ENGINEERING PHYSICS 1A By Dr. Z. Birech Department of Physics, University of Nairobi

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1 Dr. Z. Birech ENGINEERING PHYSICS (04) ENGINEERING PHYSICS By Dr. Z. Birech Department f Physics, University f Nairbi Intrductin These lecture ntes are fr rst year Engineering students (Electrical, Civil, Mechanical, Gespacial, and Envirnmental & Bisystems) fr their first semester Physics curse. The ntes cver mechanics and prperties f matter. The curse is intended t intrduce the student t the science behind physical entities that will be cvered in varius areas f engineering. Please nte that these ntes cver nly the first half f first semester engineering physics curse. The secnd half f the curse is nt cvered here. The sectin f vectrs is nt als cvered here but the student can get the backgrund n vectrs elsewhere. Curse utline SECTION I: MECHNICS ND PROPERTIES OF MTTER. Mtin in ne and tw dimensins Kinematics: Vectr & scalar quantities. Reslutin and cmpsitin f vectrs. The inclined plane. Displacement. Velcity. cceleratin. Equatins f linear mtin. Mtin under gravity. Prjectiles. Newtn s laws f mtin. Linear mmentum. Principle f cnservatin f linear mmentum. Elastic and inelastic cllisin. Impulse. Circular mtin: ngular displacement. ngular mmentum, The radian measure. ngular velcity. Perid. Frequency. cceleratin. Centripetal frce. Vertical and hrizntal circular paths. Static equilibrium. Mments. Cuples. Trque. Wrk. Energy pwer. Rtatinal dynamics Rtatin f rigid bdies. Equatins f mtin. Mment f inertia 3. Simple harmnic mtin Definitin. Relatin t circular mtin. Velcity. cceleratin. Perid. Frequency. K.e. p.e. 4. Gravitatin Newtn s laws f Gravitatin. Kepler s laws 5. Prperties f Matter Hke s law.stress. strain. Yung Mdulus. Cefficient f frictin. Pressure, Pascal principle. rchimedes principle. Cefficient f viscsity. Stke s law. Bernullis s principle. SECTION II: SOUND ND VIBRTIONS 6. Intrductin t sund Wave phenmenn, general wave equatin, sund waves, velcity f sund wave, interference, beats and beat frequency, Dppler Effect SECTION III: THERML PHYSICS 7. Heat Internal energy and temperature, phase changes f pure substance, isthermal and isbaric cmpressibility f gases, liquids and slids 8. Heat transfer Cnductin, cnvectin and radiatin. Kinetic thery f gases, perfect gas equatin, intermlecular frces, specific heats and equipartitin f energy Bks f Reference. Ohanian Physics II Editin, Nrtn & C., NY (989). Vectr nalysis, Schaum Series by Murray, R., McGraw Hill, NY (980) 3. Theretical Mechanics, Schaum Series by Murray, R., McGraw Hill, NY (986) 4. Cllege Physics, 3 rd Ed, by Miller, Harcurt B. Inc. NY (97) 5. -Level Physics 5 th Ed. by Nelkn & Parker, Heinemann (K) Ltd (99) 6. ny bk that cvers mechanics, heat and sund I

2 Dr. Z. Birech ENGINEERING PHYSICS (04) MECHNICS ND PROPERTIES OF MTTER Chapter : Mtin in One and Tw Dimensins. Linear Mtin (mtin in a straight line) The mathematical cncept f vectrs is very useful fr the descriptin f displacement, velcity and acceleratin in ne, tw r three dimensins. bdy can underg either ne f the fllwing types f mtin r a cmbinatin f tw r mre f these mtins namely: i Translatinal r rectilinear mtin i.e. mtin in a straight line ii Rtatinal r circular mtin e.g. a rtating wheel r planets arund the sun iii Vibratinal r scillatry mtin e.g. a pendulum clck, atms r electrns in a metal r slid. Displacement (S) is a physical quantity that specifies the psitin f an bject relative t the initial pint (rigin) r it is the distance mved in a given directin. Velcity ( ) is vectr that specifies the rate f change f displacement with time. It is defined by S () t The average velcity f an bject is given by Change in Displacement Time Taken Fr example, cnsider the displacement-time graph f, say, an autmbile given belw: S S P S P t t t If S S is the change ccurring in psitin in time intervalt t, then the average velcity is S S S slpe f the straight line P P. On the ther hand, the speed ( ) between t t t t and t is defined by ctual Dis tan ce Curved Dis tan ce PP t t t S t

3 Dr. Z. Birech ENGINEERING PHYSICS (04) i.e. speed is the rate f change f distance with time ( and in sme cases is equal t the magnitude f the velcity). The instantaneus velcity i.e. velcity f the bject at a given time r pint is the time rate f change f displacement i.e. Instantaneus Velcity d S dt cceleratin (a) is a vectr specifying hw fast the velcity f a bdy changes with time i.e. ChangeinVelcity a (3) Timetaken t t The instantaneus acceleratin is defined by d a inst dt Their Units are: Displacement (m), Speed (ms - ), Velcity (ms - ) and cceleratin (ms - ). Types f linear mtin (i) Mtin with unifrm velcity Velcity-time graphs and displacement-time graphs are valuable ways f depicting mtin in a straight line. The figures belw shws the displacement- and velcity-time graphs fr a bdy mving with unifrm velcity ie cnstant speed in a fixed directin. The slpe f fig (a) gives the velcity. S v (a) Displacement-time graph t (b) Velcity-time graph t Fr the displacement-time graph, the slpe gives the velcity f the bject while the area has n physical significance. In the velcity-time graph, the slpe is the acceleratin f the bject while the area is the displacement f the bject. (ii) Mtin with unifrm acceleratin (equatins f mtin) If the velcity f a unifrmly accelerating bject increases frm a value u t in time t, then frm the definitin f acceleratin that we have u a t u at (5) Fr unifrm mtin, the displacement cvered in time t is defined by S verage velcitytime (u ) t (6) Using equatin 5, equatin 6 gives S ut at (7) Eliminating t in equatin 7 by substituting fr t in 6 gives (4) 3

4 Dr. Z. Birech ENGINEERING PHYSICS (04) u as (8) Equatins 5, 7 and 8 are equatins f mtin fr an bject mving in a straight line with unifrm acceleratin. Example bdy cvers a distance f 0m in 4s it rests fr 0s and finally cvers a distance f 90m in 6s. Calculate its average speed. Slutin Ttal distance m Ttal time s verage speed 00 m 5ms 0s Example student runs 800m due nrth in 0s fllwed by 400m due suth in 90s. Calculate his average speed and his average velcity fr the whle jurney. Slutin (i) verage speed Ttaldis tan ce Ttaltime m 6ms s TtalDisplacement (ii) verage velcity ms Ttaltime 0 90 Fr velcity, since it is a vectr, yu have t chse the directin Example car mving with a velcity f 54kmhr - accelerates unifrmly at the rate f ms -. Calculate the distance traveled frm the place where the acceleratin began t that where the velcity reaches 7kmhr - and the time taken t cver this distance. Slutin Given 54kmhr 5ms ( u) 7kmhr 0ms ( ) a ms, then (i) u as implying 0 5 () S S m (ii) u at implying 0 5 t t. 5s (iii) Mtin under gravity (free falling bdies) freely falling bdy is an bject mving freely under the influence f gravity nly, regardless f its initial mtin. Objects thrwn upward r dwnward and thse released frm rest are all falling freely nce they are released. bdy released near the earth s surface will accelerate twards the earth under the influence f gravity. If air resistance is neglected, then the bdy will be in free fall and the mtin will prceed with unifrm acceleratin f a g 9.8ms. The value f this dwnward acceleratin (g) is the same fr all bdies released at the same lcatin and is independent f the bdies speed, mass, size and shape. The equatins f mtin fr freely falling bdies are similar t thse fr linear mtin with cnstant acceleratin a being replaced by g. Fr upward mtin (rising bdy), g is negative since the bdy is decelerating. Thus 4

5 Dr. Z. Birech ENGINEERING PHYSICS (04) u gt S ut gt u gs Example ball is thrwn vertically upwards with a velcity f 0 ms-. Neglecting air resistance, calculate (i) The maximum height reached (ii) The time taken t return t the grund (9) Slutin Taking upward directin as psitive, u 0ms and a g 0ms then (i) t maximum height, 0ms thus u as ( 0) S S 0m (ii) On return t grund, S becmes zer, thus frm S ut at 0 ut gt t 4sec 0t 5t. Mtin in Tw Dimensins (Prjectile Mtin) prjectile is any bdy that is given an initial velcity and then fllws a path determined entirely by the effects f gravitatinal acceleratin and air resistance. The bject mves bth vertically and hrizntally hence the mtin is in tw space dimensins. In analyzing this mtin, g is assumed cnstant and effect f air resistance cnsidered negligible. The path traced by a prjectile is knwn as its trajectry. The tw mtins (hrizntal and vertical) are independent f each ther i.e. the bject mves hrizntally with cnstant speed, and at the same time, it mves vertically in a way a similar bject nt underging hrizntal mtin wuld mve. If air resistance is neglected, then a prjectile can be cnsidered as a freely falling bject and its equatins f mtin can be determined frm the linear equatins f mtin tgether with the initial cnditins i.e. the initial velcity has cmpnents u cs alng the hrizntal- and u sin alng the vertical directin. The hrizntal and vertical mtins are analyzed as fllws u y u u x ucs u usin y Cmpnents f initial velcity u θ u x y R 5

6 Dr. Z. Birech ENGINEERING PHYSICS (04) (i)vertical mtin the vertical cmpnent f u is u sin and the acceleratin is g. When the prjectile reaches the grund at B, the vertical distance h traveled is zer. S frm h u t gt y.. () we have u sin u sint gt t.. () g which is the ttal time f flight. ls the maximum height reached can be evaluated as fllws; frm equatin we have u sin Y u sint gt.. (3) g after substituting half f the value f t frm. (ii) Hrizntal mtin since g acts vertically, it has n cmpnent in the hrizntal directin. S the prjectile mves in a hrizntal directin with a cnstant velcity u cs which is the hrizntal cmpnent f u. s frm the relatin s ut we have u sin sin cs sin R u cs u u..(4). g g g The maximum range is btained when sin r 90. In this case u R..(5). g ls at the maximum height Y f the path, the vertical velcity f the prjectile is zer. S applying the relatin v u at in a vertical directin, the time t t reach is given by y y u sin 0 u sin gt t.(6). g This is just half the time t reach B. Example n bject is thrwn hrizntally with a velcity f 0m/s frm the tp f a 0m-high building as shwn. Where des the bject strike the grund? 0ms - 0m Slutin: We cnsider the hrizntal and vertical prblems separately. (i) In the vertical case if the dwnward directin is taken t be psitive, then g 9.8ms and y 0m y 0ms,. Thus we can find the time taken t reach the grund frm y vy a yt t frm which t=.0s. 6

7 Dr. Z. Birech ENGINEERING PHYSICS (04) (ii) In the hrizntal case, we have fund that the bject will be in the air fr.0s. Therefre given that 0ms and t. 0s, then X xt 0ms.0s 0. m. x x.3 Newtn s Laws f Mtin First Law: bdy tends t remain at rest r in unifrm mtin in a straight line (with cnstant velcity) unless acted upn by a resultant frce. The tendency f a bdy t cntinue in its initial state f mtin (a state f rest r a state f unifrm velcity) is called inertia. ccrdingly, the first law is ften called the law f inertia. Secnd Law: If a net frce acts n a bdy, it will cause an acceleratin f that bdy. That acceleratin is in the directin f the net frce and it magnitude is prprtinal t the magnitude f the net frce and F inversely prprtinal t the mass f the bdy ie a s that F m a. Frm the definitin f a m Newtn, the law can be written in the frm F kma ma This (vectr) equatin is a relatin between vectr quantities F and a, and is equivalent t the three Fx ma x algebraic equatins Fy ma y Fz maz Third Law: ctin and reactin are always equal and ppsite ie when ne bdy exerts a frce n anther, the secnd exerts an equal, ppsitely directed frce n the first. Examples include when pushing n a car, the car pushes back against yur hand, when a weight is supprted by a rpe, the rpe pulls dwn n the hand; a bk resting n a table pushes dwn n the table, and the table in turn pushes up against the bk; the earth pulls n the mn hlding it in a nearly circular rbit and the mn pulls n the earth causing tides. The law differs frm the first and secnd in that, whereas the first and secnd laws are cncerned with the behavir f a single bdy, the third law invlves tw separate bdies. The inherent symmetry f the actin-reactin cuple precludes identifying ne as actin and the ther as reactin..4 Cllisins and Linear Mmentum Linear Mmentum is defined as the prduct f the bject s mass (m) and its velcity v and is a vectr. Linear mmentum P mass velcity mv M L The SI unit f linear mmentum is kgms - (Newtn secnd-ns) and its dimensin is. Frm T Newtn s secnd law ( F ma), if n external frce acts n an bject, then v u P F ma m 0 P is a cnstant. Thus its mmentum is t t cnserved. This is the principle f cnservatin f linear mmentum. It is useful in slving prblems invlving cllisins between bdies. The prduct f the frce and the time is called the impulse f the frce I ie Impulse I F t mv mu P change in mmentum The change f linear mmentum P f a particle during a time interval t equals the impulse f the net frce that acts n the particle during the interval. The SI unit f impulse is the same as that f mmentum ie Newtn-secnd r kilgram-meter-per-secnd. 7

8 Dr. Z. Birech ENGINEERING PHYSICS (04) Cllisin is any strng interactin between bdies that lasts a relatively shrt time. Examples include autmbile accidents, neutrns hitting atmic nuclei in a nuclear reactr, balls clliding, the impact f a meter n the surface f earth, a clse encunter f a spacecraft with the planet Saturn etc. In all cllisins, mmentum is cnserved. The ttal energy is als cnserved. Hwever, kinetic energy might nt be cnserved since it might be cnverted int ther frms f energy like sund, heat r wrk during plastic defrmatin. There are tw main types f cllisins: elastic and inelastic cllisins. If the frces between the clliding bdies are much larger than any external frces, then the external frces are neglected and the bdies are treated as an islated system i.e. all external frces are zer. Elastic cllisin: bth kinetic energy and mmentum are cnserved. m u a u b a mb v a m a mb v b (a) befre cllisin (b) after cllisin Frm the figure, we have m u a a m u m v m v cnservatin f linear mmentum b b a a b b m u a a mbub mava mbvb cnservatin f kinetic energy. Inelastic cllisin: mmentum is cnserved but kinetic energy is nt cnserved. Thus maua mbub mava mbvb If the clliding bdies stick tgether, the cllisin is ttally inelastic and hence we have m u m u ( m m ) V where V is the cmmn velcity. a a b b a b Special cases (i). Elastic cllisin in a straight line If the tw bjects and B have equal masses m and mass B is statinary ( u B =0) then fr elastic cllisin we have mu mv mv cnservatin f linear mmentum B m u a m B B u b = 0 m u a m B B v b (a) befre cllisin (b) after cllisin and mu mv mvb cnservatin f kinetic energy, frm which we have 8

9 Dr. Z. Birech ENGINEERING PHYSICS (04) u u v v v B v B Slving gives u vb and v 0. Thus the tw bjects simply exchange velcities ie mass cmes t rest while mass B mves ff with the riginal velcity f. this is a situatin f maximum energy transfer between tw clliding bdies and is mstly applicable in nuclear reactins where neutrns are stpped by prtns. (ii). Oblique cllisins f equal masses If mass cllides bliquely with mass B which is at rest and bth bjects are f equal masses m, then the ttal mmentum f any bject will be the sum f the respective mmentum cmpnents in the vertical and hrizntal directins respectively θ m u a m B B v B = 0 θ φ B Thus cnservatin f linear mmentum gives directin mu mv cs mv cs alng the x - B directin Cnservatin f kinetic energy gives mu mv sin mv sin alng the y 0 B mv (iii). Recil In a case where part f a cmpsite bdy suddenly flies apart eg a bullet fired frm a gun, the remaining part (the gun) must underg mmentum in the ppsite directin (recil) in rder t cnserve the mmentum. If m b and b are the mass and velcity f the bullet while m g is the mass f the gun, then the gun will recil with velcity v g given by r m v v b b m m v g v g mv B b b g vg is far much less than b mg v since m b is far much less than m g. Example car traveling at 90kmhr - slams int a tree and is stpped in 40ms. If the car has a mass f 800kg, calculate the average frce acting n the car during the cllisin. Slutin Frm v 90kmhr Ft 5ms mv mu, we have 9

10 Dr. Z. Birech ENGINEERING PHYSICS (04) 0.04sec F 800kg5ms F 50 Example persn f mass 50kg wh is jumping frm a height f 5m will land n the grund with a velcity v gh 05 0ms fr g 0ms. If he des nt flex his knees n landing, he will be brught t rest very quickly, say th secnd. The frce F acting is then given by 0 mmentum 500 F 5000N. This is a frce f abut 0 times the persn s weight and the large t 0. frce has a severe effect n the bdy. Suppse, hwever, that the persn flexes his knees and is brught t rest much mre slwly n landing, say secnd. Then the frce F nw acting is 0 times less than befre, r 500N. Cnsequently, much less damage is dne t the persn n landing..5 Circular Mtin ngular velcity We cnsider the kind f mtin where an bject mves arund a circular path abut sme fixed pint. Examples are mn and earth revlving arund the sun, the rim f a bicycle wheel, a stne being whirled n a string etc. In this chapter the system is assumed t mve in a circle with a unifrm speed arund a fixed pint O as the centre. 5 N v O θ s B θ v If the bject mves frm t B s that the radius O mves thrugh an angle θ, its angular velcity, ω, abut O may be defined as the change f the angle per secnd. Cnsidering time t taken by the bject t mve frm t B we have [units: radians per secnd]. t radian ( rad) is the angle subtended the centre f a circle by an arc whse length is equal t the radius f the circle. The perid fr this kind f mtin is given by T since π radians is the angle in ne revlutin. If s is the length f arc B, then s/r = θ, (frm when, s/r = θ) r taking s = rθ s t r gives t 0

11 Dr. Z. Birech ENGINEERING PHYSICS (04) v r (Relatinship between angular and linear velcity). ngular acceleratin (α) n bject mving in a circle at cnstant speed experiences a frce that pulls it twards the centre f the circular path. This frce is knwn as centripetal frce. If v is the unifrm speed in the circle f radius r, then the acceleratin is given by v r But v r Therefre v ( r) r r Centripetal Frces Cnsider the situatin belw r mg T B T O mg T 3 mg C The figure shws an bject f mass m whirled with cnstant speed v in a vertical circle f centre O by a string f length r. Let T be the tensin in the string at pint (the highest pint). Then since the weight mg acts dwnwards twards O, then the frce twards the centre is given by F T mg mv r T mv r mg Suppse T is the tensin when the bject is at pint B, then at this pint mg acts vertically dwnwards and has n effect n T. S mv F T r Finally cnsidering at pint C where it is at the lwest pint, mg acts in the ppsite directin t T 3 giving

12 Dr. Z. Birech ENGINEERING PHYSICS (04) leading t F T 3 mg mv r mv T3 mg r Cmparing T, T and T 3 frm the abve equatins, it is seen that maximum tensin in the string is at pint C (i.e, T 3 is the highest). This implies that T 3 must be greater than mg by mv /r t make the bject keep mving in a circular path. Banking Suppse a car is mving rund a banked rad in a circular path f hrizntal radius r. R R mv r F θ F mg If the nly frces at the wheels, B are the nrmal reactins R and R respectively, then the resultant frce twards the centre f the track (i.e. prviding the centripetal acceleratin) is (R + R ) sin θ where θ is the angle f inclinatin f the plane t the hrizntal. This frce is equal t the centripetal frce; Fr vertical equilibrium ( R R ) sin mv r ( R R)cs mg Dividing the tw equatins leads t v tan rg Implying that fr a given velcity v and radius r, the angle f inclinatin f the track fr n sided slip must be tan - (v /rg). This has als been applied n rail tracks. ngular Mmentum

13 Dr. Z. Birech ENGINEERING PHYSICS (04) Cnsider a single particle f mass m which at ne instant f time has a mmentum p and is at a distance r frm the rigin f crdinates. The angular mmentum L f the particle is defined as the vectr f magnitude L rpsin Where θ is the angle between the mmentum vectr p and the psitin vectr r. The directin f the vectr L is alng the perpendicular t the plane defined by the vectrs p and r. The directin f the vectr L alng this perpendicular is specified by the right-hand rule, i.e But p = mv, therefre L = mr x v (units: Kgms - ) The angular mmentum f a particle mving with cnstant velcity in the absence f frce is cnstant, i.e. the magnitude f L = rpsin θ is cnstant and the directin f L is als cnstant. This situatin describes a free particle and represents cnservatin f angular mmentum. Cnsider a case f cnservatin f angular mmentum, i.e. a case f a particle in a unifrm circular mtin, such as a stne whirled alng a circle at the end f a string (Fig belw) z L r y x The vectr L is perpendicular t the plane f the circle. In this case, since the psitin vectr is always perpendicular t the velcity vectr, the magnitude f the angular mmentum vectr is L rpsin mrv S the directin f the angular mmentum vectr is perpendicular t the plane f the circle. s the particle mves arund the circle, L remains cnstant in magnitude and directin..6 Equilibrium st cnditin fr Equilibrium Much f Physics has t d with bjects and systems which are at rest and remain at rest. This prtin f physics is called statics. It is f prime imprtance since the cncepts which it invlves permeate mst fields f physical sciences and engineering. n bject is in equilibrium if it is nt accelerating ie n net frce must act n it. That des nt mean that n frces may be applied t the bdy. If several frces act simultaneusly, equilibrium demands nly that the net frce ie the vectr sum f the varius frces vanish (be equal zer). This is the first cnditin f static equilibrium which can quantitatively be written as i F i 0 () 3

14 Dr. Z. Birech ENGINEERING PHYSICS (04) which is equivalent t three cmpnent equatins i i i F F F ix iy iz Fr example when an bject rests n a table, there are tw frces acting n it, namely its weight and the upward reactin frce f the table n the bject. Withut the table, the bject can nt remain at rest but wuld drp under the pull f gravity. If the bdy remains at rest, it is said t be in static equilibrium while when it is in steady mtin in straight line, it is said t be in dynamic equilibrium. Trque and the nd cnditin f Equilibrium Trque is a deciding factr in a state f equilibrium. Trque is defined as the prduct f frce and lever arm r simply the turning effects f a frce. If all the lines a lng which several frces lie intersect at the same pint, then the frces are said t be cncurrent. The secnd cnditin fr equilibrium invlves r the trque applied t the bject. This cnditin can be stated as: the resultant f all the trques acting n the bject must be t cause n turning effect ie the clckwise trques must balance the cunterclckwise trques n n 0 which can quantitatively be written as trque Leverarm Frce F d where the lever arm is the length f a perpendicular distance drpped frm the pivt t the line f the frce. ls r F rfsin Where r is the magnitude f the lever arm, F is the applied frce and θ is the angle f rtatin. In the equatin abve, τ is given by the right hand rule fr the advance f a screw rtated frm the directin f r twards that f F. (Remember the vectr crss prduct!). Trques can be classified as either clckwise r cunterclckwise. By cnventin, we take cunterclckwise trques as psitive and clckwise trques as negative. Mrever, a frce whse line passes thrugh the pivt causes zer trque. This is a reflectin f the fact the lever arm fr such a frce is zer. We therefre cnclude that an bject will be in equilibrium if the fllwing cnditins are satisfied: F ni 0 and n 0 Example lng rpe is stretched between pints and B. t each end the rpe is tied t a spring scale that measures the frce the rpe exerts n the supprts. Suppse the rpe is pulled sideways at its midpint with a frce f 400N prducing a deflectin such that the tw segments make angles f 5 with the line B. What is the reading f the spring scales? Slutin Since the bdy is in equilibrium, we have X-cmpnents () 4

15 Dr. Z. Birech ENGINEERING PHYSICS (04) T cs 5 T cs5 0 () Y-cmpnents T sin 5 T sin () Frm equatin, T T Therefre T sin 5 T sin T sin T 95N T sin 5 The tensin in the rpe and therefre the frce registered n the spring scales is 95N. frce f 400N applied perpendicular t the line B caused a tensin f nearly 300N, mre than five times the applied frce in magnitude! There is a practical lessn t be learned here. Frictinal Frces Frictinal frces play an imprtant rle in the applicatin f Newtn s Laws. There are three majr categries f frictinal frces: (i) Viscus frictinal frces ccur when bjects mve thrugh gases and liquids. n example is the frictinal frce the air exerts n a fast mving car r plane. The air exerts a retarding frce n the car as the car slides thrugh the air. (ii) Rlling frictinal frces arise as, fr example tire rlls n pavement. This type f frictin ccurs primarily because the tire defrms as the wheel rlls. Sliding f mlecules against each ther within the rubber causes energy t be lst. (iii) Sliding frictinal frces are frces that tw surfaces in cntact exert n each ther t ppse the sliding f ne surface ver the ther. We will be cncerned with sliding frictinal frces. Laws f slid frictin Experimental results n slid frictin are summarized in the laws f frictin, which state: The frictinal frce between tw surfaces ppses their relative mtin. The frictinal frce is independent f the area f cntact f the given surfaces when the nrmal reactin is cnstant. 3 The limiting frictinal frce is prprtinal t the nrmal reactin fr the case f static frictin. The frictinal frce is prprtinal t the nrmal reactin fr the case f kinetic (dynamic) frictin, and is independent f the relative velcity f the surfaces. Cnsider the fllwing simple experiment If a bk resting n the table is pushed lightly with a hrizntal frce F the bk des nt mve. pparently, the table-tp als pushes hrizntally n the bk with an equal and ppsite frce. The frictinal frce f ppses the sliding mtin f the bk and it is always directed parallel t the sliding surfaces. If the pushing is increased slwly, then when the pushing frce reaches a certain critical value f s, the bk suddenly begins t mve. fterwards t keep the bk mving a smaller frictinal frce f k is enugh. This simple experiment shws that tw frictinal frces are imprtant: the maximum static frictinal frce f s that must be vercme befre the bject can start mving and the smaller kinetic frictinal frce f k that ppses the mtin f the sliding bject. The majr reasn fr this behavir (cause f frictin) is that the surfaces in cntact are far frm smth. Their jagged pints penetrate ne anther and cause the surfaces t resist sliding. Once sliding has begun, hwever, the surfaces d nt have time t settle dwn nt each ther cmpletely. s a result, less frce is required t keep them mving than t start their mtin. The frictinal frce depends n hw frcefully the tw surfaces are pushed tgether. This situatin is described by what is called the nrmal frce F N (where nrmal means perpendicular). The nrmal frce is the perpendicular frce that the supprting surface exerts n the surface that rests n it. The situatin is shwn belw. 5

16 Dr. Z. Birech ENGINEERING PHYSICS (04) W F N W F N W W W F N W F sin (a) (b) (c) F Experiments N shw that the frictinal frces F N f s and f k are ften directly prprtinal F N t the nrmal frce. In equatin frm, we have f F f s k s k N F N W The factrs s and k are called the static and kinetic (r dynamic) cefficients f frictin, respectively. They vary widely depending n the nature f the surfaces invlved as well as the cleanliness and dryness f the surfaces. The cefficient f static frictin s can als be fund by placing the blck n the surface S and then gently tilting S until is n the pint f slipping dwn the plane. The static frictinal frce F is then equal t mg sin where is the angle f inclinatin f the plane t the hrizntal; the nrmal reactin R is equal t mg cs s that F mgsin tan R mg cs and hence s can be fund by measuring. θ F Fsinθ R F S MgSinθ θ MgCsθ Mg Figure Cefficient f frictin by inclined plane Example: Cnsider the situatin shwn f 5kg 40N 37 W P The applied frce due t the rpe is 40N, and the blck has a mass f 5kg. If the blck accelerates at 3.0m s-, hw large a frictinal frce must be retarding its mtin? Slutin free-bdy diagram fr this situatin is as shwn belw: 40N P f N W

17 Dr. Z. Birech ENGINEERING PHYSICS (04) N mtin ccurs in the y directin and s we are nt cncerned with the y frces. We have fr the x directin F ma 3N f 5.0kg3.0ms nx f 7N Can yu shw that the cefficient f frictin is 0.68 in this case? x.7 Wrk, Energy and Pwer WORK: is defined as the prduct f the magnitude the displacement and the cmpnent f the frce F parallel t the displacement. Mathematically: W F. S FS cs S f pint f applicatin f frce where W is the amunt f wrk dne by the frce f magnitude F during a small displacement f magnitude S. We can regard the prduct S cs as the cmpnent f the displacement in the directin f the frce F, r alternatively, regard the prduct F cs as the cmpnent f the frce in the directin f the displacement. Features f the definitin First wrk requires the actin f a frce. Withut a frce, n wrk is dne. Secnd, the applicatin f a frce is a necessary but nt sufficient cnditin fr wrk. Wrk is dne nly if there is displacement f the pint f applicatin f the frce, and then nly if this displacement has a cmpnent alng the line f actin f the frce. lthugh wrk is the prduct f tw vectr quantities, it is a scalar. Its SI unit is the Newtn-meter r kgm s - and is given the name Jule. One jule is the amunt f wrk dne by a frce f ne Newtn acting ver a distance f ne meter in the directin f the displacement. Example Calculate the wrk dne by a man f mass 65kg in climbing a ladder 4m high. Slutin wrk dne= frce x distance=weight x distance = mgh = 65kg x 0ms - x4m=600j =.6kJ Example Hw much wrk is dne in lifting a 3kg mass a height f m and in lwering it t its initial psitin? Slutin (i) Since the frce is directed up and the displacement is in the same directin, 0. Hence W mgs 3kg9.8ms m 58. 8J (ii) Suppse we nw slwly lwer this mass t its riginal psitin. gain we must apply an upward frce f mg t prevent it frm drpping. Hw much wrk is dne by this frce? Nw the angle between that frce and the displacement is 80 and since cs80 =-, we have W=-58.8J The negative sign tells us that sme ther agent, gravity, has dne wrk n the bdy. In this example, there are tw frces that act n the 3-kg mass: the frce f gravity, which pints dwnwards and the tensin in the string which pulls upward. If we had asked fr the wrk dne by the frce f gravity, it wuld have been negative during the lifting f weight and psitive as the weight was lwered. ENERGY: Energy is defined as the capacity t d wrk. system may have mechanical energy by virtue f its psitin, its internal structure r its mtin. There are als ther frms f energy besides 7

18 Dr. Z. Birech ENGINEERING PHYSICS (04) mechanical, namely chemical energy(fund in fds, ils, charcal, bigas etc and is due t the kinetic energy and ptential energy f the electrns within atms), electrical energy(assciated with the electric charge and can be prduced by generatrs frm hydrelectric pwer statins-waterfalls, gethermal statins, nuclear fissin etc-), nuclear energy frm a nuclear reactr, thermal energy (due t heat prduced frm burning fuels, the sun, heaters etc). It is a remarkable fact abut ur physical universe that whenever ne frm f energy is lst by a bdy/system, this energy never disappears but it is merely translated int ther frms f energy. eg Vehicles burn fuels t prduce bth thermal(heat) and mechanical energy. Mechanical Energy It is the energy f mtin-whether that energy is in actin r stred. It exists in tw frms: Kinetic energy- energy pssessed by a bdy by virtue f its mtin and it represents the capacity f the bdy t d wrk by virtue f its speed. mving bject can d wrk n anther bject it strikes. It exerts a frce n the ther bject causing it t underg a displacement. If a frce F acts n an bject f mass m such that the mass accelerates unifrmly frm initial velcity v i t a final velcity ms ver a distance S (as shwn), F S B Then the wrk dne ver the distance S is W F. S. But F ma and S ut at. Frm the relatin v u at v u then t s that the wrk dne is a u k e W ma S m v. which is the WORK-ENERGY RELTION (THEOREM). If the bdy starts frm rest, then the wrk dne n the bject equals kinetic energy gained by the bject. The net wrk n an bject is equal t the change in the bjects kinetic energy Ptential energy-energy pssessed by a bdy by virtue f its cnfiguratin (psitin) in a frce field eg gravitatinal field, electrstatic field, magnetic field etc. If an bject f mass m is lifted t a height h frm the grund, then: Wrk dne n the mass W F h mgh ie wrk dne n the bject gain in the ptential energy by the bject. Whether a bdy falls vertically r slides dwn an inclined plane, the wrk dne n it by gravity depends nly n its mass and n the difference in height between the initial and final psitins. Ptential energy f an bject depends nly n its lcatin and nt n the rute by which it arrived at that pint. It fllws that if a bdy is transprted arund a clsed path, the change in ptential energy vanishes ie ptential energy is independent f the previus histry because the gravitatinal frce is cnservative. frce is said t be cnservative if the wrk W B dne by the frce in mving a bdy frm t B depends nly n the psitin vectrs r and r B. In particular, a cnservative frce must nt depend n time, r n the velcity r acceleratin f the bdy. 8

19 Dr. Z. Birech ENGINEERING PHYSICS (04) Example 00kg crate f milk is pushed up a frictinless 30 inclined plane t a.5m-high platfrm. Hw much wrk is dne in the prcess? Slutin The x cmpnent f mg is mgsin30. This frce must be balanced by the applied frce F t prevent the crate frm slipping dwn the plane. The wrk dne by the frce F is W Fd cs Since F acts in the directin f mtin, θ=0 and csθ =. The distance d ver which the frce acts is the length f the incline namely.5m.5m d Hence W ( mgsin 30 ) 470J sin 30 sin 30 Example Suppse in the previus example the inclined plane is nt frictinless and that the cefficient f frictin is 0.. Hw much wrk is dne in pushing the crate t the.5m-high platfrm? Slutin The wrk dne against the frce f gravity is the same as befre 470J. Hwever, the applied frce must be greater than mgsin30 s as t vercme the frce f frictin which als acts in the x directin (ppsite t the directin f mtin). The frce f frictin is f k k R kmg cs 30 0.(980N)(0.866) 70N The wrk dne against this frictinal frce is then.5m Wn f kd 0.(980N)(0.866) 50J sin 30 where n indicates nncnservative frces. The ttal wrk dne in bringing the crate t the platfrm is W pe Wn 470J 50J 980J POWER is the rate f ding wrk ie it is the rate at which energy is cnverted frm ne frm t anther. Mathematically, W P ( averagepw er) t W S ls since W F. S, then P F F. v Frce velcity t t The unit f pwer is the Watt (W) which is the rate f wrk (transfer f energy) f ne jule per secnd. Pwer is als measured in hrsepwer (hp) where hp=746w. The efficiency f a machine r system is the rati f the pwer utput t pwer input ie pwerutput Efficiency pwerinput Example manually perated winch is used t lift a 00kg mass t the rf f a 0m tall building. ssuming that yu can wrk at a steady rate f 00W, hw lng will it take yu t lift the bject t the rf? Neglect frictinal frces. Slutin The wrk dne equals the increase in ptential energy f the 00kg mass, namely W mgh 00kg9.8ms 0m 9600J Since this wrk is dne at a cnstant rate f 00W, then 9600J 9600J 00W t 98sec t 00W Let us see hw large an errr may have been made by neglecting the kinetic energy f the mass during the ascent. The average speed f the mass is 0 m v 0.0ms 98sec 9

20 Dr. Z. Birech ENGINEERING PHYSICS (04) Kinetic energy during ascent is therefre k. e mv (00kg)(0.0ms ). 04J an amunt negligibly small cmpared with the change f 9600J in ptential energy. We can therefre safely neglect this small amunt f k.e in the prblem s slutin. 0

21 Dr. Z. Birech ENGINEERING PHYSICS (04) Chapter : Rtatinal Dynamics. Rigid Bdy rigid bdy is an bject with a definite shape that des nt change s that the particles cmprising it stay in fixed psitins relative t ne anther (i.e. a bdy having a perfectly definite and unchanging shape and size). Recall that the angular mmentum fr a single particle is expressed as; L = r x p = mr x v where r is the psitin vectr f the particle (relative t rigin) and p is the mmentum. Ttal angular mmentum fr a rigid bdy is the sum f the angular mmentum f all particles in the bdy. If these particles have masses m, velcities v i and the psitin vectr r i (relative t a given rigin f crdinates), then the ttal angular mmentum is L n i m r v i i i Where n is the ttal n. f particles. Nte that the angular mmentum btained frm the abve frmula depends n the chice f rigin crdinates. Mment f Inertia (I) Mment f inertia, (r angular mass, SI units kg m ) is a measure f an bject's resistance t changes in its rtatin rate. This is the rtatinal analgue f mass in linear mtin. Cnsider a rigid bdy rtating abut a fixed axis O, and a particle f the bject makes an angle θ with a fixed line OY in space at sme instant. O r r ω The angular velcity ω f every particle is same everywhere. Fr a particle at, the velcity is v = r ω (r = O). The ttal kinetic energy fr the whle bdy is the sum f individual ke given by ke ( mr ) Cmparing this with ke fr linear mtin (ke = /mv ) shws that the magnitude f Σmr can be dented by the symbl I and is knwn as the mment f inertia f the bject abut its axis. Trque n a rtating bdy Cnsider a rigid bdy rtating abut a fixed axis O. Y

22 Dr. Z. Birech ENGINEERING PHYSICS (04) O θ r m r ω Y The frce acting n a particle = m x acceleratin given by d d m ( r ) mr dt dt mr The mment f this frce abut the axis O = frce x perpendicular distance frm O which is mr r mr The ttal mment f all frces (r ttal trque) is ( mr ) I Cnservatin f ngular Mmentum The angular mmentum f ne particle is L = r x P The rate f change f this mmentum is dl d dr dp r P P r dt dt dt dt Taking the first term n RHS f the abve equatin we have dr P v ( mv) m( v v) 0 dt While the secnd term is equal t frce F (frce is equal t rate f change f mmentum). This implies that dp r r F and dt dl r F dt Fr a rigid bdy, the ttal angular mmentum is the sum f angular mmentum f individual particles and the rate f change f the ttal angular mmentum is the sum f the rates f change f individual angular mmentum.

23 Dr. Z. Birech ENGINEERING PHYSICS (04) i.e. dl dt n i r i F i Where F i is the frce acting n the particle i. But r i x F i is the trque f the sum F i n particle i. If the frces acting n the particle are external and if the ttal external frces are such that the ttal external trque is zer, then the angular mmentum is cnserved, i.e. L = [cnstant]. This is the law f cnservatin f angular mmentum. (i.e. The ttal angular mmentum f a rtating bject remains cnstant if the net trque acting n it is zer) Summary Quantity Linear Rtatinal Psitin x θ Velcity v ω cceleratin a α Equatins f Mtin v= u + at s = ut + /at v = u + as ω = ω + αt θ = ω t +/αt ω = ω +α θ Mass m I Newtn s nd law F = ma τ = Iα Mmentum p= mv L = I ω Wrk Fd τθ KE /mv /Iω 3

24 Dr. Z. Birech ENGINEERING PHYSICS (04) Chapter 3: Simple Harmnic Mtin (SHM). Intrductin pendulum swinging back and frth, the vibratin f a guitar string, a mass vibrating at the end f the spring- these and all bjects that vibrate have ne thing in cmmn: each system is subject t a restring frce that increases with increasing distrtin. restring frce is ne that tries t pull r push a displaced bject back t its equilibrium psitin. Whenever the system is displaced frm equilibrium, F r urges the system t return. (a) (b) (c) F r F r x F r The recrd f its vibratry mtin, a displacement versus time graph is at least sinusidal r csinusidal in frm. y O a λ c mplitude 0 O d t There are certain terms used t describe vibratry systems and we shall illustrate them by reference t the figure abve. One cmplete vibratin r cycle f the mass ccurs when the mass vibrates frm the psitin indicated by pint t the pint indicated by C r by any tw ther similar pints that are in phase. It is called the wavelength and the type f mtin is peridic r vibratry mtin. The time taken fr the system t underg ne cmplete vibratin is the perid T f the system. Since the system will underg T cmplete vibratins in unit time, this quantity is the frequency f the vibratin and we have f T The dimensins f frequency are (time) -. Smetimes frequency is expressed in cycles r vibratins per secnd. One cycle per secnd is dented as ne hertz (Hz) which is the SI unit f frequency. The distance D is the amplitude f the vibratin. It is the distance frm the equilibrium psitin (dashed line) t the psitin f maximum displacement. It is nly half as large as the ttal vertical distance traveled by the mass.. Simple Harmnic Mtin Cnditin(s) a system must satisfy if its vibratin is t be sinusidal b 4

25 Dr. Z. Birech ENGINEERING PHYSICS (04) Cnsider the system shwn belw Fr m 0 If the spring beys Hke s law then F r = -kx. t equilibrium, the mass is at pint O. Suppse it is displaced a distance X as shwn and released. The restring frce F r will pull the mass back tward pint O and the mass will vibrate arund O as centre. If this mtin is t be sinusidal r csinusidal in this case, then the displacement X f the mass will be given by X X cs t T The functin cs t is the scillatry functin traced in the figure belw. Ntice that cs ges T thrugh ne cmplete cycle as θ ges frm 0 t. In the abve equatin, the angle ges frm 0 t as t ges frm 0 t T. hence T is the time taken fr ne cmplete cycle and is the perid. nther feature abut equatin is the factr X. Because cs scillates between + and - as θ keeps increasing, the displacement X scillates between + vibratin. X and - X X as time ges n. Therefre X is the amplitude f the +x 0 y O T 0 O t -x 0 π What srt f frce acts n the mass t prduce this sinusidal mtin? It is simply the frce F r exerted n the mass by the spring. We can find F frm ur equatin fr X by using equatin t cmpute the r acceleratin f the mass and then by using F ma t find F r. T carry ut this, we differentiate equatin with respect t time in rder t find the velcity f the mass. We have d x d v X cs t X sin t 3 dt dt T T T The velcity we have fund is the velcity f the mass in figure 3. If we nw differentiate v with respect t t, we btain the acceleratin f the mass. It is dv d 4 4 a X sin t X cs t X 4 dt T dt T T T T 5

26 Dr. Z. Birech ENGINEERING PHYSICS (04) The acceleratin f the mass is prprtinal t the negative displacement. This acceleratin is caused by the unbalanced restring frce F r. Therefre F ma becmes 4 F r m X 5 T The restring frce is ppsite in directin t the displacement X, a cnditin that is inherent in the nature f restring frces. In additin, the restring frce is prprtinal t the displacement and this is equivalent t saying that the system beys Hke s law. Thus we have arrived at the fllwing result: a system that vibrates sinusidally beys Hke s law in which the restring frce is prprtinal t the distrtin. Such a system beys equatin and is called simple harmnic mtin. Hence t test whether r nt a vibratry system beys a sinusidal equatin, we simply check t see that the system beys Hke s law. Frm Hke s law, the sprig cnstant k is the prprtinality cnstant s that F r kx (Hke s law) This relatin simply states that the restring frce is prprtinal t the distrtin and is directed ppsite t the directin. If we cmpare this frm f Hke s law and equatin 5, we see that 4 m k m r T 6 T k This is a very imprtant relatin because it gives the perid f vibratin f the mass m in terms f the spring cnstant k. the perid f vibratin is lng fr large mass(fr large inertia) and fr small k (because the frce exerted by the spring is small). The frequency f vibratin can als be fund frm f and T frm equatins 3 and 6 we have, fr velcity, k v X sin T 7 m T Squaring this equatin and adding it t the square f equatin gives m k v X X v ( X X ) 8 k m Ntice that v is maximum when X O, that is, when the system passes thrugh equilibrium. Example When a 30g mass is hung frm the end f a spring, the spring stretches 8.0cm. this same spring is used in the experiment with a 00g mass at its end. The spring is stretched 5.0cm and released. If we assume that the spring slides withut frictin, find the fllwing fr the mass: (a) perid f vibratin (b) frequency (c) acceleratin as a functin f X (d) speed f the mass as it ges thrugh the equilibrium psitin. Slutin First we must find the spring cnstant. In a stretching experiment, stretching frce 0.30kg9.8ms k 3.7Nm stretch 0.080m m kg (a) T s k 3.7Nm (b) f 0. 68Hz T Fr kx 3.7Nm (c) a X 8.5s X m m 0.00kg Why is the acceleratin greatest at the extreme f the vibratin? Why is it zer when X=0? 6

27 Dr. Z. Birech ENGINEERING PHYSICS (04) (d) When X=0, v X k m 0.5ms Energy Cnsideratins (a) F r X U s K v a (b) (c) F r F r = 0 X 0 /kx 0 0 -a m 0 0 /kx v m 0 (d) (e) F r = 0 -X /kx 0 0 a m 0 0 /kx v m 0 /kx 0 0 -a m F r X When a spring is stretched a distance X and the mass is at rest, the energy (all the energy is ptential energy) stred in the stretched spring is U s kx Because the mass is at rest at (a), kinetic energy K is zer. The restring frce f the spring accelerates the mass tward the left in (a) by the time the system reaches the cnfiguratin in (b), the spring is n lnger distrted s U 0. The energy riginally stred in the spring has been changed t kinetic s energy. Thus in (b) we have K kx. Since K has nw achieved its maximum value, the velcity f the mass is largest. But because F r 0 in (b), the acceleratin f the mass is zer. In general. The energy changes back and frth frm U s t K. t the ends f the path, K 0 and U s is maximum. t the centre, U s 0 and K is maximum.. The velcity f the mass is largest as the mass passes thrugh the centre. It is zer when the mass is at either end f the path. 3. The acceleratin is zer at the midpint and is maximum at the tw ends. t any pint X the ttal energy f the system is Ttal energy U s K kx mv Since the ttal energy stred in the spring is kx, we arrive at a very imprtant relatin stating hw the energy f the system is apprtined: kx kx mv 9 If we slve this equatin fr v we btain equatin 8. 7

28 Dr. Z. Birech ENGINEERING PHYSICS (04) nther vibratry system f interest is the pendulum. The pendulum mass scillates back and frth between the psitins shwn. t psitins and C, the energy is all ptential U mgy. t any ther psitin where the bb s height is y, we have Ttal energy U g K mgy mgy mv This is the basic energy equatin fr a pendulum. Other vibratry systems can be analyzed in a similar way. In all f them, interchange between U and K ccurs. The mass is mving fastest when the system is mving thrugh its equilibrium cnfiguratin because then all the energy is kinetic. Example Suppse the pendulum belw is released frm a psitin where y 0cm. Find the speed f the mass at (a) pint B and (b) when the value f y. 30cm. g C y Slutin Using the energy methd, we can write mgy mgy mv frm which we have v g( y y) (a) Using y 0 gives (b) In the same way with v 9.8ms (0.0 0) m 0.66ms y m we find v 0.37ms Equatin f mtin fr SHM Simple harmnic mtin ccurs if the system beys Hke s law where the restring frce is prprtinal t the distrtin. Fr a distrtin X, this requires that F r kx where k is the spring cnstant and the negative sign is arising since it is a restring frce. When X is psitive, F r will be directed tward X. the equatin f mtin fr a mass-spring system is btained by writing F ma fr the system. Since F Fr in this case, we have kx ma Hwever the acceleratin itself depends n X because dv d d X d X a dt dt dt dt Thence the equatin f mtin becmes d X k X. dt m This is the typical equatin f mtin fr SHM. Mathematicians call it a differential equatin. When yu study differential equatins, yu will learn that in term f peridic mtin the slutin f this equatin is X X sin( ft ) 8

29 Dr. Z. Birech ENGINEERING PHYSICS (04) k where f m The cnstant X is the amplitude f the vibratin and is an arbitrary phase cnstant. The result cnfirms that a system beying Hke s law gives rise t sinusidal vibratin with the frequency given abve. Example The spring-mass system shwn belw is vertical and is therefre influenced by the frce f gravity. Des it still underg SHM? y F r Slutin The frce acting n the mass is F ky mg Where dwn is taken as psitive. Using the F ma gives d y ky mg m dt Let us change variables frm y t y where mg y y k mg Ntice frm the figure that is simply the amunt the spring stretches due t the weight f the mass. k By changing variables, we subtract this change in spring length. The quantity y is simply the displacement frm the equilibrium psitin f the laded spring. With this new variable, the equatin f mtin becmes d y m ky dt nd s the crdinate y underges simple harmnic mtin with the frequency as thugh gravity were nt present. We therefre cnclude that gravity shifts the equilibrium pint but des nt therwise affect the vibratin. 9

30 Dr. Z. Birech ENGINEERING PHYSICS (04) Chapter 4: Gravitatin 4. Intrductin The frce that binds tgether prgressively larger structures frm star t galaxy t supercluster and may be drawing them all tward the great attractr is the gravitatinal frce. This frce nt nly hlds yu n Earth but als reaches ut acrss intergalactic space. Physicists like t study seemingly unrelated phenmena t shw that a relatinship can be fund if they are examined clsely enugh. This search fr unificatin has been ging n fr centuries. Fr example, in 665 Isaac Newtn made a basic cntributin t physics when he shwed that the frce that hlds the Mn in its rbit is the same frce that makes an apple fall. We take this s much fr granted nw that it is nt easy fr us t cmprehend the ancient belief that the mtins f earthbund bdies and heavenly bdies were different in kind and were gverned by different laws. Newtn cncluded that nt nly des Earth attract an apple and the Mn but every bdy in the universe attracts every ther bdy: this tendency f bdies t mve tward each ther is called gravitatin. Newtn s cnclusin takes a little getting used t, because the familiar attractin f Earth fr earthbund bdies is s great that it verwhelms the attractin that earthbund bdies have fr each ther. Fr example, Earth attracts an apple with a frce magnitude f abut 0.8N. Yu als attract a nearby apple (and it attracts yu), but the frce f attractin has less magnitude than the weight f a speck f dust. Quantitatively, Newtn prpsed a frce law that we call Newtn s law f gravitatin: Every particle attracts any ther particle with a gravitatinal frce whse magnitude is given by mm F G r Here m and m are the masses f the particles, r is the distance between them and G is the gravitatinal cnstant, with a value that is given by 3 G Nm kg m kg s s the figure belw shws, a particle m attracts a particle m with a gravitatinal frce F that is directed tward particle m and particle m attracts particle m with a gravitatinal frce directed tward m. F that is -F m m F r The frces F and F frm a third law frce pair; they are ppsite in directin but equal in magnitude. They depend n the separatin f the tw particles, but nt their lcatin: the particles culd be a deep cave r in deep space. ls the frces F and F are nt altered by the presence f ther bdies, even if thse bdies lie between the tw particles we are cnsidering. The strength f the gravitatinal frce ie hw strngly tw particles with given masses at a given separatin attract each ther, depends n the value f the gravitatinal cnstant G. If G, by sme miracle, were suddenly multiplied by a factr f 0, yu wuld be crushed t the flr by earth s attractin. If G were divided by this factr, earth s attractin wuld be weak enugh that yu culd jump ver a building. 30

31 Dr. Z. Birech ENGINEERING PHYSICS (04) lthugh Newtn s law f gravitatin applies strictly t particles, we can als apply it t real bjects as lng as the sizes f the bjects are small cmpared t the distance between them. The mn and earth are far enugh apart s that t a gd apprximatin, we can treat them bth as particles. But what abut an apple and earth? Frm the pint f view f the apple, the brad and level earth stretching ut t the hrizn beneath the apple certainly des nt lk like a particle. Newtn slved the apple-earth prblem by prving an imprtant therem called the shell therem: unifrm spherical shell f matter attracts a particle that is utside the shell as if all the shell s mass were cncentrated at its centre. Earth can be thught f as a nest f such shells, ne within anther and each attracting a particle utside earth s surface as if the mass f that shell were at the center f the shell. Thus frm the apple s pint f view, earth des behave like a particle ne that is lcated at the centre f earth and has a mass equal t that f earth. Example ssuming the rbit f the earth abut the sun t be circular with radius.5 0 m, find the mass f the sun. Slutin The centripetal frce needed t hld the earth in an rbit f radius R is furnished by the gravitatinal attractin f the sun. We therefre have Centripetal frce gravitatinal frce mev GmEmS Rv m S R R G where v is the speed f the earth in its rbit arund the sun. Since the earth travels arund its rbit nce each year r in a time f s then we have (.5 0 m) 4 v ms s frm which m 30 S.0 0 kg 4. Gravitatin near Earth s Surface Escape velcity Is the maximum initial velcity that will cause a prjectile t mve upward frever, theretically cming t rest nly at infinity. If yu fire a prjectile upward, usually it will slw, stp and return t Earth. There is, hwever, a certain minimum initial velcity that will cause it t mve upward frever, theretically cming t rest nly at infinity. This initial velcity is called the (Earth s) escape velcity. Cnsider a prjectile f mass m leaving the surface f a planet r sme ther astrnmical bdy r system with escape velcity v. It has a kinetic energy K given by mv and ptential energy U given by GMm U R In which M is the mass f the planet and R is its radius. When the prjectile reaches infinity, it stps and thus has n kinetic energy. It als has n ptential energy because this is ur zer-ptential-energy cnfiguratin. Its ttal energy at infinity is therefre zer. Frm the principle f cnservatin f energy, its ttal energy at the planet s surface must als have been zer s that GMm GM K U mv 0 v R R The escape velcity s des nt depend n the directin in which a prjectile is fired frm a planet. Hwever, attaining that speed is easier if the prjectile is fired in the directin the launch site is mving as 3

32 Dr. Z. Birech ENGINEERING PHYSICS (04) the planet rtates abut its axis. Fr example, rckets are launched eastward at Cape Canaveral t take advantage f the Cape s eastward speed f 500km/h due t Earth s rtatin. The equatin abve can be applied t find the escape velcity f a prjectile frm any astrnmical bdy prvided we substitute the mass f the bdy fr M and the radius f the bdy fr R. the table belw shws sme escape velcities frm sme astrnmical bdies. Bdy Mass (kg) Radius (m) Escape speed (km/s) Ceres a Earth s mn Earth Jupiter Sun Sirius B b Neutrn star c a- the mst massive f the asterids b- a white dwarf(a star in a final stage f evlutin) that is a cmpanin f the bright star Sirius c- the cllapsed cre f a star that remains after that star has explded in a supernva event. 4.3 Planets nd Satellites: Kepler Laws The mtins f the planets, as they seemingly wander against the backgrund f the stars, have been a puzzle since the dawn f histry. The lp-the-lp mtin f Mars was particularly baffling. Jhannes Kepler (57-630), after a lifetime f study, wrked ut the empirical laws that gvern these mtins. Tych Brahe (546-60), the last f the great astrnmers t make bservatins withut the help f a telescpe, cmpiled the extensive data frm which Kepler was able t derive the three laws f planetary mtin that nw bear his name. Later Newtn ) shwed that his law f gravitatin leads t Kepler s laws. In this sectin we discuss each f Kepler s law in turn. lthugh here we apply the laws t planets rbiting the sun, they hld equally well fr satellites, either natural r artificial, rbiting Earth r any ther massive central bdy. THE LW OF ORBITS: ll planets mve in elliptical rbits, with the sun at ne fcus. The figure belw shws a planet f mass m mving in such an rbit arund the sun, whse mass is M. We assume that M m s that the centre f mass f the Planet-Sun system is apprximately at the centre f the Sun. the rbit in the is described by giving its semimajr axis a and its eccentricity e, the latter defined s that ea is the distance frm the centre f the ellipse t either fcus F r F`. n eccentricity f zer crrespnds t a circle, in which the tw fci merge t a single central pint. The eccentricities f the planetary rbits are nt large s that the rbits lk circular.eg the eccentricity f the Earth s rbit is nly Ra M r θ m F a ea F ea 3

33 Dr. Z. Birech ENGINEERING PHYSICS (04) THE LW OF RES: line that cnnects a planet t the Sun sweeps ut equal areas in the d plane f the planet s rbit in equal times; that is the rate at which it sweeps ut area is dt cnstant. Qualitatively, this law tells us that the planet will mve mst slwly when it is farthest frm the sun and mst rapidly when it is nearest t the sun. s it turns ut Kepler s secnd law is ttally equivalent t the law f cnservatin f mmentum as prved belw. dθ m θ r rδθ m r θ P P r The area f the shaded wedge in the figure belw clsely apprximates the area swept ut in time t by a line cnnecting the sun and the planet, which are separated by a distance r. the area f the wedge is apprximately the area f a triangle with base r and height r. since the area f a triangle is ne-half f the base times the height, then r d d In the limit t 0, r r () dt dt in which is the angular speed f the rtating line cnnecting Sun and Planet. Figure (b) shws the linear mmentum p f the planet, alng with its radial and perpendicular cmpnents. The magnitude f the angular mmentum L f the planet abut the Sun is given by the prduct f r and cmpnent f p perpendicular t r. here, fr a planet f mass m, p, the L rp ( r)( mv ) ( r)( mr ) mr () where we have replaced v with its equivalent r. Frm equatin we have d L (3) dt m If d dt is cnstant, as Kepler said it, then equatin 3 means that L must als be cnstant-angular mmentum is cnserved. Kepler s secnd law is indeed equivalent t the law f cnservatin f angular mmentum. 3 THE LW OF PERIODS: The square f the perid f any planet is prprtinal t the cube f the semimajr axis f its rbit Cnsider the circular rbit with radius r (the radius f a circle is equivalent t the semimajr axis f an ellipse) in the figure belw. 33

34 Dr. Z. Birech ENGINEERING PHYSICS (04) m r M θ pplying Newtn s secnd law ( F ma) t the rbiting planet yields GMm ( m)( r) (4) r If we use, where T is the perid f the mtin, then we btain Kepler s third law T 4 3 T r GM (5) The quantity in parentheses is a cnstant that depends nly n the mass M f the central bdy abut which the planet rbits. Equatin 5 hlds als fr elliptical rbits, prvided we replace r with a, the T semimajr axis f the ellipse. This law predicts that the rati has essentially the same value fr every 3 a planetary rbit arund a given massive bdy. The table belw shws hw well it hlds fr the rbits f the planets f the slar system. Table: Kepler s Law f Perids fr the Slar System Planet semimajr axis a (0 0 m) perid T (yrs) T 34 3 (0 y m ) 3 a Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Plut Satellites: Orbits nd Energy s a satellite rbits Earth n its elliptical path, bth its speed, which fixes its kinetic energy K, and its distance frm the center f Earth, which fixes its gravitatinal ptential energy U, fluctuates with the fixed perids. Hwever, the mechanical energy E f the satellite remains cnstant. (Since the satellite s mass is s much smaller than Earth s mass, we assign U and E fr the Earth-satellite system t the satellite a lne. The ptential energy f the system is give by 34

35 Dr. Z. Birech ENGINEERING PHYSICS (04) GMm U (0) r (with U 0 fr infinite separatin). r is the radius f the rbit, assumed fr the time being t be circular, and M and m are the masses f Earth and the satellite, respectively. T find the kinetic energy f a satellite in a circular rbit, we write Newtn s secnd law as GMm v m () r r v where is the centripetal acceleratin f the satellite. Then frm, the kinetic energy is r GMm K mv () r which shws that fr a satellite in a circular rbit, U K (3) The ttal mechanical energy f the rbiting satellite is GMm GMm GMm E K U (4) r r r This tells us that fr a satellite in a circular rbit, the ttal energy E is the negative f the kinetic energy K: E K (5) Fr a satellite in an elliptical rbit f semimajr axis a, we can substitute a fr r in equatin 4 t find the mechanical energy as GMm E (6) a Equatin 6 tells us that the ttal energy f an rbiting satellite depends nly n the semimajr axis f its rbit and nt n its eccentricity e. eg fur rbits with the same semimajr axis are shwn in figure a belw. The same satellite wuld have the same ttal mechanical energy E in all fur rbits. Figure b shws the variatin f K, U and E with r fr a satellite mving in a circular rbit abut a massive central bdy. Energy e = 0 e = 0.5 e = 0.8 e = 0.9 K(r) m e = 0 E(r) r U(r) Example playful astrnaut releases a bwling ball, f mass m = 7.0kg, int circular rbit abut Earth at an altitude h f 350km. (a) What is the mechanical energy E f the ball in its rbit? (b) What is the mechanical energy E f the ball n the launch pad at say Cape Canaveral? Frm there t the rbit, what is the change E in the ball s mechanical energy? Slutin (a) the key idea here is that we can get E frm the rbital energy, given by equatin 4, if we first find the rbital radius r. that radius is 35

36 Dr. Z. Birech ENGINEERING PHYSICS (04) 6 r R h 6370km 350km 6.70 m in which R is the radius f Earth. Then frm equatin 4, the mechanical energy is 4 GMm ( Nm kg )(5.980 kg)(7.0kg) 8 E.40 J 4MJ 6 r 6.70 m Slutin (b) the key idea is that, n the launch-pad, the ball is nt in rbit and thus equatin 4 des nt apply. Instead we must find E0 K U, where K is the ball s kinetic energy and U is the gravitatinal ptential energy f the ball-earth. T find U, we use equatin 0 t write 4 GMm ( Nm kg )(5.980 kg)(7.0kg) 8 U 4.50 J 45MJ 6 R m The kinetic energy K f the ball is due t the ball s mtin with Earth s rtatin. Yu can shw that K is less than MJ, which is negligible relative tu. Thus the mechanical energy f the ball n the launch pad is E K U 0 45MJ 45MJ The increase in the mechanical energy f the ball frm launch pad t rbit is E E E ( 4MJ) ( 45MJ) 37MJ Chapter 5: Prperties f Matter 5. Pressure Pressure is defined as the average frce per unit area at the particular regin f fluid (liquid r gas) ie F P where F is the nrmal frce due t the liquid n the side f area. t a given pint in a liquid, the pressure can act in any directin ie pressure is a scalar quantity. If the pressure were nt the same, there wuld be an unbalanced frce n the fluid at that pint and the fluid wuld mve. The lgical basis fr the statement that pressure is exerted equally in all directins, then, is simply that therwise the parts f the fluid wuld nt be in equilibrium. ls pressure increases with depth, h, belw the liquid surface and with its density s that P hg When g is in ms -, h is in m and ρ is in kgm -3, then the pressure is in Newtn per meter squared (Nm - ). The bar is a unit f pressure used in meterlgy and by definitin, bar 0 5 Nm - The Pascal (Pa) is the name given t a pressure f Nm -. Thus bar 0 5 Pa Pressure is ften expressed in terms f that due t a height f mercury (Hg). One unit is the trr (after Trricelli); trr mmhg 33.3Nm - Frm P hg it fllws that the pressure in a liquid is the same at all pints n the same hrizntal level in it. Thus a liquid filling the vessel shwn belw rises t the same height in each sectin if BCD is hrizntal. 36

37 Dr. Z. Birech ENGINEERING PHYSICS (04) B C D Fig Pressure in a vessel is independent f the crss-sectin tmspheric Pressure barmeter is an instrument fr measuring the pressure f the atmsphere which is required in weather frecasting. It cnsists f a vertical tube abut a meter lng cntaining mercury with a vacuum at the clsed tp. The ther end f the tube is belw the surface f mercury cntained in a vessel B. 760mm = h vacuum x H = 760mm B B The pressure n the surface f the mercury in B is atmspheric pressure and since the pressure is transmitted thrugh the liquid, the atmspheric pressure supprts the clumn f mercury in the tube. Suppse the clumn is a vertical height H abve the level f the mercury in B. then if H 760mm 0.76m and ρ = 3600kgm -3, we have P Hg Nm - The pressure at the bttm f a clumn f mercury 760mm high fr a particular density and value f g is knwn as standard pressure r ne atmsphere. By definitin, atmsphere Nm - Standard temperature and pressure is 0 C and 760mmHg. It shuld be nted that the pressure P at a place X belw the surface f a liquid is given by P Hg where H is the vertical distance f X belw the surface. In fig ii abve, a very lng barmeter tube is inclined at an angle f 60 t the vertical. The 37

38 Dr. Z. Birech ENGINEERING PHYSICS (04) length f mercury alng the slanted side f the tube is x mm say. If the atmspheric pressure is the same as in i, then the vertical height t the mercury surface is still 760mm. S xcs x 760 cs mm 5. rchimedes Principle n bject immersed in a fluid experiences a buyant (upthrust) frce equal t the weight f the fluid that it displaces pplicatin f the principle The buyant frce f a fluid n an bject depends n the weight f fluid displaced and thus n the density f the fluid and the vlume f the fluid displaced (since M DV ). In the case f a ttally immersed bject, the vlume f fluid displaced is just equal t the vlume f the bject, and therefre if the buyant frce is measured and the fluid density knwn, the density f the bject can be readily calculated. Frm this knwn vlume and its mass, the density f the bject may be fund. It is nt easy t measure directly the vlume f irregularly shaped bjects with great accuracy, but rchimedes principle prvides a way t find vlume accurately, since nly balance measurements are needed. If tw substances have densities D and D, then the density f the secnd substance relative t the first is mass m m D /D. Since density, then D and D vlume V V D m If ne cmpares equal vlumes f the tw substances s that V = V, then the relative density, D m shwing that the relative density will equal the rati f their masses r their weights. The density f a substance relative t that f water is called the specific gravity f the substance ie the specific gravity f a substance is equal t the density f the substance divided by the density f water r the specific gravity f a substance is equal t the weight f a certain vlume f the substance divided by the weight f an equal vlume f water. In accurate wrk it is necessary t specify the temperature at which the measurements are made. Example chunk f cpper suspended frm a balance weighs 56.8g in air. When it is cmpletely surrunded by pure water at 0 C, the reading n the balance is 39.g. Calculate the specific gravity f cpper. Slutin The weight f cpper in air = 56.8g The apparent weight f cpper in water = 39.g Therefre the buyant frce f the water = = 7.6g By rchimedes principle, the weight f the displaced water = 7.6g weight f Cpper The specific gravity f the cpper = weight f equal vlume f water But the vlume f the displaced water is equal t the vlume f the cpper which displaces it. Therefre 56.8g The specific gravity f cpper = g 38

39 Dr. Z. Birech ENGINEERING PHYSICS (04) Therefre the density f the cpper in this sample is 8.9 times the density f pure water at 0 C. When the specific gravity f a substance is knwn, its density in any units can be calculated frm the knwn density f water. Wherever the frce f gravity acts n a fluid, the fluid exerts a buyant frce as a result f difference in pressure at different levels. Every fish and submarine in the sea is buyed up by a frce equal t the weight f water displaced. T remain submerged these bjects must have a weight equal t r greater than the buyant frce. If they wish t mve frm ne level t anther, the balance between the frce f gravity and the buyant frce must be disturbed. Sme fish can rise by expanding their bdies, thereby displacing mre water. Submarines are made t rise by decreasing their weight by frcing water ut f their ballast tanks. The air f the earth s atmsphere als exerts a buyant frce n all bjects immersed in it eg Ballns utilize the buyancy f air. If a gas such as hydrgen r helium is used t inflate a lightweight plastic envelpe, the buyancy frce f the air can be cnsiderably greater than the weight f the balln. Such ballns are used in making high altitude measurements f varius prperties f the atmsphere. Example weather balln has a vlume f 0.5m 3 when inflated. The weight f the envelpe is 350g. If the balln is filled with helium, what weight f instrument can it carry a lft? The density f air is.9kgm -3 and the density f helium is 0.38 times the density f air. Slutin By rchimedes principle, the buyant frce f the air is =the weight f air displaced=the weight f 0.5m 3 f air =0.5m 3.9kgm -3 =0.65kg=650g Since the relative density f helium is 0.38(air ), the weight f helium in the balln is = g=90g Therefre the weight f the balln and helium =350g+90g=440g The excess f the buyant frce ver the frce f gravity=650g - 440g=0g Therefre if an unbalanced frce f 0g is left available t prduce upward acceleratin, the weight f the instrument lad can be 00g. Flating at the surface f liquids ccrding t rchimedes principle, if a liquid is displaced by a slid, the liquid exerts a buyant frce n the slid. This is true fr any fractin f the slid that displaces liquid eg when a by is lifting a stne ut f water, he des nt have t supprt the full weight f the stne until it is cmpletely clear f the water. When half f the stne s vlume is submerged, the buyant frce is half f the buyant frce when it is cmpletely submerged. Suppse that an bject is placed in a liquid f density greater than that f the bject. The buyant frce f the liquid will equal the frce f gravity befre the slid is cmpletely submerged. T g beynd this pint f equilibrium, requires then an extra dwnward frce t be applied. If it is nt, the bject remains in equilibrium under the actin f balanced frces: it flats. When an bject that will flat is placed int a liquid, it will sink int the liquid until the weight that it displaces is equal t its wn weight. This relatin, a direct cnsequence f rchimedes principle, is smetimes called the law f fltatin. Example blck f wd weighing 0g has a vlume f 80cm 3. What fractin f its vlume wuld be submerged when flating in alchl f density 0.80gcm -3? Slutin Fr the blck t flat, it must displace 0g f alchl. The vlume f 0g f alchl= 0g 3 50cm gcm T displace 50cm3 f alchl, the blck will have t have 50cm3 f its vlume submerged. 39

40 Dr. Z. Birech ENGINEERING PHYSICS (04) The fractin submerged rchimedes principle prvides the basis fr the design f ships made f steel. Fr a steel vessel t flat it is nly necessary t spread the steel arund s that it can displace an amunt f water having a weight that exceeds the weight f the steel. Example steel bx is cnstructed t make a cube 0cm n a side, frm material 0.0cm thick. What weight f cntents is pssible befre the bx sinks in a liquid f specific gravity.? The density f steel is 7.0gcm -3 Slutin The vlume f the bx is (0cm) 3 =000cm The maximum buyant frce that the liquid can prvide is 000cm..0gcm 00g The weight f the bx which has 6 sides, each 0cm square and 0.cm thick, and which is made f steel f density 7gcm -3 is 60cm 0cm 0.cm 7gcm 3 840g The excess f buyant frce ver weight is 00g 840g 360g If material is added t the bx, it will cntinue t flat until the weight f the cntents exceeds g. Hydrmeters The relatin between fluid density and flating prvides basis fr the cnstructin and use f hydrmeters. Hydrmeters usually cnsist f a hllw tube weighted at ne end and having a graduated scale at the ther end. In a liquid, the weighted end ensures that the instrument flats upright. The depth t which it sinks will depend n the density f the fluid. The higher the density f the liquid, the greater will be its buyant frce per unit vlume f the hydrmeter immersed. When the scale is graduated using liquids f knwn density, the hydrmeter may be used t measure the density f unknwn liquids. number f specialized hydrmeters are used fr the determinatin f specific gravity: in dairies fr milk, in autmbile service statins fr antifreeze and battery acids, in chemical labratries fr determining the cmpsitin f aqueus slutins (in a.5% sugar slutin at 3 C, a hydrmeter wuld indicate a specific gravity f.05). 5.3 Fluids in mtin stream / river flws slwly when it runs thrugh pen cuntry and faster thrugh narrw penings r cnstrictins This is due t the fact that water is practically an incmpressible fluid ie changes f pressure cause practically n change in fluid density at varius parts. Figure belw shws a tube f water flwing steadily between X and Y where X has a bigger crss-sectinal area than the party f crss-sectinal area. The streamlines f the flw represent the directins f the velcities f the particles f the fluid and the flw is unifrm r laminar. l S Q l P R 40

41 Dr. Z. Birech ENGINEERING PHYSICS (04) ssuming the liquid is incmpressible, then if it mves frm PQ t RS, the vlume f liquid between P and R is equal t the vlume between Q and S. Thus l l l l where l is PR and l is QS. Hence l is greater thanl. Cnsequently the velcity f the liquid at the narrw part f the tube, where the streamlines are clser tgether, is greater than at the wider part Y where the streamlines are further apart. Fr the same reasn, slw-running water frm a tap can be made int a fast jet by placing a finger ver the tap t narrw the exit. Bernulli s principle Bernulli btained a relatin between the pressure and velcity at different parts f a mving incmpressible fluid. If viscsity is negligibly small, there are n frictinal frces t vercme. Hence the wrk dne by the pressure difference per unit vlume f a fluid flwing a lng a pipe steadily is equal t the gain in kinetic energy per unit vlume plus the gain in ptential energy per unit vlume. The wrk dne by a pressure in mving a fluid thrugh a distance=frce distance mved=pressure area distance mved=pressure vlume. t the beginning f the pipe where the pressure is P, the wrk dne per unit vlume n the fluid is P ; at the ther end the wrk dne per unit vlume is P. Hence the net wrk dne n the fluid per unit vlume =P -P. The kinetic energy per unit vlume= mass per unit velcity = ρ velcity where ρ is the density f the fluid. Thus if v and v are the final and initial velcities respectively at the end and the beginning f the pipe, the the kinetic energy gained per unit vlume= ( v v ). Further, if h and h are the respective heights measured frm a fixed level at the end and beginning f the pipe, the ptential energy gained per unit vlume =mass per unit vlume g h h = gh h. Thus frm the cnservatin f energy P P ( v v ) g( h h ) P v gh p v gh Therefre P v gh Cnstant where P is the pressure at any part and v is the velcity there. Hence, fr streamline mtin f an incmpressible nn-viscus fluid, The sum f the pressure at any part plus the kinetic energy per unit vlume plus the ptential energy per unit vlume there is always a cnstant. This is knwn as Bernulli s principle. The principle shws that at pints in a mving fluid where the ptential energy change gh is very small, r zer as in flws thrugh a hrizntal pipe, the pressure is lw where the velcity is high; cnversely, the pressure is high where the velcity is lw. Example s a numerical illustratin, suppse the area f crss-sectin f X in the figure abve is 4cm, the area f Y is cm and water flws past each sectin in laminar flw at the rate f 400cm 3 s -, then vlumeper sec nd at X speed v f water= area 3 400cm s 4cm 00cms ms 4

42 Dr. Z. Birech ENGINEERING PHYSICS (04) at Y speed v f water= cm s 400cms 4ms cm 3 The density f water 000kgm. S if P is the pressure difference, then 3 P v v Nm 3 P 7.50 Therefre P hg h 0. 77m g The pressure head is thus equivalent t 0.77m f water. pplicatins f Bernulli s principle. suctin effect is experienced by a persn standing clse t the platfrm at a statin when a fast train passes. The fast-mving air between the persn and the train prduces a decrease in pressure and excess air pressure n the ther side pushes the persn twards the train.. Filter pump. filter pump has a narrw sectin in the middle, s that a jet f water frm the tap flws faster here. This causes a drp in pressure near it and air therefre flws in frm the side tube t which a vessel is cnnected. The air and water tgether are expelled thrugh the bttm f the filter pump. 3. erfil lift. The curved shape f an aerfil creates a faster flw f air ver its tp surface than the lwer ne. This is shwn by the clseness f the streamlines abve the aerfil cmpared with thse belw. Frm Bernulli s principle, the pressure f the air belw is greater than that abve, and this prduces the lift n the aerfil. 4. Flw f a liquid frm wide tank. Cnsider the figure belw. t the tp X f the liquid in the tank, the pressure is atmspheric say B, the height measured frm a fixed level such as the hle H is h, and the kinetic energy is negligible if the tank is wide s that the level falls very slwly. t the bttm, Y near H, the pressure is again B, the height is zer and the kinetic energy is v where is the density and v is the velcity f emergence f the liquid. Thus frm Bernulli s principle, B gh B v v gh Thus the velcity f the emerging liquid is the same as that which wuld be btained if it fell thrugh a height h and this is Trricelli s therem. In practice the velcity is less than that given by gh wing t viscus frces and the lack f streamline flw. 3 Example Water flws steadily a lng a hrizntal pipe at a vlume rate f80 m s. If the area f crss-sectin f the pipe is 40cm, calculate the flw velcity f the water. Find the ttal pressure in the pipe if the static pressure in the hrizntal pipe is Pa, assuming the water is incmpressible, nn-viscus and its density is 000kgm-3. What is the new flw velcity if the ttal pressure is Pa. 3 vlumeper sec nd 80 i Velcity f water= ms 4 area ii Ttal pressure=static pressure v Pa iii v ttal pressure- static pressure Therefre 000 v v ms. 4

43 Dr. Z. Birech ENGINEERING PHYSICS (04) 5.4 Elastic Prperties f Matter perfect rigid bdy has cnstant distance between tw particles (nt true in practice), i.e mst bdies get defrmed under an applied frce and the bdy has a tendency t regain its riginal size and shape when the frce is remved. This prperty f the bdy that tends t regain its shape r size when defrming frces are remved is called elasticity. perfectly elastic bdy retains its riginal shape/size very quickly, while if it regains slwly then it is called a perfectly plastic bdy. Slids tend t resist change f bth shape and vlume and hence they psses rigidity r shear elasticity as well as vlume elasticity. Liquids n the ther hand tend t resist change in vlume and nt shape (they psses nly vlume elasticity). Stress When a bdy experiences a defrming external frce, different particles in it are displaced, and they try t ccupy their riginal psitins. This restring frce per unit area taking place inside the bdy is called the stress. s lng as there is n permanent change in shape r vlume f the bdy, the restring frce is always equal t the applied frce. mathematically Stress F / a where a is the crss sectinal area f the bdy. Strain When an external frce acts n a bdy, it displaces varius particles and the bdy is said t be under strain. Usually defined as the ratin f change in length, vlume r shape t the riginal length, vlume r shape. Hke s Law The relatinship between stress and strain. stress E(cnst) strain E is the cefficient f elasticity r mdulus f elasticity (value depends n the nature f the material. Graphical representatin f this relatinship is shwn belw. stress Plastic range Elastic limit strain Frm the graph, hke s is nly valid in the regin belw the elastic limit, beynd which the bdy has un-prprtinal variatin in strain and stress (the change in stress leads t a rapid change in strain). Yung s Mdulus This is the rati f stress t lngitudinal strain within the elastic limits. Cnsider a wire f length L and let it change by l under an applied frce F acting n a crss sectinal area a. Frm this, lngitudinal strain is l/l and stress is F/a. Therefre Yung s mdulus f Elasticity will be 43

44 Dr. Z. Birech ENGINEERING PHYSICS (04) Y stress strain F / a l / L FL al [N/m ] Bulk Mdulus This is the rati between stress and vlumetric strain. If a frce is applied nrmally ver a surface f a bdy and it changes nly in vlume takes place, then the strain caused here is vlumetric strain. It is measured by change in vlume per unit vlume (v/v). i.e F / a v / V FV av 5.5 Viscsity Viscsity is a prperty by which a liquid ppses relative mtin between its different layers. Liquids like kersene, alchl, water, etc flw easily while thers like tar, glycerin, etc, flw with difficulty and are called viscus. Cefficient f viscsity Cnsider a layer B f a liquid mving with velcity v w.r.t. a parallel layer CD which is at a distance r frm it. Cnsider als that the frce required t prduce the mtin be F acting n an area and this frce is acting alng the directin B (i.e in the directin f mtin). n equal frce will act in the ppsite directin due t viscsity and it will depend n the fllwing: i. F α v ii. F α iii. F α /r Cmbining the three we have v F r Where η is the cefficient f viscsity and it depends n the nature f the liquid. If the tw layers are very clse t each ther, then dv F dr Where dv/dr is the velcity gradient. If = cm and dv/dr = then F = η which gives the definitin f cefficient f viscsity as the tangential frce per unit area required t maintain a unit velcity gradient. If F = then η = and the unit is Pise. Stkes Law If a bdy falls thrugh a fluid (liquid r gas) then it carries alng with it a layer f the fluid in cntact and hence it will tend t prduce sme relative mtin between the layers f the fluid. This relative mtin is ppsed by frces f viscsity and the ppsing frce increases as the velcity f the bdy increases. If the bdy if small enugh, then the ppsing frce becmes equal t the driving frce that prduces the mtin. t that instance then the bdy mves with cnstant velcity knwn as terminal velcity. Cnsider a small sphere falling thrugh a viscus medium, then the ppsing frce is directly prprtinal t the velcity f the sphere and depends n i. cefficient f viscsity f the medium ii. radius f the sphere iii. density f the medium 44

45 Dr. Z. Birech ENGINEERING PHYSICS (04) cmbining the three we have F k. v. r Where k is a cnstant (calculated and fund t be 6π). II 45

46 Dr. Z. Birech ENGINEERING PHYSICS (04) SOUND ND VIBRTIONS Chapter 5: Wave Mtin Intrductin Wave mtin is a frm f disturbance that travels thrugh the medium due t the repeated peridic mtin f particles f the medium abut their mean psitins. This disturbance is transferred frm ne particle t the next, e.g water waves, sund. This als invlves the transfer f energy frm ne pint t the ther. Characteristics f a wave. It s a disturbance prduced in the medium by repeated peridic mtin f the particles f the medium.. The wave travels frward while particles f the medium vibrate abut their mean psitins 3. there is a regular phase change between varius particles f the medium 4. Velcity f the wave is different frm the velcity with which the particles f the medium are vibrating abut their mean psitins. Types f wave mtin. Transverse wave: particles f the medium vibrate abut their mean psitins in the directin perpendicular t the directin f prpagatin, e.g. light waves, water waves etc. Lngitudinal: particles f the medium vibrate abut their mean psitin in the directin f the wave, e.g sund wave General wave equatin Since wave mtin invlves vibratin f particles abut the mean psitin, they are als a SHM type f wave mtin. The displacement f a particle P in SHM is given by y asint Suppse anther particle Q is at a distance x frm P and the wave is traveling with velcity v in psitive x directin (Fig belw). a P Q x Then the displacement f the particle Q will be given by y asin( t ) (*) Where φ is the phase difference between the particles P and Q. The phase difference crrespnding t the path difference (wavelength, λ) is π, which leads t the relatin x Where x is the path difference between P & Q. Hence the phase difference φ between P and Q is x and ngular frequency ω is 46

47 Dr. Z. Birech ENGINEERING PHYSICS (04) v f T Hence equatin (*) becmes v x y asin( t ) (**) Or y asin ( vt x) (***) This is the general wave equatin fr a traveling wave in psitive x directin with velcity v. Differential frm f general wave equatin The general wave equatin can als be represented in differential frm. Differentiating the equatin (***) w.r.t. t gives dy av cs ( vt x) dt Further differentiatin f equatin (a) w.r.t. t gives d y av dt sin ( vt x) (a) (b) T get the cmpressin f the wave (in x space), we differentiate equatin (**) w.r.t. x, i.e. dy a cs ( vt x) (c) dx Further differentiatin f equatin (c) w.r.t. x will give the cmpressin in terms f distance (in x- directin), i.e d y a dx sin ( vt x) Cmparing equatins (a) and (c) we see that dy dy dx v dt nd frm (b) and (d) we have (d) d y dx v d y dt 47

48 Dr. Z. Birech ENGINEERING PHYSICS (04) Which represents the differential frm f a wave equatin. Particle velcity f a wave If the velcity f a particle is dented by u then dy av u cs ( vt x) dt a nd umax v, which implies that Maximum particle velcity is πa/λ times the wave velcity (v). Likewise maximum particle acceleratin is given by v fmax a (Shw this) Distributin f velcity & Pressure in a wave If a wave is prgressive (i.e, new waves are cntinuusly frmed) then there is a cntinuus transfer f energy in the directin f prpagatin f the wave. Remember particle velcity is given by dy av u cs ( vt x) dt The strain in the medium is given by dy/dx, i.e, if it is psitive, it represents a regin f rarefactin and when negative, represents regin f cmpressin. Fr such a medium, the mdulus f elasticity is given by change in pressure dp K vlume strain ( dy / dx) dy nd dp K. dx Implying that dp is psitive in regins f cmpressin and ve in rarefactin regin y T u πav/λ T dp C R C P T t 48

49 Dr. Z. Birech ENGINEERING PHYSICS (04) Interference When tw sund waves are mving alng a straight line in a medium, then every particle f the medium is simultaneusly acted upn by bth f the waves. If the tw waves arrive at a pint in phase (tw crests r tw trughs) superimpse and the resultant amplitude is equal t the sum f the respective waves (this is the principle f superpsitin). If the waves arrive at a pint when they are cmpletely ut f phase (a crest f ne falls ver the trugh f anther, then the resultant amplitude is equal t the difference f the individual amplitudes. This implies that at pints where the tw waves meet in phase will give maximum amplitude (hence maximum sund intensity) while where they meet ut f phase gives minimum amplitude (minimum sund intensity). The phenmenn described abve is called interference. cmbined Wave Wave (a) (b) Fig. (a) cnstructive and (b) destructive interference Let tw waves having amplitudes a and a be represented by equatins and y a sin ( vt x) y a sin ( vt x) Where φ is the phase difference between the tw waves after sme time. The resultant displacement will be Y y y a sin ( vt x) a sin ( vt x) a sin ( vt x) a sin ( vt x) cs a cs ( vt x) sin sin ( vt x) a a cs cs ( vt x) a sin (using the trig identity Sin ( + B) = Sin cs B + sin B cs ) Letting a + a cs φ = cs θ, a sin φ = sin θ and using trignmetric identities leads t 49