Chapter II Newtnian Mechanics Sinle Particle Recended prbles: -, -5, -6, -8, -9, -, -, -, -6, -, -, -, -5, -6, -7, -9, -30, -3, -3, -37, -38, -39, -, -, -3, -, -7, -5, -5, -53, -5..
. Newtn s Laws The First Law: A bd reains at rest r in unifr tin unless acted upn b a frce. The Secnd Law: A bd acted upn b a frce ves in such a anner that the frce is equal t the tie rate f chane f entu, i.e., dp F (.) with P v (.) The Third Law: If tw bdies eert frce n each ther, these frce are equal in anitude and ppsite in directin. In anther wrd, if tw bdies cnstitute an ideal, islated sste, then the acceleratins f these bdies are alwas in ppsite directins, and the rati f the anitudes f the acceleratins is cnstant and equal t the inverse rati f the asses f the bdies. F F (.3) Usin Eq.(.) we et
a dp dp (.) dv dv ( with cnstant ass) a (.5) a a Eq.(.) can be rearraned as d P P 0 P P cnstant That is, fr islated sste, the ttal entu is cnserved.
.3 Fraes f Reference Fr Newtn's laws f tin t have eanin, the tin ust be easured relative t se reference frae. A reference frae is called an inertial frae if Newtn s laws are valid in that frae. If Newtn s laws are valid in ne reference frae, then the are als valid in an reference frae in unifr tin (nt accelerated) with respect t the first frae. If Newtn s laws are valid in ne reference frae, then the are als valid in an reference frae in unifr tin (nt accelerated) with respect t the first frae. This is because the Equatin F v invlves a tie derivative f velcit: a chane f crdinates invlvin cnstant velcit desn t influence the equatin.
. The Equatin f Mtin f a Particle Newtn s secnd law can be epressed as F v r (.6) This is a secnd rder differential equatin, which can be slved t find r if the frce functin F(v,r,t) and the initial values f r and v are knwn. Eaple. If a particle slides withut frictin dwn a fied, inclined plane with =30. What is the blck s acceleratin. Slutin There are tw frces actin n the blck: The ravitatinal frce and the nral frce. If the blck is cnstrained t ve n the plane, and takin the +ve - ais dwn the plane the nl directin the blck can ve is the -directin. Eq.(.6) nw reads F N r F cs F N F sin
Applin the last equatin in the tw directins, we have -directin: F cs N 0 -directin: F sin sin sin.9 / s T find the velcit after tie t, we have d sin v v d sin t 0 T find the velcit after it ves a distance dwn the plane we have d d sin d sin d v v d sin sin d v v sin t
v v sin If at t=0, and v are zer, then v sin Eaple. If the cefficient f static frictin between the blck and the plane is s =0., at what anle will the blck start slidin if it is initiall at rest? acceleratin. Slutin We have nw an additinal frce, the static frictinal frce which is parallel t the plane. Applin Newtn s we have F cs -directin: F cs N 0 f s F N F sin -directin: F sin f s Knwin that f s f a N s At the vere f slippin f s reaches its aiu value, then we write
F sin f a F sin sn Substitutin fr F = and fr N fr the first equatin we et sin s cs sin cs Just befre the blck starts t slide, the acceleratin is zer s sin cs 0 s tan tan 0. s
Eaple.3 After the blck in the previus eaple bein t slide, the cefficient f kinetic frictin beces k =0.3. Find the acceleratin fr the anle =30. Slutin Applin Newtn s aain we have -directin: cs N 0 -directin: Knwin that f k N sin fk cs k sin k cs 0. In eneral s > k. This can be bserved b lwerin the anle belw 6.7, we find that 0 and the blck stps. If we increase the anle abve 6.7, the blck desn t slidin aain until eceeds. This is because nw the frce that retardin the tin is n lner the kinetic frictinal frce but rather it is the static frictinal frce which is reater than the kinetic frictinal frce.
Effects f Retardin Frces If a bd is acted upn b a resistin frce F(v) in additin, fr instance, t the ravitatinal frce, the ttal frce is then. F r F F F r F kv F v v n v (.7) (.8) Where k is a psitive cnstant that specifies the strenth f the retardin frce and v/v is a unit vectr in the directin f v. Eaple. Find the displaceent and the velcit f hrizntal tin f a particle in a ediu in which the retardin frce is prprtinal t the velcit. Slutin Knwin that the nl hrizntal frce actin n the particle is the retardin frce, applin Newtn s aain we have -directin: kv dv kv dv v k lnv kt C
If the initial velcit (at t=0) v=v then v v e kt T find the displaceent we have d kt kt v v v e v e kt e C k v kt e k If (t=0)=0 then C =v /t We can find the velcit as a functin f displaceent b writtin dv dv dv d d v dv v d dv kv dv d k dv k d v k C3 If the initial vel (t=0)=0 and v(t=0)=v then C 3 =v v v k
Eaple.5 Find the displaceent and the velcit f a particle underin vertical tin in a ediu with a retardin frce is prprtinal t the velcit. Slutin Cnsiderin the particle is fallin dwnward with a initial velcit v fr a heiht h in a cnstant ravitatinal field. The equatin f tin is z-directin: F dv kv The inus sin in the retardin frce (which is upward frce) is due t the fact that the velcit is dwnward. The last equatin can be written as dv kv Knwin that (at t=0) v=v then interatin the last equatin we et kv kv t C k ln kv ln t k kv kt kv kt kv e v e It is clear that as t, the velcit apprachin the terinal value (-/k). At this value the net frce vanish. dz k k
If v eceeds the terinal velcit in anitude, then the bd beins t slw dwn and v appraches the terinal speed fr the ppsite directin. T find the displaceent we interate aain, with (at t=0) z=h t et z kv kt e t h k k
Eaple.6 Let us stud the prjectile tin in -diensins withut cnsiderin air resistance. v =0 Slutin The equatins f tin are -directin: -directin: 0 v H a The rane R Assuin (t=0)= (t=0) we et v cs v cs t v sin t v sin t t Eliinatin t fr the abve -equatins we et v tan v cs Which is the equatin f a parabla
The speed and the ttal displaceent are fund t be v v t v r vt t v tsin t 3 sin The rane can b fund b deterinin the value f when the prjectile falls back t rund, i.e., (=0) v sin t t 0 Nw the rane R is fund b R t T v sin cs v sin v t t T sin 0 & It is eas t shw that the aiu rane ccurs at =5.
Eaple.7 Let us stud the effect f the air resistance t the prjectile tin in the previus eaple, assuin that the retardin frce is directl prprtinal t the prjectile s velcit (F r =-kv). Slutin The equatins f tin are in this case -directin: k -directin: k Assuin aain (t=0)= (t=0) we et cs kt t kv sin kt e e v k k k T find the rane we need the tie T when =0 T kv sin kt e k This is a transcendental equatin s we can t btain an analtical epressin fr T. It can be slved b appriatin (perturbatin) r b nuerical technique.
T appl the perturbatin ethd we assue that k is relativel sall. Nw rewrite the transcendental equatin as T kv sin 3 3 kt k T k T T k 6 3 kv sin T kv sin kt kv sin k T kv sin kv sin T kv sin 0 kt T Usin theidentit kv sin v sin kv sin 3 kt 3 kv T first rder f k we then have sin 6 6 kv sin T v sin T 3 v sin k O k
With n air resistance (k=0) we recver the sae result as in the previus eaple, i.e., T T v sin If k is sall (but nnvanishin), the fliht tie will be appriatel equal t T. Usin this appriated value we et T v sin v sin v sin k 3 T v sin kv sin 3 Nw t find the rane we have v cs kt k k t 6 k 3 t 3 T first rder f k the rane is btained fr R v cs T kt Substitutin fr T in the last equatin
R v sin kv sin 3 R kv sin 3 With R is the rane withut air resistance.
Eaple.9 Atwd s achine cnsists f a sth pulle with -asses suspended fr a liht strin at each end. Find the acceleratin f the asses and the tensin f the strin (a) when the pulle center is at rest and (b) when the pulle is descendin in an elevatr with cnstant acceleratin. Slutin (a) The equatins f tin, fr each ass, are T T If the strin is inetensible, i.e., cnstant
Slvin the first tw equatins fr T we et T (b) The crdinate sste with the riin at the pulle center is n lner an inertial. S we select the riin f the crdinates t be at the tp f the elevatr shaft. The equatins f tins in such a sste are T T But, as it is clear fr the fiure,, T T Knwin that & T T
Slvin the last -equatins fr the acceleratin and the tensin we et T Nte that the result are just as if the acceleratin f ravit were reduced b an aunt f the elevatr acceleratin. If the elevatr is ascendin rather than descendin we epect T
Eaple.0 Cnsider a chared particle enterin a rein f unifr anetic field. Deterine its subsequent tin. Slutin Let the anetic field be parallel t the -ais. The anetic frce is F qv B The equatin f tin reads qb k ˆ zi ˆ iˆ ˆj z kˆ qbz 0 z qb Interatin Eq.() we et Interatin Eq.(&3) we et () () (3) cnstant () t (5)
qb z C z C z C (7) (6) Substitutin fr z Eq.(7) int Eq.() we et C3 A A (8) a Rcs t (9) Differentiatin Eq. (9) w.r.t tie and substitutin int Eq. (6) t z C Rsin z b Rsin t (0) Squarin Eqs(9+0) and then addin we et a z b R ()
Then the path f the tin is a circle f radius R and centered at (a,b). Nw fr z cnstant the tin is heli with its ais in the directin f B. Nw fr Eqs.(0&) we have R sint R cst Squarin the abve tw equatins and then addin we et v R v qb R w v
.5 Cnservatin Theres Recallin Eq.() and assuin that the net frce is zer we et dp F 0 P cnstant (.9) I. The ttal linear entu f a particle is cnserved when the ttal frce n it is zer. Let s be se cnstant vectr such that F s F s P s 0 P s cnstant (.0) 0. Then The cpnent f linear entu in a directin in which the frce vanishes is cnstant in tie.
Definin the anular entu f a particle with respect t riin as L r P (.) The trque with respect t the sae riin is defined as N r F (.) Where r is the psitin vectr fr the riin t the pint where the frce acts. Nw substitutin fr F fr Eq.(.) int Eq.(.) we et N r P (.3) Nw fr Eq.(.) we have dr P L But r P r 0 L r P N r (.) r P r P If n trque actin n a particle then the anular entu is cnstant, i.e.,
II. The anular entu f a particle subject t n trque is cnserved. Definin the wrk dne n a particle b a frce in transfrin the particle fr pint t pint as With W F dr (.5) dv dr dv Nw F dr v d v v F dr d v W T v v T (.6) v T is the kinetic ener f the particle. If the wrk f a frce is independent n the path, such a frce is called cnservative. Fr ever cnservative frce we assciate a ptential ener accrdin t
W U F dr U U (.7) Fr the last equatin we cnclude that the frce can be written as F U (.8) T prve Eq.(.8) we have fr Eq.(.7) F dr U dr du U U Ptential ener has n abslute eanin; nl differences f ptential ener are phsicall eaninful. Nw definin the ttal ener as the su f kinetic and ptential eneries, i.e.,. E T U (.9) de dt du (.0)
But dt d v F dr dt F r (.) ) And du U t U t U U t U r U t i i i i i i (. Substitutin Eqs.(. &.) int Eq.(.0) we et Since F U de U F U r t F U r 0 de U t If U is nt an eplicit functin f tie then the ttal ener is cnstant, i.e., III. The ttal ener E f a particle is a cnservative field is cnserved.
Eaple. A use f ass jups n the utside ede f a freel turnin ceilin fan f rtatinal inertia I and radius R. B what rati des the anular velcit chane? Slutin Here the anular entu is cnserved befre and after the use's jupin. Recallin that the anular entu can be written as L I L i L f I I vr Knwin that v R I I R I R I I R
.6 Ener In tdas phsics, ener is re ppular than Newtn s laws. Mst f phsical prbles are slved b eans f ener. Cnsider a particle under the influence f a cnservative, -diensinal, frce. The ttal ener is written as t E T v t U d d v U (.3) E U (.) d E U (.5) If we knw U we can slve Eq.(.5) t et as a functin f tie. We can knw a lt abut the tin f a particle b eainin the plt f U(). Let cnsider the plt f the fllwin fiure. It is clear, fr Eq.(.9), and since T is alwas psitive E T U U (.6)
The tin is bunded fr E & E, i.e., it can't ve ff t. Fr E the tin is peridic between a & b, i.e., a b. Fr E the tin is peridic in tw pssible reins: between c d and e f. Fr E the particle is at rest since here E=U. Fr E 3 the particle ces fr, stps and turns at = and returns t. Here the tin is unbunded. Fr E the tin is unbunded and the particle a be at an psitin. The pint = is called an equilibriu pint. In eneral, the equilibriu state is characterized b du d 0 (.7)
The equilibriu is said t be stable if d U d 0 (.8) The equilibriu is said t be unstable if d U d 0 (.9) An equilibriu is cnsidered stable if the sste alwas returns t equilibriu after sall disturbances. If the sste ves awa fr the equilibriu after sall disturbances, then the equilibriu is unstable.
Eaple. Cnsider the sste f liht pulles, asses, and strin shwn. A liht strin f lenth b is attached at pint A, passes ver a pulle at pint B lcated a distance d awa, and finall attaches t ass. Anther pulle with ass attached passes ver the strin, pullin it dwn between A & B. Calculate the distance when the sste is in equilibriu, and deterine whether the equilibriu is stable r unstable. Slutin Let U=0 aln the line AB. U c But b b d c d U T deterine the equilibriu psitin we set du 0 b d b 0 d
b d b 6 b d b 6 d b 6 d b 0 d b Ntice that a real slutin eists nl when Nw t deterine wither the equilibriu is stable r n we have 3 6 d b b d b d U d Nw substitutin fr = we et 0 3 fr d d U d The equilibriu is stable fr real slutiin.
Eaple. Cnsider the ptential Sketch the ptential and discuss the tin at varius values f. Is the tin bunded r unbunded? Where are the equilibriu values? Are the stable r unstable? Find the turnin pints fr E=-W/8. W is a +ve cnstant. 8d d Wd U Slutin Rewrite the ptential as d with W U Z 8 Let us first find the equilibriu pints usin Eq.(.7) 0 8 8 3 d dz 0 8 8 8 8 3 0 8 0,
Usin d 0 d 3 d We have 3-equilibriu pints. Nw sketch Z() versus we et As it is clear fr the fiure, the equilibriu is stable at & 3 but unstable fr. The tin is bunded fr all eneries E<0. AT the turnin pints the speed is zer. S T=0, i.e. E U E U W 8 8 8 8 W 8 8 0 0,