Classical Mechanics Virtual Work & d Alembert s Principle

Size: px
Start display at page:

Download "Classical Mechanics Virtual Work & d Alembert s Principle"

Transcription

1 Classcal Mechancs Vrtual Work & d Alembert s Prncple Dpan Kumar Ghosh UM-DAE Centre for Excellence n Basc Scences Kalna, Mumba August 15, Constrants Moton of a system of partcles s often constraned, ether geometrcally or knematcally. Constrants reduce the number of degrees of freedom of a gven body. Consder the moton of a sngle partcle n space. For free, unconstraned moton, t has three degrees of freedom whch are usually expressed by three coordnates such as x, y, z or r, θ, ϕ etc. If, however, the partcle s constraned to move on the surface of a sphere, we must have (takng Cartesan coordnates), x 2 + y 2 + z 2 = R 2 whch reduces the number of degrees of freedom by one. Consder two masses connected by a rgd rod, lke a dumbbell. Two partcles have 6 degrees of freedom, Snce the dstance between the two bodes remans constant, we have the constrant (x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 ) 2 = d 2 whch reduces the degree of freedom to 5. These are examples of geometrc or holonomc constrants whoch are expressble as algebrac equatons nvolvng the coordnates. There are other constrants whch restrct the moton of bodes. Some of these are expressble as dfferental equatons whch constran the coordnates and components of veloctes. These are called knematc constrants. Non-ntegrable knematc constrants whch cannot be reduced to holonomc constrants are called non-holonomc constrants. Thus, f m s the dmenson of the confguraton space (.e., the number of generalsed coordnates), holonomc constrants are expressble as equatons of the form f (t, q 1, q 2,..., q m ) = 0, 1 k

2 c D. K. Ghosh, IIT Bombay 2 where k s the number of constrants. Holonomc constrants are called scleronomc f they do not explctly depend on tme. Tme dependent constrants are called rheonomc. Knematc constrants are expressed as equatons n the phase space f (t, q 1, q 2,..., q m ; q 1, q 2,..., q m ) = 0, 1 k Both the constrants are classfed as rheonomc f they explctly depend on tme. Sometmes a constrant may appear to be knematc but may be n realty holonomc. For nstance, a constrant of the type Aẋ + B = 0 may actually be holonomc f there exsts a functon f such that A = f/ x and B = f/ t. We then have, df = f df x + f t whch gves a holonomc constrant f = constant. Example 1: Consder two masses connected by pulleys, as shown. In general two partcles have 6 degrees of freedom. However, m 1 can only move along the x drecton and m 2 along the z drecton. Thus y 1 = z 1 = 0 and x 2 = y 2 = 0. We are left wth two degrees of freedom. However, f m 1 moves along the x drecton by a dstance d, m 2 would have to move along z drecton by 2d,.e., we get another holonomc constrant, z 2 2x 1 = 0 whch reduces the degree of freedom further by one. The problem s essentally a one dmensonal problem. m 1 P 1 z x P 2 m Non-holonomc constrant could be n the form of dfferental equatons or algebrac nequaltes. For nstance, a partcle constraned to move nsde a sphere of radus R satsfes x 2 + y 2 + z 2 < R 2. Consder a dsk rollng on a horzontal surface, on x-y plane.

3 c D. K. Ghosh, IIT Bombay 3 z y P φ θ x If the dsk s rollng wthout slppng, keepng ts plane vertcal, we need four coordnates to descrbe the poston of the dsk. These are the x and y coordnates of the centre of the dsk, the angle ϕ by whch a fxed pont on the rm of the dsk has rotated about the axs of rotaton and the angle θ that the axs of the dsk makes wth the x axs. If R s the radus of the dsk, the velocty of the dsk s gven by v = Rω = R ϕ (1) Snce the dsk remans vertcal, the components of the velocty of the centre of the dsk are gven by Usng eqs. (1) to (3) we get dx = v sn θ (2) dy = v cos θ (3) dx dy = R sn θ dϕ = R cos θ dϕ the mnus sgn s due to the sense of rotaton beng opposte to the postve angle. Combnng these two we get the followng par of dfferental equatons: dx = R sn θdϕ dy = R cos θdϕ These equatons cannot be further reduced and we cannot connect x, y, θ and ϕ by an algebrac equalty, showng that the constrant s non-holonomc.

4 c D. K. Ghosh, IIT Bombay 4 Tacklng problems wth non-holonomc constrants s more dffcult and no general prescrpton can be provded for ther soluton. Constrants ntroduce two new elements nto problem solvng. Snce the generalsed coordnates are no longer ndependent, the equatons of moton are not ndependent ether. Constrants arse from forces between elements whose nature s unknown. These forces are known only by the effect they have on moton of the system. 2 Vrtual Dsplacement A real dsplacement of partcles consttutng a system happens over a fnte tme. Such a dsplacement of, say, the -th partcle s, n general, functon of all the generalsed coordnates as well as of tme. If the poston of the -th partcle s wrtten as r = r (q 1, q 2,..., q s ) The total dfferental of the poston vector s then wrtten as d r = r dq j + r (4) q j t j=1 A vrtual or an magned dsplacement s nstantaneous and s consstent wth the constrants on the system. The dsplacements, whether real or vrtual, result n admssble geometrcal confguraton of the system. In case of a vrtual dsplacement, we have, d r = j=1 r q j δq j (5) where we have use δq to ndcate a vrtual dsplacement whle reservng dq for real dsplacement. For a real dsplacement, the forces of constrants may change. The velocty s gven by so that we have, ṙ = d r = ṙ q k = j=1 r q j + r q j t (Note the structure of the above equaton - as f the dots cancel!) j=1 (6) r q j δ jk = r q k (7) 2.1 Prncple of Vrtual Work Consder a system of N partcles under tme dependent holonomc constrants. If q 1,..., q s be a set of generalzed coordnates, the vrtual dsplacement of the -th partcle s gven

5 c D. K. Ghosh, IIT Bombay 5 by eqn. (5). Suppose the system, under the acton of appled forces as well as those of constrants, s n equlbrum. The total force actng on each partcle s then zero, F = 0; ( = 1,..., N) We then have, for the vrtual work done by F n the dsplacement δ r s so that the total work done s F δ r = 0 δw = F δ r = 0 The total force actng on any partcle can be splt nto two: an appled part and a part due to the constrants, F = F a + F c then we have δw = F a δ r + F c δ r = 0 (8) The forces of constrants (e.g. normal reacton, tenson, rgd body constrants etc.) do not do any work. Ths s general true of scleronomc holonomc constrants and ths statement s central hypothess n the prncple of vrtual work. Two examples llustrate the hypothess. Consder two types of dsplacements consstent wth the constrants on a rgd body. A dsplacement along the lne jonng two partcles does not do any net work because the reactons are equal and opposte. In order to be consstent wth rgd body constrants, for a par of partcles j and k, we must have δr k = δr j. The work done s f kj δr j + f jk δr k = (f kj + f jk ) δr j = 0 where we have used δr j = δr k. Thus there s no work done. The second type of dsplacement s along the arc of a crcle normal to the lne jonng the partcles.as the forces of constrants are normal to the drecton of dsplacement, the work done s once agan zero. Consder the pulley arrangement n Example 1. When m 1 moves by an amount δx to the rght, m 2 moves by 2δx downwards n order to keep the sum of the lengths of the two ropes constant. The only appled forces are the gravty and frcton. Thus by the prncple of vrtual work, we have, µmgδx + mg2δx = 0 whch shows that for statc equlbrum m 2 = µm 1 /2. Example 2: Two frctonless blocks of mass m each are connected by a massless rgd rod. The system s constraned to move n the vertcal plane.

6 c D. K. Ghosh, IIT Bombay 6 R mg δx 1 mg θ N δx 2 F If the block on the vertcal track undergoes a vrtual dsplacement δx 1 and that on the horzontal plane has a vrtual dsplacement δx 2, we have δx 1 sn θ = δx 2 cos θ whch s the constrant whch keeps the rod length constant. The gravty does work on the mass on the vertcal track whle the appled force F 2 s responsble for work on the horzontally movng block, mgδx 1 + F 2 δx 2 = 0. Thus we have F 2 = mg δx 1 δx 2 = mg cot θ Example 3: Consder an Atwood s machne n equlbrum. In ths case we have the constrant y 1 + y 2 = l = constant m m 2 1 Thus we have δy 1 = δy 2. The appled forces are m 1 g and m 2 g, both actng downwards. The vrtual work done s δw = m 1 gδy 1 + m 2 gδy 2 = (m 1 m 2 )gδy 1 = 0, whch gves the condton for equlbrum to be m 1 = m Generalzed Force We recall that n the presence of holonomc constrants, all the components of generalsed coordnates can be made ndependent. If the system has s degrees of freedom (s 3N), the poston vector of the -th partcle ( = 1, N) can be wrtten as

7 c D. K. Ghosh, IIT Bombay 7 r = r (q 1, q 2,..., q s ) The velocty of the -th partcle s gven by ṙ = d r = j=1 r q j + r q j t { q } are known as the generalsed veloctes. (If the constrants are scleronomc, we would have r t = 0. ) Let us return to the expresson for vrtual work and express t n terms of the generalsed coordnates. (9) δw = F a δ r = F a r δq α q α=1 α Q α δq α (10) α=1 where we have defneed a generalsed force correspondng to the coordnate q α by the relaton Q α = F a r q α (11) Note that the generalsed force depends only on appled forces and not on forces of constrants. In equlbrum, we have δw = α Q α δq α = 0 Snce the generalsed coordnates are ndependent, we have Q α = 0. However, ths does not apply that appled forces vansh. The condton for vanshng of F a δ r = 0 s applcable only to statc stuatons. d Alembert extended ths to nclude general moton of the system. 3 d Alembert s Prncple d Alembert s prncple, developed from an dea orgnally due to Bernoull, s to use the fact that accordng to Newton s law, the force appled on a partcle results n a rate of change of ts momentum, F = ṗ

8 c D. K. Ghosh, IIT Bombay 8 ṗ s known as the nertal force or pseudo force actng on the partcle. One can then thnk of brngng the body to equlbrum by applyng a pseudo force ṗ on the -th partcle of the system (1 N), F ṗ = 0 Note that F contans both appled and constrant forces actng on the -th partcle. Once agan, we can splt F nto two parts and wrte the above equaton for vrtual dsplacements as ( F a + F c ṗ ) δ r = 0 Snce the force of constrants do not do any work, we get ( F a ṗ ) δ r = 0 (12) Equaton (12) s the statement of d Alembert s prncple. It may be noted that the only force appearng n ths equaton s the appled force. Further, F a refers to the force on the -th partcle and the sum n (12) s over all partcles and not over the ndependent generalsed coordnates. Consequently, (12) does not mply F a ṗ = 0. An nterestng consequence follows f the dsplacement n (12) happens to be real dsplacement nstead of vrtual ones. In that case the dsplacement can be wrtten as d r = r If the force s conservatve and s dervable from a potental V,.e. F = V, can rewrte (12) as follows: ( F a ṗ )d r = [ V m r ] ṙ = [ V d r d (1 2 mṙ2 )] = d( T + V ) = Example 4: Consder a mass restng on a frctonless nclne. The mass would slde down wth an acceleraton when released. A horzontal acceleraton s appled on the mass to keep the mass from sldng. We need to fnd the acceleraton. The problem s

9 c D. K. Ghosh, IIT Bombay 9 reasonably smple and s readly solved by Newton s laws. N 90 θ a N = mg cos θ + ma sn θ mg ma cos θ = mg sn θ θ whch gves a = g tan θ Let us look at the problem from d Alembert s prncple. Suppose the mass has an nstantaneous (vrtual) dsplacement δl along the nclne. We then have δx = δl cos θ and δy = δl sn θ. The only appled force s mg along the y axs: F = mgŷ. From the prncple of vrtual work t follows that F x δx + F y δy ma x δx ma y δy = 0 We only apply a horzontal acceleraton so that we have, a y = 0. Snce F x = 0, we get mgδy ma x δx = 0.e., mgδl sn θ maδl cos θ = 0. Thus we get, a = g tan θ. Example 5: Consder the arrangement shown n the fgure. The pulley s fxed on the fxed wedge. Fnd the acceleraton of the masses when released. l l 1 2 m 1 m 2 α m g m g 2 1 β From d Alembert s prncple we have Snce l 1 + l 2 = constant, we have ( F 1 ṗ 1 ) δ l 1 + ( F 2 ṗ 2 ) δ l 2 = 0 (13) δl 1 = δl 2 l1 = l 2 (14a) (14b) The nertal forces are ṗ 1 = m 1 l1 and ṗ 2 = m 2 l2 = m 2 l1 and the only appled forces are the weght of the masses. Takng the components of (13) along the nclne, we have, Usng (14a) and (14b) we get, (m 1 g sn α m 1 l1 )δl 1 + (m 2 g sn β m 2 l2 )δl 2 = 0 (m 1 g sn α m 1 l1 m 2 g sn β m 2 l1 )δl 1 = 0

10 c D. K. Ghosh, IIT Bombay 10 so that l1 = m 1g sn α m 2 g sn β m 1 + m 2 4 Euler- Lagrange Equaton We wll now derve the Euler-Lagrange equaton from d Alembert s prncple. Recall eqn. (7) where we showed ṙ r = δ jk = r (15) q k q j q k j=1 we also had, from (6), for scleronomc constrants, ṙ = j=1 r q j q j (16) Usng chan rule dfferentaton, we can wrte, d ( ) r = q j 2 r dq k q j q k ( r q k q k k=1 = q j k=1 ) = q j ṙ (17) where n the last lne we have used (16) and have used the fact that q s ndependent of q. Let us return to d alembert s prncple k=1 N ( F ṗ ) δ r = 0 The nertal force term may be smplfed as follows: ( ) m r δ r = m r r δq k q k=1 k [ ( ) d = m ṙ r q k=1 k [ ( ) d = m ṙ ṙ q k =1 ( d r m ṙ q k ] m ṙ ṙ q k where n the last lne we have used the dot cancelaton relatonshp (17). ) ] δq k δq k (18)

11 c D. K. Ghosh, IIT Bombay 11 The rght hand sde of the above expresson can be smplfed and the equaton can be wrtten as follows: [ ( ) ( )] d 1 m r δ r = q k 2 m ṙ 2 1 q k=1 k 2 m ṙ 2 δq k [ ( ) d T = T ] δq k (19) q k q k =1 k=1 where T s the knetc energy of the system of partcles. Substtutng (19) n d Alembert s equaton, we have N ( ( ) d T F a δ r T ) δq k = 0 (20) q k q k In terms of generalzed coordnates, we could wrte the frst term as s k=1 Q kδq k. Thus we have, [ ( ( ) d T Q k T )] δq k = 0 (21) q k q k k=1 Snce the generalzed coordnates are ndependent, we may vary each coordnate ndependently and get ( ) d T T = Q k (22) q k q k Equaton (22) s the form of Euler-Lagrange equaton whch s derved from d Alembert s prncple. If, however, the external forces actng on the system are conservatve,.e., f we can express F = V, where mples gradent taken wth respect to the coordnates of the -th partcle, we have N Q k = F r q =1 k N = V r q k =1 = V q k (23) If the potental s a functon only of the poston q k, we have V = 0 whch enables us q k to wrte the generalzed force as Q k = V + d ( ) V (24) q k q k On brng ths expresson for the generalzed force to the left hand sde of (23) and recognzng that L = T V, we recover the Euler-Lagrange equaton d ( ) L q k L q k = 0 (25)

12 c D. K. Ghosh, IIT Bombay 12 5 Velocty Dependent Potental - Lorentz Force The expresson for the generalzed force n the expresson (24) can be used to defne a potental functon for velocty dependent forces as well. Only n such a stuaton, the second term on the rght hand sde of (24) s not dentcally zero but would depend on the exact dependence of the force on velocty. Thus n such stuatons, we could derve Euler-Lagrange equaton form an approprately defned Lagrangan functon. It s an mportant generalsaton for us because Lorentz force whch a charge partcle experences n an electromagnetc feld s a velocty dependent force. The force on an electrc charge q (not to be confused wth generalzed coordnate whch s also usually wrtten by the same notaton), s gven by F = q( E + v B) (26) Usng the expressons for the electrc and the magnetc felds n terms of the scalar and the vector potentals ϕ and A respectvely, we have E = ϕ A t B = A we can wrte the expresson for the force as [ F = q ϕ A t + v ( A) ] (27) The scalar potental depends only on poston and we wll need to deal only wth the remanng two terms. Snce v does not depend on poston, we have the followng dentty, Proof of (28) v ( A) = ( v A) ( v ) A (28) Proof. To prove the dentty (28), we consder the x-component of ts left hand sde. [ v ( A)] x = v y ( A) z v z ( A) y ( = v y x A y ) ( y A x v z z A x ) x A z ( ) ( ) A y = v y x + v A z A x z v y x y + v A x z z ( ) ( A x = v x x + v A y y x + v A z A x z v x x x + v y where n the last lne we have added and subtracted v x A x x. A x y + v z ) A x z

13 c D. K. Ghosh, IIT Bombay 13 Snce v or ts components do not depend on poston coordnates, we can wrte the frst term as x ( v A) and the second term as ( v )A x. Addng three components of v ( A) then yelds the dentty (28). We wll now rewrte the second term of (28) usng a smart trck. Note that the total dervatve of any component of A can be wrtten as follows: da x = A x x dx + A x y = v x A x x + v y = ( v )A x + A x t Combnng three components, we can wrte, dy + A x x A x y + v z dz + A x t A x x + A x t (29) da = ( v ) A + A t Usng (28) and substtutng the expresson for A from (30) nto (27), we get t [ F = q (ϕ v A) d A ] (30) (31) Now, we can wrte usng the fact that A does not depend on velocty components da x = d ( ) (A x v x ) v x = d [ ] (A x v x + A y v y + A z v z ) v x = d ( ) ( v v A) x = d ( ) ( v v A ϕ) x where, n the last step, we have added a term ϕ/ v x whch s zero because the scalar potental does not depend on the velocty ether. Thus da = d ( v ( v A ) ϕ) where v = î + ĵ + v x v ˆk s gradent wth respect to the velocty vector. Substtutng ths n the expresson (31), we get y v z [ F = q (ϕ v A) + d ( A)) ] v (ϕ v (34) (32) (33)

14 c D. K. Ghosh, IIT Bombay 14 Thus f we defne a velocty dependent potental as U = ϕ v A (35) the component of Lorentz force would be gven by F j = U + d ( ) U q j q j (36) whch s of the form (24). Wth ths modfcaton, the Euler-Lagrange equaton s stll vald for velocty dependent potental for whch the Lagrangan s gven by. L = T U

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1 P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the

More information

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

Physics 5153 Classical Mechanics. Principle of Virtual Work-1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal

More information

CHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)

CHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics) CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O

More information

Lagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013

Lagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013 Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned

More information

Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints

Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or

More information

coordinates. Then, the position vectors are described by

coordinates. Then, the position vectors are described by Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,

More information

PHYS 705: Classical Mechanics. Calculus of Variations II

PHYS 705: Classical Mechanics. Calculus of Variations II 1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary

More information

PHYS 705: Classical Mechanics. Newtonian Mechanics

PHYS 705: Classical Mechanics. Newtonian Mechanics 1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

First Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.

First Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force. Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act

More information

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force.

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force. The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,

More information

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is. Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and

More information

Physics 181. Particle Systems

Physics 181. Particle Systems Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system

More information

Study Guide For Exam Two

Study Guide For Exam Two Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 01-06 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force

More information

Part C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis

Part C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta

More information

Moving coordinate system

Moving coordinate system Chapter Movng coordnate system Introducton Movng coordnate systems are mportant because, no materal body s at absolute rest As we know, even galaxes are not statonary Therefore, a coordnate frame at absolute

More information

Spin-rotation coupling of the angularly accelerated rigid body

Spin-rotation coupling of the angularly accelerated rigid body Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s

More information

CHAPTER 10 ROTATIONAL MOTION

CHAPTER 10 ROTATIONAL MOTION CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The

More information

11. Dynamics in Rotating Frames of Reference

11. Dynamics in Rotating Frames of Reference Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons

More information

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding. Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays

More information

Quantum Mechanics I Problem set No.1

Quantum Mechanics I Problem set No.1 Quantum Mechancs I Problem set No.1 Septembe0, 2017 1 The Least Acton Prncple The acton reads S = d t L(q, q) (1) accordng to the least (extremal) acton prncple, the varaton of acton s zero 0 = δs = t

More information

A Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph

A Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular

More information

Physics 207: Lecture 20. Today s Agenda Homework for Monday

Physics 207: Lecture 20. Today s Agenda Homework for Monday Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems

More information

So far: simple (planar) geometries

So far: simple (planar) geometries Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector

More information

Lecture 20: Noether s Theorem

Lecture 20: Noether s Theorem Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

Notes on Analytical Dynamics

Notes on Analytical Dynamics Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame

More information

The Feynman path integral

The Feynman path integral The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space

More information

Canonical transformations

Canonical transformations Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,

More information

Newton s Laws of Motion

Newton s Laws of Motion Chapter 4 Newton s Laws of Moton 4.1 Forces and Interactons Fundamental forces. There are four types of fundamental forces: electromagnetc, weak, strong and gravtatonal. The frst two had been successfully

More information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look

More information

Modeling of Dynamic Systems

Modeling of Dynamic Systems Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how

More information

Chapter 8. Potential Energy and Conservation of Energy

Chapter 8. Potential Energy and Conservation of Energy Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and non-conservatve forces Mechancal Energy Conservaton of Mechancal

More information

In this section is given an overview of the common elasticity models.

In this section is given an overview of the common elasticity models. Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp

More information

Chapter 11 Angular Momentum

Chapter 11 Angular Momentum Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle

More information

LAGRANGIAN MECHANICS

LAGRANGIAN MECHANICS LAGRANGIAN MECHANICS Generalzed Coordnates State of system of N partcles (Newtonan vew): PE, KE, Momentum, L calculated from m, r, ṙ Subscrpt covers: 1) partcles N 2) dmensons 2, 3, etc. PE U r = U x 1,

More information

12. The Hamilton-Jacobi Equation Michael Fowler

12. The Hamilton-Jacobi Equation Michael Fowler 1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and

More information

Chapter 11: Angular Momentum

Chapter 11: Angular Momentum Chapter 11: ngular Momentum Statc Equlbrum In Chap. 4 we studed the equlbrum of pontobjects (mass m) wth the applcaton of Newton s aws F 0 F x y, 0 Therefore, no lnear (translatonal) acceleraton, a0 For

More information

Poisson brackets and canonical transformations

Poisson brackets and canonical transformations rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order

More information

Mathematical Preparations

Mathematical Preparations 1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the

More information

1 Matrix representations of canonical matrices

1 Matrix representations of canonical matrices 1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

More information

Physics 111: Mechanics Lecture 11

Physics 111: Mechanics Lecture 11 Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 Rgd-Body Rotaton

More information

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the

More information

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 NMT EE 589 & UNM ME 48/58 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 48/58 7. Robot Dynamcs 7.5 The Equatons of Moton Gven that we wsh to fnd the path q(t (n jont space) whch mnmzes the energy

More information

ENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15

ENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15 NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound

More information

Iterative General Dynamic Model for Serial-Link Manipulators

Iterative General Dynamic Model for Serial-Link Manipulators EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for Seral-Lnk Manpulators In ths set of notes, we are gong to develop a method for computng a general

More information

ˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m)

ˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m) 7.. = = 3 = 4 = 5. The electrc feld s constant everywhere between the plates. Ths s ndcated by the electrc feld vectors, whch are all the same length and n the same drecton. 7.5. Model: The dstances to

More information

Physics 207 Lecture 13. Lecture 13

Physics 207 Lecture 13. Lecture 13 Physcs 07 Lecture 3 Goals: Lecture 3 Chapter 0 Understand the relatonshp between moton and energy Defne Potental Energy n a Hooke s Law sprng Develop and explot conservaton of energy prncple n problem

More information

Physics 207 Lecture 6

Physics 207 Lecture 6 Physcs 207 Lecture 6 Agenda: Physcs 207, Lecture 6, Sept. 25 Chapter 4 Frames of reference Chapter 5 ewton s Law Mass Inerta s (contact and non-contact) Frcton (a external force that opposes moton) Free

More information

Linear Momentum. Center of Mass.

Linear Momentum. Center of Mass. Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html

More information

Module 1 : The equation of continuity. Lecture 1: Equation of Continuity

Module 1 : The equation of continuity. Lecture 1: Equation of Continuity 1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum

More information

Rotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa

Rotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa Rotatonal Dynamcs Physcs 1425 Lecture 19 Mchael Fowler, UVa Rotatonal Dynamcs Newton s Frst Law: a rotatng body wll contnue to rotate at constant angular velocty as long as there s no torque actng on t.

More information

COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD

COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,

More information

Lesson 5: Kinematics and Dynamics of Particles

Lesson 5: Kinematics and Dynamics of Particles Lesson 5: Knematcs and Dynamcs of Partcles hs set of notes descrbes the basc methodology for formulatng the knematc and knetc equatons for multbody dynamcs. In order to concentrate on the methodology and

More information

Physics 2A Chapters 6 - Work & Energy Fall 2017

Physics 2A Chapters 6 - Work & Energy Fall 2017 Physcs A Chapters 6 - Work & Energy Fall 017 These notes are eght pages. A quck summary: The work-energy theorem s a combnaton o Chap and Chap 4 equatons. Work s dened as the product o the orce actng on

More information

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2 Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to

More information

Celestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg

Celestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg PHY 454 - celestal-mechancs - J. Hedberg - 207 Celestal Mechancs. Basc Orbts. Why crcles? 2. Tycho Brahe 3. Kepler 4. 3 laws of orbtng bodes 2. Newtonan Mechancs 3. Newton's Laws. Law of Gravtaton 2. The

More information

Integrals and Invariants of Euler-Lagrange Equations

Integrals and Invariants of Euler-Lagrange Equations Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,

More information

Mechanics Cycle 3 Chapter 9++ Chapter 9++

Mechanics Cycle 3 Chapter 9++ Chapter 9++ Chapter 9++ More on Knetc Energy and Potental Energy BACK TO THE FUTURE I++ More Predctons wth Energy Conservaton Revst: Knetc energy for rotaton Potental energy M total g y CM for a body n constant gravty

More information

EN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics

EN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics N40: ynamcs and Vbratons Homewor 7: Rgd Body Knematcs School of ngneerng Brown Unversty 1. In the fgure below, bar AB rotates counterclocwse at 4 rad/s. What are the angular veloctes of bars BC and C?.

More information

Chapter 3. r r. Position, Velocity, and Acceleration Revisited

Chapter 3. r r. Position, Velocity, and Acceleration Revisited Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector

More information

THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions

THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George

More information

How Differential Equations Arise. Newton s Second Law of Motion

How Differential Equations Arise. Newton s Second Law of Motion page 1 CHAPTER 1 Frst-Order Dfferental Equatons Among all of the mathematcal dscplnes the theory of dfferental equatons s the most mportant. It furnshes the explanaton of all those elementary manfestatons

More information

Chapter 20 Rigid Body: Translation and Rotational Motion Kinematics for Fixed Axis Rotation

Chapter 20 Rigid Body: Translation and Rotational Motion Kinematics for Fixed Axis Rotation Chapter 20 Rgd Body: Translaton and Rotatonal Moton Knematcs for Fxed Axs Rotaton 20.1 Introducton... 1 20.2 Constraned Moton: Translaton and Rotaton... 1 20.2.1 Rollng wthout slppng... 5 Example 20.1

More information

PHYS 705: Classical Mechanics. Canonical Transformation II

PHYS 705: Classical Mechanics. Canonical Transformation II 1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m

More information

The classical spin-rotation coupling

The classical spin-rotation coupling LOUAI H. ELZEIN 2018 All Rghts Reserved The classcal spn-rotaton couplng Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 louaelzen@gmal.com Abstract Ths paper s prepared to show that a rgd

More information

Week 9 Chapter 10 Section 1-5

Week 9 Chapter 10 Section 1-5 Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,

More information

Conservation of Energy

Conservation of Energy Lecture 3 Chapter 8 Physcs I 0.3.03 Conservaton o Energy Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcsall.html 95.4, Fall 03,

More information

You will analyze the motion of the block at different moments using the law of conservation of energy.

You will analyze the motion of the block at different moments using the law of conservation of energy. Physcs 00A Homework 7 Chapter 8 Where s the Energy? In ths problem, we wll consder the ollowng stuaton as depcted n the dagram: A block o mass m sldes at a speed v along a horzontal smooth table. It next

More information

Spring 2002 Lecture #13

Spring 2002 Lecture #13 44-50 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallel-as Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the md-term

More information

Lagrangian Field Theory

Lagrangian Field Theory Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,

More information

Professor Terje Haukaas University of British Columbia, Vancouver The Q4 Element

Professor Terje Haukaas University of British Columbia, Vancouver  The Q4 Element Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to

More information

Important Dates: Post Test: Dec during recitations. If you have taken the post test, don t come to recitation!

Important Dates: Post Test: Dec during recitations. If you have taken the post test, don t come to recitation! Important Dates: Post Test: Dec. 8 0 durng rectatons. If you have taken the post test, don t come to rectaton! Post Test Make-Up Sessons n ARC 03: Sat Dec. 6, 0 AM noon, and Sun Dec. 7, 8 PM 0 PM. Post

More information

Calculus of Variations Basics

Calculus of Variations Basics Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y

More information

Spring Force and Power

Spring Force and Power Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi IN THIS CHAPTER, you wll learn how to solve problems

More information

Indeterminate pin-jointed frames (trusses)

Indeterminate pin-jointed frames (trusses) Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all

More information

A Review of Analytical Mechanics

A Review of Analytical Mechanics Chapter 1 A Revew of Analytcal Mechancs 1.1 Introducton These lecture notes cover the thrd course n Classcal Mechancs, taught at MIT snce the Fall of 01 by Professor Stewart to advanced undergraduates

More information

Translational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f.

Translational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f. Lesson 12: Equatons o Moton Newton s Laws Frst Law: A artcle remans at rest or contnues to move n a straght lne wth constant seed there s no orce actng on t Second Law: The acceleraton o a artcle s roortonal

More information

Angular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004

Angular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004 Angular Momentum and Fxed Axs Rotaton 8.01t Nov 10, 2004 Dynamcs: Translatonal and Rotatonal Moton Translatonal Dynamcs Total Force Torque Angular Momentum about Dynamcs of Rotaton F ext Momentum of a

More information

Rigid body simulation

Rigid body simulation Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum

More information

Classical Mechanics ( Particles and Biparticles )

Classical Mechanics ( Particles and Biparticles ) Classcal Mechancs ( Partcles and Bpartcles ) Alejandro A. Torassa Creatve Commons Attrbuton 3.0 Lcense (0) Buenos Ares, Argentna atorassa@gmal.com Abstract Ths paper consders the exstence of bpartcles

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

PHYS 1443 Section 003 Lecture #17

PHYS 1443 Section 003 Lecture #17 PHYS 144 Secton 00 ecture #17 Wednesda, Oct. 9, 00 1. Rollng oton of a Rgd od. Torque. oment of Inerta 4. Rotatonal Knetc Energ 5. Torque and Vector Products Remember the nd term eam (ch 6 11), onda, Nov.!

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Chapter Seven - Potential Energy and Conservation of Energy

Chapter Seven - Potential Energy and Conservation of Energy Chapter Seven - Potental Energy and Conservaton o Energy 7 1 Potental Energy Potental energy. e wll nd that the potental energy o a system can only be assocated wth specc types o orces actng between members

More information

Physics 2A Chapter 3 HW Solutions

Physics 2A Chapter 3 HW Solutions Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C

More information

Classical Field Theory

Classical Field Theory Classcal Feld Theory Before we embark on quantzng an nteractng theory, we wll take a dverson nto classcal feld theory and classcal perturbaton theory and see how far we can get. The reader s expected to

More information

PY2101 Classical Mechanics Dr. Síle Nic Chormaic, Room 215 D Kane Bldg

PY2101 Classical Mechanics Dr. Síle Nic Chormaic, Room 215 D Kane Bldg PY2101 Classcal Mechancs Dr. Síle Nc Chormac, Room 215 D Kane Bldg s.ncchormac@ucc.e Lectures stll some ssues to resolve. Slots shared between PY2101 and PY2104. Hope to have t fnalsed by tomorrow. Mondays

More information

Three views of mechanics

Three views of mechanics Three vews of mechancs John Hubbard, n L. Gross s course February 1, 211 1 Introducton A mechancal system s manfold wth a Remannan metrc K : T M R called knetc energy and a functon V : M R called potental

More information

EPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski

EPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on

More information

2D Motion of Rigid Bodies: Falling Stick Example, Work-Energy Principle

2D Motion of Rigid Bodies: Falling Stick Example, Work-Energy Principle Example: Fallng Stck 1.003J/1.053J Dynamcs and Control I, Sprng 007 Professor Thomas Peacock 3/1/007 ecture 10 D Moton of Rgd Bodes: Fallng Stck Example, Work-Energy Prncple Example: Fallng Stck Fgure

More information

χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body

χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown

More information

Kernel Methods and SVMs Extension

Kernel Methods and SVMs Extension Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general

More information

Comparative Studies of Law of Conservation of Energy. and Law Clusters of Conservation of Generalized Energy

Comparative Studies of Law of Conservation of Energy. and Law Clusters of Conservation of Generalized Energy Comparatve Studes of Law of Conservaton of Energy and Law Clusters of Conservaton of Generalzed Energy No.3 of Comparatve Physcs Seres Papers Fu Yuhua (CNOOC Research Insttute, E-mal:fuyh1945@sna.com)

More information

Chapter Eight. Review and Summary. Two methods in solid mechanics ---- vectorial methods and energy methods or variational methods

Chapter Eight. Review and Summary. Two methods in solid mechanics ---- vectorial methods and energy methods or variational methods Chapter Eght Energy Method 8. Introducton 8. Stran energy expressons 8.3 Prncpal of statonary potental energy; several degrees of freedom ------ Castglano s frst theorem ---- Examples 8.4 Prncpal of statonary

More information

One Dimensional Axial Deformations

One Dimensional Axial Deformations One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the

More information

EMU Physics Department

EMU Physics Department Physcs 0 Lecture 8 Potental Energy and Conservaton Assst. Pro. Dr. Al ÖVGÜN EMU Physcs Department www.aovgun.com Denton o Work W q The work, W, done by a constant orce on an object s dened as the product

More information