion. In a ansvese wave he oscillaion of he souce is pependicula o he diecion of ansmission, in a longiudinal wave he oscillaion is paallel o he diecion of ansmission. 30

31 ampliude Wavelengh, λ The souce oscillaes in SHM wih he displacemen equaion y () A sin (ω + δ), whee A is he maximum displacemen (ampliude), ω he angula fequency of he SHM, and he ime of he oscillaion and δ he phase shif/consan/diffeence. The quaniy (ω + δ) is called he phase of he oscillaion. The elaionship of he SHM and he wavefom popagaed can be shown as below. y Wavelengh, λ displacemen ime gaph T T y Occus a 0 T T x displacemen-posiion gaph 3

32 Peiod he ime aken fo one oscillaion of he SHM he ime aken o make one wavefom T Fequency he numbe of oscillaion in one second he numbe of wavefom in one second To fom N wave fom will ake a ime of NT. Thus he numbe of wave in a ime NT is N, N he he fequency is, f NT T wavelengh he size of one wavefom in he diecion of popagaion he disance a wave avel in ime T, wavelengh, λ vt whee v is he speed of popagaion of he wave. A complee oscillaion (o one wave fom) is made in ime T, he angula displacemen of he ω SHM in his ime is π. Theefoe angula displacemen θ π ωt f Thus ω πf Thus he oscillaion can be wien as y () A sin (πf + δ) Now if he souce is a x 0, and he equaion of he SHM is y () A sin (πf ), hen he oscillaions of he paicles along he posiive x axis all lags behind he souce. Ie, y (x, ) A sin (πf - δ(x)) δ(x) x, whee is he disance he wave avel in ime, hus x v Bu wavelengh, λ π ad δ x Thus δ π π λ x π, hen δ kx, whee k λ λ The popagaion of Waves The popagaion of wave is he movemen of he wavefom fom he souce ouwads. Fo mechanical wave, he popagaion will follows he ype of medium. i.e. in a sing he wave will avel along he sing, wae waves may spead adially if i is fom a poin souce. Spheical waves o plana waves can be poduced depending on he souce. The speed a which he wavefom avels is called he wave velociy o wave speed. As a complee wave lengh moves a disanceλ in a ime T, hus he speed a which i moves is v λ/t f.λ Wave equaions Hamonic wave equaion The equaion fo a wave moving o he igh is y (x, ) A sin (πf - kx), while fo a wave moving o he lef is y (x, ) A sin (πf + kx), elaive o he souce of he oscillaion (he oigin) which has he equaion y () A sin (πf ), Lae in ime y Diecion of popagaion Ealie in ime x y (x, ) A sin (πf kx +φ), 3

33 Ealie in ime y Diecion of popagaion Lae in ime x y (x, ) A sin (πf + kx +φ), Noe : A phase consan φ is included in he geneal fom in case he souce of oscillaion has an iniial displacemen (ie y(o) A sin (φ) 33

34 Tempeaue, Themal expansion and he Ideal Gas Law (4 hs) Tempeaue and hemomees Tempeaue he measue of he degee of honess of a body. Lecue Noes PHY43 Tempeaue scale. Celsius scale (mos commonly used) 0 o C - Feezing Poin of wae 00 o C Boiling poin of wae Kelvin Scale (Absolue empeaue scale)(si uni) 0K absolue zeo, lowes empeaue, all molecules sops vibaing equals -73 o C, 73K equals 0 o C, 373K equals 00 o C Change of K change of o C θ(k) (θ( o C) +73) (K) Fahenhei Scale 3 o F equals 0 o C o F equals 00 o C θ( o F) (9/5)θ( o C) +3) ( o F) Types of hemomees Mecuy in glass Uses he expansion/conacion of mecuy due o absopion/loss of hea. The level of mecuy ises/fall agains a calibaed scale. Ex : Room hemomee, Clinical hemomee Resisance hemomee Uses he change in he esisance of a wie as a measue of empeaue change Themocouple EMF geneaed beween he wo wie juncions kep a diffeen empeaue is measued. The e.m.f. is popoional o he diffeence in he juncions empeaue. Consan Volume gas The empeaue is popoional o he change in pessue of he gas. Calibaion of a empeaue scale A measuable physical change wih empeaue is obained. The change should be linea in naue. Two known ends is equied.. Usually he meling poin of ice and he boiling poin of wae. The new hemomee is use a he wo known ends. The values of he physical change is noed. The diffeence is divides ino 00 pas. Themal expansion Linea expansion 34

35 The incease in lengh when a od o ube is heaed is called linea expansion. coefficien of linea The coefficien of linea expansion is defined as he aio of he change in lengh pe uni change in empeaue o is oiginal lengh. α l θ lo uni : K o o C - Fom he definiion l o α θ l change in lengh Thus new lengh l l o + l l o + l o α θ l o ( + α θ ) he new lengh Aea expansion (solid) The change in aea due o a change in empeaue The new aea A is given by A Ao ( +β θ ) Whee β is he coefficien of aea expansion, β A θ Relaionship beween α, coefficien of linea expansion and β, coefficien of aeal expansion If a lamina has dimension X o x Y o, hen afe a change in empeaue, X X o ( + α θ ) and Y Y o ( + α θ ) The new aea XY is hen (X o ( + α θ )).(Y o ( + α θ )) A X o Y o ( + α θ ). ( + α θ ) X o Y o ( + (α θ) + (α θ) +(α θ)(α θ)) X o Y o ( + (α θ) + (α θ)(α θ)) Ao if we disegad (α θ)(α θ) as α θ l we ll have A X o Y o ( + (α θ) ) A o ( + α θ) A o ( + β θ), whee β α lo l is less hen, he ( ) << lo Volume expansion (solid) The change in volume due o a change in empeaue. The new volume V is given by Vo ( +γ θ ) 35

36 V Whee γ is he coefficien of volume expansion, γ Vo θ Relaionship beween α, coefficien of linea expansion and γ, coefficien of volume expansion If a cuboid has dimension X o x Y o x Z o, hen afe a change in empeaue X X o ( + α θ ), Y Y o ( + α θ ) and Z Z o ( + α θ ) The new volume XYZ can be shown as XYZ X o Y o Z o ( + 3(α θ) ) afe disegading smalle ems, Thus γ, he coefficien of volume expansion is hee imes α, coefficien of linea expansion. γ 3α 36

37 The gas laws and Absolue empeaue Mole The mole is defined as he amoun of subsance of a sysem which conains as many elemenay eniies as hee ae aoms in 0.0 kg of cabon. Avogado Numbe The numbe of molecule in one mole o gam molecula weigh of a subsance is x 0 3. Mol volume The volume occupied by a mol o a gam molecula weigh of any gas a sandad condiions is.44 l Boyle s Law A a consan empeaue he volume of a given quaniy of any gas vaies invesely as he pessue o which he gas is subjeced. Fo a pefec gas, changing fom pessue P and volume V o pessue P and volume V wihou change of empeaue PV P V o PV consan hen P /V P P V /V Chales Law (Chales Gay Lussac Law) An empiical genealizaion ha in a gaseous sysem a consan pessue, he empeaue incease and he elaive volume incease sand in appoximaely he same popoion fo all so-called pefec gases. T / v is a consan o T V If he absolue empeaue scale is used (i.e. Kelvin scale) Then V T V 37

38 Pessue Law Fo a gas a consan mass and volume, is pessue P is popoional o is empeaue T P T P T The ideal gas law The ideal gas law is he equaion of sae of a hypoheical ideal gas, fis saed by Benoî Paul Émile Clapeyon in 834. The sae of an amoun of gas is deemined by is pessue, volume, and empeaue accoding o he equaion: whee is he absolue pessue [Pa], is he volume [m 3 ] of he vessel conaining moles of gas, is he amoun of subsance of gas [mol], is he gas consan [ m 3 Pa K mol ], is he empeaue in Kelvin [K]. The ideal gas consan (R) depends on he unis used in he fomula. The value given above, , is fo he SI unis of pascal cubic mees pe mole pe Kelvin, which is equal o joule pe mole pe kelvin (J mol - K - ). Anohe value fo R is L am mol K ) 38

39 "R" has a diffeen value fo each diffeen uni of pessue used. The values ae... R (pascals/kpa) R.08 (ams) R 6.4 (o/mmhg) R. (psi) The ideal gas law is he mos accuae fo monoaomic gases a high empeaues and low pessues. This follows because he law neglecs he size of he gas molecules and he inemolecula aacions. Obviously he neglec of molecula size becomes less impoan fo lage volumes, i.e., fo lowe pessues. The elaive impoance of inemolecula aacions diminishes wih inceasing hemal kineic enegy 3kT/, i.e., wih inceasing empeaues. The moe accuae Van de Waals equaion akes ino consideaion molecula size and aacion. The ideal gas law mahemaically follows fom saisical mechanics of pimiive idenical paicles (poin paicles wihou inenal sucue) which do no ineac, bu exchange momenum (and hence kineic enegy) in elasic collisions. 39

40 Hea and The Fis Law of Themodynamics (4 hs) Hea as enegy ansfe Hea, symbolized by Q, is enegy ansfeed fom one body o sysem o anohe due o a diffeence in empeaue. Specific hea Hea capaciy and specific hea capaciy. Hea capaciy - he amoun of hea ansfeed o/fom a subsance fo evey uni change in empeaue. C H θ Specific hea capaciy he amoun of hea ansfeed o/fom a uni mass of subsance fo evey uni change in empeaue. H c m θ Laen hea of fusion and evapoaion Laen hea he amoun of hea needed o change a subsance fom one sae of mae o anohe. (i.e. fom solid o liquid vice vesa o fom liquid o gas vice vesa) Specific Laen hea - he amoun of hea needed o change a uni mass of subsance fom one sae of mae o anohe H L m The fis law of hemodynamics The fis law of hemodynamics is an expession of he univesal law of consevaion of enegy, and idenifies hea ansfe as a fom of enegy ansfe. The mos common fom of he fis law of hemodynamics is: The incease in he inenal enegy of a hemodynamic sysem is equal o he amoun of hea enegy added o he sysem minus he wok done by he sysem on he suoundings wok done by a gas 40