SpaceTime Structure, Newton s Laws and Galilean Transformation


 Christopher Todd
 10 months ago
 Views:
Transcription
1 Chapte 1 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation What is motion? The phase change of position with time expesses the basic idea of motion. We say that an object is moving if it occupies diffeent positions in space at diffeent times. In contast, if an object does not change its position in space as time flows, we say that the object is at est. Inheent in this desciption of est and motion is the obseve (that is we ) and the concepts of a sepaate space and time. Ou peception tells us that space exists as a stationay, pemanent and absolute backgound in which we can place an object, o though which an object can move without any inteaction with it. Newton fomulated this idea by asseting, Absolute space, in its own natue, without elation to anything extenal, emains always simila and immovable. Time, accoding to Newton, is also absolute and flows on without elation to the pesence of any physical object o event. In his wods, Absolute, tue, and mathematical time, of itself, and fom its own natue, flows equably without elation to anything extenal, and by anothe name is called duation. In this Newtonian absolute space and time, motion of an object is chaacteized by change of its position in space as time evolves. Howeve, since this abstact absolute space cannot be seen and/o no pat of it can be distinguished fom anothe, how do we fix the position of an object in this space? Newton egaded the distant stas in the sky at est and defined the positions of othe objects in elation to them. Howeve, astonomes now tell us that none of these cosmological objects in the Univese ae at est. Thee is no object at est in this absolute space with espect to which we can define the position of anothe object. Theefoe, at any instant of time, the position of an object is always defined by the obseve with espect to himself. An obseve, theefoe, fist defines a local space, in which positions of othe objects ae then descibed. This local space, to which is attached the obseve, is called a fame of efeence. The concepts of est and motion ae theefoe elative, that is, these ae in elation to a fame of efeence o obseve. A house built on Eath does not change its position in the local space o fame of efeence associated with Eath; howeve, as obseved fom a fame of efeence attached to Moon fo example (o in othe wods, in the local space associated with Moon), its (house on Eath) position continuously changes. Thus, even within the Newtonian point of view, space does not exist on its own but is a elative space. Time, on
2 Mechanics the othe hand, is same and flows equably eveywhee, i.e., fo example its ate of flow is same both on Eath and Moon. 1.1 FRAMES OF REFERENCE AND COORDINATE SYSTEMS A fame of efeence is usually a collection of objects, togethe with the obseve, which ae at est elative to each othe*. Thus, fo example, we at est elative to Eath along with Eath and laboatoies built on Eath, etc., constitute a fame of efeence. It is with espect to this Eath s fame of efeence that we obseve and measue the changes of positions of othe objects. Simila fames of efeence can be associated with othe obseves (o objects) who may not be at est elative to us. Within a fame of efeence, we set up a coodinate system which is used to measue the position of an object. Measuements indicate that the space we inhabit is a theedimensional space. This means that we need thee numbes to specify the position of any (point) object in the space. The choice of numbes is detemined by the type of coodinate system that we use. The geneally used coodinate systems ae ectangula (Catesian) coodinates (x, y, z), spheicalpola coodinates (, θ, ), and cylindical coodinates (ρ,, z). A coodinate system may be set up anywhee in the fame of efeence, the choice of oigin has complete feedom. Given the oigin, a Catesian (ectangula) system is fomed by having thee mutually pependicula staight lines, the X, Y, and Z axes. Thee ae two independent ectangula systems possible in space, the lefthanded and the ighthanded, which cannot be made to coincide with each othe by any means of tanslation o otation. Catesian system, by convention, is chosen to be ighthanded, as shown in Fig The position of a paticle** P elative to oigin O is given by position vecto, chaacteized by its specific length and diection: = x i+ y j+ z k...(1.1) X Z O i k P (x, y, z) j Y whee i, j, kdenote constant unit vectos in the diections of X, Y, Z axes. We talk of a point, staight line, pependicula staight lines, length of line segment, etc. in the Euclidean sense. Ou expeience tells us that Euclidean geomety is applicable in ou odinay Newtonian space. This is valid if the objects ae not vey massive, o ae not moving with vey high speeds of the ode of speed of light. If the position of the paticle P changes, the diffeential change in its position vecto is, in geneal, given by, Fig. 1.1: Catesian coodinates. d = dx i+ dy j+ dz k...(1.) * The collection of such point objects constitute the concept of a igid body. ** We define paticle as a mass concentated at a point. The paticle is theefoe a point object.
3 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 3 The instantaneous velocity and acceleation of P, in Catesian system, theefoe ae d dx dy dz v = = i+ y+ k dt dt dt dt = v i+ v j+ v k...(1.3) x y z and dv dv dv x y dvz a = = i+ j+ k dt dt dt dt = a i+ a j+ a k...(1.4) x y z SpheicalPola Coodinates The position of paticle P elative to oigin O, in spheical pola coodinate system, is descibed by adial distance, pola angle θ, and azimuth, as shown in Fig. 1.a. Geomety of the figue shows the elations between Catesian and spheical pola coodinates: x = sin θcos = x + y + z...(1.5a) y = sinθsin tanθ = x + y z...(1.5b) z = cosθ tan = y x...(1.5c) Z Z S cos θ = z P (, θ, ) P e P e e θ θ θ X x O sin θ y Y X T Y (a) (b) Fig. 1.: Spheical coodinates. The unit vecto e points towads axis, i.e., in the diection of vecto along which only coodinate changes, θ and emain fixed. Hence, we can wite = e. Unit vecto e θ is tangent at P to cicle SPT; displacement along cicle SPT changes only coodinate θ, distance and angle emain fixed. Unit vecto e is tangent at P to cicle PP poduced
4 4 Mechanics by the otation of OP about Zaxis; displacement along this cicle changes only coodinate, distance OP = and angle θ emain fixed; see Fig. 1. (b). The geneal diffeential displacement of paticle P in spheical pola coodinates is given by, d = de + dθe + sin θde...(1.6) θ One can satisfy himself that the above elation is tue by consideing special cases: (i) if dθ = d = 0, d = d e gives displacement vecto when distance changes by d along line OP; similaly, (ii) if d = d = 0, and angle θ changes by dθ, displacement vecto d = ( dθ e θ ) and (iii) if d = dθ = 0, change of angle by d poduces a displacement d = (ρ d ) e, whee ρ = sin θ is the adius of the cicle as we otate P about Zaxis. The unit vectos e, e θ and e can be expessed in tems of i, j, k as follows (see Example 1.1 on p.6): e = sinθ cos i+ sinθsin j+ cosθ k...(1.7a) eθ = cosθ cos i+ cosθsin j sin θ k...(1.7b) e = sin i+ cos j...(1.7c) The unit vectos (e, e θ, e ), unlike (i, j, k), ae not constant vectos but change in diection as coodinates θ and change. Howeve, at each point, they constitute an othogonal ighthanded coodinate system, that is, we have e e = e e = eθ e = 0...(1.8a) e e = e, e e = e, e e = e...(1.8b) θ θ θ 1.1. Motion in Dimensions Quite often, we find a paticle constained to move in a plane, athe than in space in geneal. If we call this plane as XY plane, then position vecto of the paticle is given by, = x i + y j because z = 0 always fo the paticle duing motion. The spheical pola coodinates (, θ, ) then educe to cicula pola coodinates (, ); θ = 90 fo the paticle. That is, we define the position of the paticle in tems of adial distance fom oigin and angle such that, Y e e P (, ) O Fig. 1.3: Cicula pola coodinates. X = x + y x = cos...(1.9a) tan =y x y = sin...(1.9b) The displacement vecto d in cicula pola coodinate is obtained by putting θ = 90 and d θ = 0 in Eq. 1.6:
5 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 5 whee unit vectos e, e ae given by, d = de + de...(1.10) e = cos i + sin j...(1.11a) e = sini+ cos j...(1.11b) Velocity and Acceleation in Cicula Pola Coodinates The velocity and acceleation of a point object, moving in a plane, can be obtained in tems of pola coodinates (, ) by diffeentiating position vecto = e with espect to time: d d e v = = e + d...(1.1) dt dt dt Note that e is not a constant vecto; its time ate can be obtained by using Eqs. 1.11a, b: d e d d d = sin i+ cos j= e...(1.13a) dt dt dt dt Similaly, we obtain d e d = e...(1.13b) dt dt The velocity vecto, theefoe, is witten as v = ve + ve...(1.14) d whee v v d = dt =, = and dt = ae the adial and tangential (o azimuthal) components of velocity vecto v. To find the expession fo acceleation, we have F F HG F I HG K J F + v + HG d e dt d v dv e I e a = = e + v e HG d dv K J + + v d dt dt dt dt dt dv = v d dt dt o a = ae + ae...(1.15) whee a...(1.15a) = and a = +...(1.15b) ae the adial and tangential components of acceleation vecto a. dv dt I KJ e I KJ
6 6 Mechanics In paticula, if the object is moving in the plane, in a cicle of adius R (i.e., = R, constant), we find v = 0, v = R...(1.16a) and a = R, a = R...(1.16b) a denotes the centipetal acceleation (towads e ), and a is tangential acceleation esponsible fo incease in tangential speed v. Example 1.1: Deive Eqs. 1.7 a, b, c. Solution: We have, = xi+ y j+ zk Hence, and and Also, fom Eq. 1.6, we find = sinθcosi+ sinθsinj+ cosθk θ i θ j θk = sin cos + sin sin + cos b = cosθcos i+ cosθsinj sinθk θ b = sinθ sini+ cosj b = e θ, const. b = eθ, const. = sin θe, θ const. b g g Compaing above two sets of equations, we get Eqs. 1.7a, b, c Cylindical Coodinates In cylindical coodinates, the position of paticle P elative to oigin O is descibed by the two cicula pola coodinates (ρ, ) defined in the plane Z = 0, and the thid being the Zcoodinate itself. The cylindical coodinates (ρ,, z) ae elated to Catesian coodinates as, ρ = x + y y,tan =, = x z z...(1.17a) Convesely, x = ρcos, y = ρsin, z = z....(1.17b) The equation ρ = constant descibes a ight cicula cylinde of adius ρ about Zaxis. The equation = constant descibes the plane containing Zaxis and making an angle to g g g
7 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 7 the XZplane. The geneal diffeential displacement of paticle P in cylindical coodinates is given by d = dρe + ρde + dze Z ρ z whee e ρ and e ae elated to (i, j, k) as given by Eqs. 1.11a, b, and e z = k. The unit vectos (e ρ, e, e z ) constitute an othogonal (ighthanded) coodinate system, i.e., e e = e, e e = e, e e = e ρ z z ρ z ρ and e e = e e = e e = ρ ρ z z 0 In situations whee thee is an axis of symmety, cylindical coodinates become the ight choice e ρ X to descibe the system. A typical example is the Fig. 1.4: Cylindical coodinates. (plana) otation of a paticle about an axis. In geneal, if a paticle moves in a plane, we descibe its motion in pola coodinates (ρ, ), whee ρ gives the adial position of the paticle. (We can call the plane as Z = 0, o XYplane). The velocity and acceleation of the paticle ae then given by Eqs. 1.1 to 1.15, as discussed in last section. In paticula, if the paticle moves in a cicle of adius ρ, we have (Eq. 1.14): v= ρ e = ω ρ whee ρ = ρe ω ρ, and = ez. The vecto ω epesents angula velocity whose magnitude is and diection is about the axis of otation. The acceleation of the paticle otating about Zaxis can be ewitten as, d v d ω d ρ a = = ρ + ω dt dt dt = α ρ+ ω ω ρ whee, b g d i c ω ω ρ = ρ ρ = ρ ez ez e eρ is the centipetal acceleation, and d α ρ = ρ e e = ρ e z ρ is the tangential acceleation. Note that if the paticle is otating in the XYplane, ρ coesponds to position vecto of the paticle. Hence, one usually denotes ρ by and hoping no confusion, we wite v = ω...(1.18a) and a = ω ω + α i h c h...(1.18b) We shall fequently need above elations when descibing the motion in otating fame, and/o otating motion of an object. O ρ P ( ρ, θ, z) z e Y
8 8 Mechanics 1. NEWTON S LAWS OF MOTION In 1687, Isaac Newton aticulated the laws upon which all classical mechanics is based. In pesentday teminology these laws may be stated as follows: I Law: An object continues in its state of est, o of unifom motion in a staight line, unless it is compelled to change that state by extenal foces acting on it. II Law: The time ate of change of linea momentum of an object is popotional to the net extenal foce acting on it and occus in the diection of the esultant foce. That is, p F = k d = km a...(1.19) dt whee F is esultant extenal foce; m is inetial mass of the object, and p = m v is its linea momentum. k is constant of popotionality which is taken as 1 by defining appopiate units. III Law: When two objects inteact, the mutual foces of inteaction on each object ae equal in magnitude and opposite in diection. That is, to evey action thee is an equal and opposite eaction Comments on Newton s Laws I Law and fames of efeence: Newton s I law has two ingedients: fist, that of foce (o the absence of it) and second, that of measuement of est o unifom motion of the object. Foce is the expession fo inteaction between diffeent objects. Thee is a pioi, no way to know that thee is no foce acting on an object except by putting it so fa away fom all othe objects that we would believe that it is not acted by an extenal foce. Having such an object, Newton s I law assets that it would eithe emain at est o continue to move with constant velocity. But, with espect to whom? Obviously, if the obseve himself is sitting on an acceleated fame continuously changing his velocity, the object cannot always emain at est o in unifom motion with espect to him. A efeence fame with espect to which I law holds is called an inetial fame of efeence. Thus, Newton s fist law defines the citeion fo an inetial fame. The law says that thee exists cetain fames of efeence (called inetial fames) with espect to which an object, (believed to be) not acted upon by any extenal foce, stays eithe at est o in unifom motion in a staight line. Obviously, an inetial fame itself must be a fame moving with constant velocity, i.e., nonacceleating. The question whethe a given fame of efeence is inetial o not then becomes a matte of obsevation and expeiment. Fo most of the obsevations made on the suface of Eath, the fame of efeence attached to the suface of Eath behaves as an inetial fame to a vey good appoximation. Moe efined obsevations, which we shall discuss late, shows that it is not. Howeve, unless othewise stated, we shall efe to Eath s fame as the inetial fame. All othe fames moving unifomly in staight line elative to Eath theefoe also constitute inetial fames of efeence. Newton s I law is an independent law and is not contained in his II law: In fact, both II and III laws hold only in an inetial fame suitably chosen on the basis of I law. 1.. II Law Newton s II law is not a definition of foce: The equation, F = m a, is an equality
9 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 9 which says that the foce acting on a body detemines not the velocity but ate of change of velocity of the body. What is the natue and value of F is left unansweed in II law. Newton s II law is valid fo an object having constant inetial mass and moving with velocity small compaed to that of light. If the velocity of the object is high (i.e., a significant faction of velocity of light), we ente into the domain of special elativity. Thee we find inetial mass changes with velocity; diection of acceleation is geneally not same as that of foce; hence Newton s law needs to be evised in elativistic dynamics III Law and Pinciple of Momentum Consevation Newton s III law is the only statement made by Newton about the natue of foce. It says that no matte what is the exact fom of a foce (i.e., inteaction) between two objects, it behaves in such a way that at any instant of time, if an object 1 exets a foce F 1 on object, then object exets an equal and opposite foce F 1 on object 1. That is, at any instant which implies F + F = 0...(1.0) 1 1 dp1 dp dp + = = 0 dt dt dt o P = p 1 + p is constant in time....(1.1) Thus, when two objects inteact, the sum of thei momenta is constant. Newton s III law, altenatively, is a statement of momentum consevation fo a system of inteacting objects (in absence of any extenal foce). In its oiginal fom, Newton s III law has pofound limitations. Fo example, it is not satisfied by two sepaate chages inteacting though electomagnetic field. Popagation of electomagnetic inteaction takes place at a finite speed c, speed of light. The tansfe of momentum (i.e., tavel of foce field) fom one chage to anothe involves a time inteval equal to distance between the chages divided by c. Thus, at any instant, a change in momentum occus in fist chage without an equal and opposite change in momentum in the othe chage. That is, instant by instant, we find F F o p p Momentum consevation pinciple is eventually peseved by associating the instantaneous change in momentum of paticles with that of the electomagnetic field which caies the inteaction. In fact, no inteaction (whethe electomagnetic, o gavitational, o any othe) can popagate with speed geate than that of light. Hence, in geneal, momentum consevation pinciple fo an isolated, inteacting system of paticles is valid only when we conside paticles as well as fields that cay the inteactions as the complete system. Howeve, in this book, we shall not go into these complications and assume Newton s III law to be valid in its stong fom as given by Eq SPACETIME SYMMETRY AND CONSERVATION LAWS Fom ou expeience, we aticulate the concepts of space and time to descibe the motion of an object. It is fom the same expeience and obsevation, we ascibe cetain fundamental popeties to the concepts of space and time. One such popety is homogeneity and the othe is isotopy of space and time.
10 10 Mechanics Homogeneity of Space Space is homogeneous means one pat (o point) of space is identical o equivalent to any othe pat (o point) of space. Any physical pocess will occu the same way (o identically) unde identical conditions, no matte whee (i.e., in whateve pat of space) it occus; a paticula expeiment, whethe pefomed in Kanpu o New Yok, will yield same esult. (Incidentally Kanpu and New Yok epesent same inetial fame.) Let us apply the basic popety of homogeneity of space to an isolated system of two inteacting paticles: if both the paticles of the system ae displaced by same amount δ, thee should be no change in the state of the system o in its intenal motion. That is, the total wok done by intenal foces when the system is displaced by δ must be zeo: F δ+ F δ = 0 Since δ is abitay, we get 1 1 F1 + F1 = 0 This is Newton s III law which leads to law of momentum consevation fo the isolated system of two paticles. Thus, momentum consevation emeges fom the fundamental popety of homogeneity of space. [The above agument can be extended to an isolated system of ninteacting paticles. No wok done by intenal foces imply: b 1 1g b 13 31g... 0 δ F + F + δ F + F + = whee the seies includes all pais of two paticles out of npaticle system. That is, in shot, we can wite δ F + F =, i j =... n i j d ij jii 0 1 j Since δ is abitay, we find d F ij + F jii = 0, i j = 1,..., n; i j] i j The comments made in Sec on Newton s III law also apply to the above poof of momentum consevation (of paticles) fom homogeneity of space. If we conside the foces popagating with finite speed, homogeneity of space leads to consevation of momentum of complete system, viz, both paticles and the caie of foce. Since space is homogeneous, we can choose the oigin of ou coodinate system anywhee we wish. Shifting the oigin means displacing the system; and it does not affect the pocesses Homogeneity of Time Time is homogeneous means one instant of time (o duation) is identical to any othe instant (o duation) of time. An expeiment, whethe pefomed today o tomoow o a yea late will yield same esult unde othewise same conditions. Let us discuss the effect of concept of homogeneity of time on Newton s laws of motion. Homogeneity of time implies that the foces in natue should not depend explicitly on
11 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 11 time. [Fo example, the foce F of gavitation between two point masses m 1 and m sepaated by distance emains the same at time t (today) and t (tomoow).] That is, fo all kinds of foces, we have F t = 0 Now if a foce is consevative, we can define a potential enegy of inteaction U such that the foce is negative gadient of U, i.e., we have U F x F U y F U x =, y =, z = z This means that if F does not depend explicitly on t, then equivalently, potential enegy U also is not an explicit function of t, o, b U = U x, y, z whee x, y, z efe to coodinates of the paticle unde consideation in a given inetial fame. The explicit independence of U on t leads to consevation of mechanical enegy of the paticle. To show this, let us wite the total enegy of the paticle as, 1 E = mv + U Hence, we get de m d du = v v + dt dt dt F HG d g U dx U dy U dz = v F+ + + x dt y dt z dt = v F+ Fv Fv Fv x x y y z z = 0 O, E emains constant in time. Thus the law of consevation of enegy, in Newtonian mechanics, follows diectly as a consequence of homogeneity of time. Homogeneity of time implies that we can choose the zeo of time at any instant fo the obsevation of a physical pocess Isotopy of Space Space is isotopic means one diection in space is identical o equivalent to any othe diection. A paticula expeiment (i.e., physical pocess) will yield the same esult whethe ou laboatoy faces Noth o West. That is, an angula displacement of an isolated system does not change the intenal state of the system, no its intenal motion. When applied to an isolated system of two inteacting paticles, isotopy of space implies that the net wok done by intenal foces must be zeo when the system is otated by an angle d. i I KJ
12 1 Mechanics That is, d F + d F = whee d1 and d ae displacement vectos of paticles 1 and espectively. We know that v = ω ; that implies, d = d, and d = d 1 1 Thus, we find dd 1i F 1 + dd i F 1 = 0 o d F + d F = Since d is abitay, we get b 1 1g b 1g 0 1 F1 + F1 = 0 that is, the net toque poduced on the system by intenal foces is zeo. This implies that the angula momentum of the system emains constant. The above analysis can be extended to an npaticles system. Thus, consevation of angula momentum of an isolated system emeges as a consequence of a fundamental popety of space, viz., its isotopy. Isotopy of space implies we can oient ou coodinate axes in space in any way we wish Isotopy of Time Isotopy means equivalence of diections; isotopy of time means equivalence of diections of time. Howeve, we usually efe to only one diection of time, the fowad diection, fom pesent to futue. Nevetheless, we can conceive of a backwad diection of time, i.e., fom pesent to past. Theefoe, the question whethe time is isotopic o not means whethe time is evesible o not? Opeationally, it means to find out what happens if we change t by t? Newton s laws of motion do not change if we eplace t by t in it: m d d t b g m d = = F dt Since F does not explicitly contain t, it emains unaffected. Thus, the motion descibed by Newton s laws emains same when we eplace t by t. What it means is the following: conside a physical pocess govened by Newton s laws, as fo example, motion of a paticle thown vetically upwads. It goes up, stops, and then falls back unde gavity. Let us take a movie of the pocess as it poceeds fowad in time. If we now view the movie in evese, we look at the pocess occuing backwad in time. The entie pocess looks nomal: paticle goes up, stops, and falls back exactly accoding to Newton s laws. Thus, Newton s laws ae evesible in time. Pocesses in systems compising of lage numbe of paticles do not show time evesibility. As a typical example, conside a glass which falls fom a table and shattes into small pieces on the floo. If we un a motion pictue of the above pocess in evese, we see the pieces gatheing togethe on the floo into a whole glass, which then jumps onto the
13 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 13 table. This does not happen in natue. Thus, while each piece of glass follows Newton s laws which ae time evesible, the collective system beaks time evesibility. This is explained by II law of themodynamics which pohibits a macoscopic system to move fom less odeed to a moe odeed state (law of incease of entopy). Isotopy of time does not lead to any specific consevation pinciple in classical mechanics. To sum up, in this section, we descibed the spacetime symmeties and thei consequences in undestanding the natue of foces and consevation laws, within a given inetial fame. Now, we shall see how Newtonian spacetime stuctue connects motion as obseved fom two diffeent inetial fames. 1.4 PRINCIPLE OF RELATIVITY AND GALILEAN TRANSFORMATION It was fist obseved by Galileo, in multitude of expeiments, that mechanical phenomena occu identically in all inetial fames. (Late, this was expeimentally veified fo electomagnetic phenomena as well.) These obsevations of Galileo wee stated by Newton in the following wods: The motions of bodies included in a given space ae the same among themselves, whethe that space is at est o moves unifomly fowad in a staight line. The above statement, known as Galilean pinciple of special elativity, means that all phenomena in natue should be elated in same way in all inetial fames. In moden teminology, it says that the laws of natue must emain fominvaiant as we go fom one inetial fame to anothe. The measuements of the motion of a paticle P (i.e., its position, velocity, and acceleation at any instant) fom two diffeent inetial fames S and S leads to two sets of values. The elationships between these two sets of measuements ae called Galilean tansfomation equations. One of the efeence fames S is (abitaily) called stationay while the othe S is called moving. The two fames must be moving with a constant velocity V elative to each othe. Othewise, the two fames become same. Both the fames (o obseves) S and S may use Catesian coodinate systems fo measuements. Let us denote the coodinates of paticle P in fame S by = (x, y, z) and time t, and in fame S by = (x, y, z ) and time t ; t and t denote the time ead by clocks attached in fames S and S, at the instant when position coodinates of P ae espectively and. X Z O S Z R P Y S Y O X Z Y O R Z Y O P X Fig. 1.5 (a) Fig. 1.5 (b)
14 14 Mechanics In paticula, if we assume that oigins O and O of both the coodinate systems coincide at t = t = 0, then at any late time t = t, the oigin O of coodinate system S will be at a distance R = Vt fom the oigin O of coodinate system S, whee V is the constant velocity with which S moves elative to S. Hence, if and ae position vectos of P as obseved fom S and S at that instant, then we have (see Fig. 1.5a). = R+ = Vt +...(1.1a) o = Vt...(1.1b) The above elation, along with t = t, ae known as Galilean tansfomations. As a special case, if the fame S moves paallel elative to fame S with X axis coinciding with Xaxis, as shown in Fig. 1.5b, the Galilean tansfomations ae given in components as following: x = x Vt y = y...(1.) z = z t = t Galilean tansfomations ae not the tansfomation equations between two coodinate systems within a single inetial fame, like e.g., Eq. 1.5 which elate ectangula to spheical pola coodinates; these ae elations between two diffeent inetial fames. 1.5 INVARIANTS OF GALILEAN TRANSFORMATIONS The numeical values of vaious physical quantities geneally change unde a coodinate tansfomation. Tivial example is the value of position coodinates (x, y, z) of a paticle P, which change to (x, y, z ) unde the tansfomation. It means that the value of position coodinates itself has no objective eality, o in othe wods, it is not an objective popety of the paticle. It is simply the geometical position of the paticle elative to a paticula coodinate system. The physical quantities whose values do not change unde a tansfomation ae called the invaiants of the tansfomation. Such physical quantities eflect the objective popeties of the object (o phenomenon) unde consideation in the sense that these popeties emain independent of the choice of the coodinate system. In the following discussion, we look into the invaiants of Galilean tansfomation Invaiance of Acceleation Diffeentiating Eq. 1.1 with espect to time, we find that d dr d = + dt dt dt o v = V + v...(1.3) d d whee v = = is the velocity of P elative to S ; V is velocity of S elative to S, and dt dt v is velocity of P elative to S. Eq. 1. descibes well known elation of velocity addition in Newtonian mechanics.
15 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 15 Diffeentiating once moe, we get d v d v a = = = a...(1.4) dt dt because V is constant. That is, acceleation of paticle P is same when obseved fom two diffeent inetial fames. Acceleation of the paticle P, theefoe, emains invaiant unde Galilean tansfomation Invaiance of Space and Time Intevals Suppose thee ae two points P 1 and P whose position vectos ae ( 1, ) and b1, g as obseved fom two fames S and S espectively, at any given instant of time t = t. Then, accoding to Galilean tansfomation, we have 1 = 1 Vt and = Vt whee V is the constant velocity of S elative to S. Hence, we get 1 = 1 o l = 1 = 1 = l...(1.5) Eq. 1.5 means that, bx x1 g + by y1 g + bz z1 g b 1g b 1g b 1g = x x + y y + z z l (o ) denotes the space inteval, o spatial distance between the two points. It may also epesent the length of a igid od, fo example, whose end points ae P 1 and P. Thus, we find that space inteval, o length of a igid object in geneal, emains invaiant unde Galilean tansfomation. Similaly, if two events occu at times t1 = t1 and t = t, the time inteval between these events as obseved fom two inetial fames is same, that is, t t1 = t t1 In othe wods, time inteval is also an invaiant of Galilean tansfomation Galilean Invaiance of Newton s Laws and Natue of Foces Pinciple of (special) elativity demands that all laws of natue should be valid in all inetial fames. This is obviously satisfied by Newton s I law: if the velocity of a paticle is constant (v) in fame S, it emains constant (v ) in fame S as well, v = v V...(1.6) because elative velocity (V) between the two inetial fames is a constant. Let us now look into Newton s II law. Suppose a paticle of mass m moves with the acceleation a due to an extenal foce F, as obseved fom fame S.
16 16 Mechanics Then we have, F = m a...(1.7) If the II law has to be fominvaiant unde Galilean tansfomation, then in fame S, we should get the following elation: F = m a...(1.8) whee F and a ae the foce and acceleation as obseved fom S. It is assumed that inetial mass m, being a scala, is a fame independent quantity. We know that acceleation is an invaiant of Galilean tansfomation, i.e., a = a. Hence, if both the Eqs. 1.7 and 1.8 have to hold, then we must have, F = F...(1.9) The above condition imposes a sevee condition on the natue of foces; that is, foces appeaing in natue must be invaiants of Galilean tansfomation if Newton s II law has to be valid in all inetial fames. How do we know that F = F? What kind of function F (o F ) should be in ode to satisfy above equality? It tuns out that all the foces we encounte in Newtonian dynamics depend only on the elative position o elative velocity of two inteacting bodies (and not on thei absolute position o velocities in space). That is, as obseved fom fame S, a foce exeted by paticle on paticle 1 is found to have the following functional dependence, b g...(1.30) F = f, v v Hence, as seen fom S, the same foce would tansfom to, b F = f, v v Since, unde Galilean tansfomation, we have 1 = 1 and v v1 = v v1...(1.31) we find F = F 1 1 That is, the foces found in natue happen to be invaiants of Galilean tansfomation, thus making Newton s II law valid in all inetial fames. Futhe, if foces in natue have a functional dependence given by Eq. 1.30, then we find F = f, v v = f, v v = F g b g b g...(1.3) which is the statement of Newton s III law. Thus, the III law implicitly ensues that both II and III laws emain valid in all inetial fames. (Note that F1 = F1 = F1 = F1.) All the Newton s laws of motion theefoe emain fominvaiant unde Galilean tansfomation in a selfconsistent manne. Example 1.: Show that the consevation of total momentum of an isolated system of two colliding paticles in the pocess of collision follows fom the pinciple of Galilean invaiance and the law of consevation of enegy.
17 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 17 Solution: To show this, let us conside the collision of two paticles 1 and as obseved fom an inetial fame S. Suppose initially 1 and, moving with velocities u 1 and u, ae so fa apat that they do not possess any potential enegy of inteaction. The total enegy of the system of both paticles is just thei kinetic enegy, 1 1 mu mu Afte they collide and sepaate fa apat, let thei velocities be v 1 and v 1. Accoding to law of consevation of enegy in fame S, we get mu mu = mv mv + E...(1.33) whee E is the change (loss o gain) in intenal enegy of the paticles duing collision. If E = 0, collision is elastic; othewise inelastic. Now let us view the pocess fom anothe inetial fame S moving with velocity V elative to S. Accoding to Galilean tansfomation, the velocities of paticles as obseved fom S ae, u = u V, u = u V (befoe)...(1.34a) 1 1 and v1 = v1 V, v = v V (afte)...(1.34b) Now we intoduce the idea that pinciple of consevation of enegy is fame independent. In fact, consevation of mechanical enegy being a consequence of homogeneity of time, is fundamentally an invaiant pinciple in Newtonian mechanics. As shown by expeiments, we futhe assume that intenal enegy change E also emains invaiant unde Galilean tansfomations. Hence, consevation of enegy in collision pocess as obseved in S, gives b g b g b g b g...(1.35) m1 u1 + m u = m1 v1 + m v + E Using Eq. 1.34, we find bg u = u V u1 V b 1g b g b 1g b g etc. Substituting fo u, u, v, v in Eq. 1.35, we get b g b g mu mu m1u1 + mu V = mv mv m1v1 + mv V + E Compaing above with Eq. 1.33, we get m u + m u = m v + m v which is the law of consevation of momentum in collision pocess. Thus, the pinciple of consevation of enegy and the concept of Galilean invaiance shows that momentum emains conseved duing collision, whethe elastic o inelastic. Of couse, momentum consevation in collision pocess follows athe staightfowad fom Newton s III law. If we conside the two colliding paticles as an isolated combined system on which no extenal foce acts, then the momentum of the system must emain
18 18 Mechanics conseved, no matte whethe they collide o do something else. Moeove, momentum consevation being a consequence of homogeneity of space must also be a fundamentally Galilean invaiant pinciple. That is, in fame S, we have m u + m u = m v + m v SOME BASIC FORCES IN NATURE A foce expesses the inteaction between two diffeent objects. We ae familia with a lage vaiety of foces such as gavitational foce, electic and magnetic foce, and intemolecula foces giving ise to dy fiction, viscosity, suface tension, tension in the sting, o elastic foce in a sping, etc. We ae also familia with nuclea foces esponsible fo binding of potons and neutons togethe in a nucleus, o the splitting of a neuton into a poton, electon and antineutino in betadecay. Among this lage divesity of tems, it tuns out that thee ae only fou types of basic foces (known at pesent) in natue, viz., 1. Gavitational foce. Electomagnetic foce 3. Stong nuclea foce 4. Weak foce. We shall now biefly outline the basic chaacteistic of these foces. 1. Gavitational foce: The gavitational inteaction is a univesal phenomenon which acts between all foms of matte and enegy. The quantitative natue of gavitational foce was fist given by Newton; accoding to Newton s law of gavitation, between any two mateial paticles thee exists an attactive foce of gavitation, given by F 1 G mm 1 = 1 whee m 1 and m ae the gavitational masses and 1 is the elative distance between the (inteacting) paticles. The value of gavitational constant G is about Nm /kg. The gavitational inteaction plays significant ole only when masses involved ae lage. It becomes the decisive foce when we study motions of planets, stas, o othe such cosmic objects. We shall study moe about the natue of gavitational inteaction in Ch 7. The gavitational inteaction between eath and othe elatively smalle objects nea the suface of eath poduces an almost unifom acceleation due to gavity g = 9.8 m/s (appoximately) in all these objects. Hence, in consideing the motion of any such pototype object, e.g., a block o a sphee of mass m on the suface of eath, it is implicit that gavity foce mg is acting on the object. On the othe hand, if we ae consideing the motion of micoscopic paticles like electons etc., the gavity foce is completely neglected because of the tiny masses of these paticles (as we discuss in Ch. 3; also see Poblem 5, set A, Ch. 1).. Electomagnetic foce: The second kind of fundamental foce is the electomagnetic foce that acts between chaged paticles. The fist law of electomagnetism is the Coulomb s law of electic foce between two chages q 1 and q fixed (i.e., kept at est) at elative distance 1. The Coulomb s law states that the electic foce between these static chages is given by 1 qq 1 F1 = 4 πε 0 1
Revision Guide for Chapter 11
Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More information12. Rolling, Torque, and Angular Momentum
12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.
More informationHour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and
Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon
More informationChapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43
Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.
More information(a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of
Homewok VI Ch. 7  Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationrotation  Conservation of mechanical energy for rotation  Angular momentum  Conservation of angular momentum
Final Exam Duing class (13:55 pm) on 6/7, Mon Room: 41 FMH (classoom) Bing scientific calculatos No smat phone calculatos l ae allowed. Exam coves eveything leaned in this couse. Review session: Thusday
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More informationVoltage ( = Electric Potential )
V1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More information2  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1
 ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation
More informationPY1052 Problem Set 8 Autumn 2004 Solutions
PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ighthand end. If H 6.0 m and h 2.0 m, what
More information14. Gravitation Universal Law of Gravitation (Newton):
14. Gavitation 1 Univesal Law of Gavitation (ewton): The attactive foce between two paticles: F = G m 1m 2 2 whee G = 6.67 10 11 m 2 / kg 2 is the univesal gavitational constant. F m 2 m 1 F Paticle #1
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationPHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013
PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0
More informationPHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013
PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,
More information81 Newton s Law of Universal Gravitation
81 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationCh. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth
Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More informationUNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100minute sessions
Name St.No.  Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,
More informationProblem Set 6: Solutions
UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 164 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationExam 3: Equation Summary
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P
More informationTORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION
MISN034 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................
More informationMagnetic Field and Magnetic Forces. Young and Freedman Chapter 27
Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew  electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field
More informationGauss Law. Physics 231 Lecture 21
Gauss Law Physics 31 Lectue 1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationGravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2
F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationProblems on Force Exerted by a Magnetic Fields from Ch 26 T&M
Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuentcaying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to
More informationForces & Magnetic Dipoles. r r τ = μ B r
Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent
More informationCharges, Coulomb s Law, and Electric Fields
Q&E 1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 14 Intoduction: Pehaps one of the most unusual
More informationMultiple choice questions [60 points]
1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions
More information7 Circular Motion. 71 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary
7 Cicula Motion 71 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o
More informationThe Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = W/q 0 1V [Volt] =1 Nm/C
Geneal Physics  PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit
More informationExperiment 6: Centripetal Force
Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee
More informationChapter 2. Electrostatics
Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.
More informationNotes on Electric Fields of Continuous Charge Distributions
Notes on Electic Fields of Continuous Chage Distibutions Fo discete pointlike electic chages, the net electic field is a vecto sum of the fields due to individual chages. Fo a continuous chage distibution
More informationAlgebra and Trig. I. A point is a location or position that has no size or dimension.
Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite
More informationF G r. Don't confuse G with g: "Big G" and "little g" are totally different things.
G1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationProblems of the 2 nd International Physics Olympiads (Budapest, Hungary, 1968)
Poblems of the nd ntenational Physics Olympiads (Budapest Hungay 968) Péte Vankó nstitute of Physics Budapest Univesity of Technical Engineeing Budapest Hungay Abstact Afte a shot intoduction the poblems
More informationChapter 13. VectorValued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates
13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. VectoValued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along
More informationLesson 7 Gauss s Law and Electric Fields
Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More information2. Orbital dynamics and tides
2. Obital dynamics and tides 2.1 The twobody poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body
More informationChapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom
Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationUniform Rectilinear Motion
Engineeing Mechanics : Dynamics Unifom Rectilinea Motion Fo paticle in unifom ectilinea motion, the acceleation is zeo and the elocity is constant. d d t constant t t 111 Engineeing Mechanics : Dynamics
More informationGravity. A. Law of Gravity. Gravity. Physics: Mechanics. A. The Law of Gravity. Dr. Bill Pezzaglia. B. Gravitational Field. C.
Physics: Mechanics 1 Gavity D. Bill Pezzaglia A. The Law of Gavity Gavity B. Gavitational Field C. Tides Updated: 01Jul09 A. Law of Gavity 3 1a. Invese Squae Law 4 1. Invese Squae Law. Newton s 4 th law
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationIntroduction to Fluid Mechanics
Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationIntroduction to Electric Potential
Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic
More informationSolution Derivations for Capa #8
Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass
More informationAP Physics Electromagnetic Wrap Up
AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation
More informationChapte 3 Is Gavitation A Results Of Asymmetic Coulomb Chage Inteactions? Jounal of Undegaduate Reseach èjurè Univesity of Utah è1992è, Vol. 3, No. 1, pp. 56í61. Jeæey F. Gold Depatment of Physics, Depatment
More information10. Collisions. Before During After
10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: F(t)
More informationGauss s law relates to total electric flux through a closed surface to the total enclosed charge.
Chapte : Gauss s Law Gauss s Law is an altenative fomulation of the elation between an electic field and the souces of that field in tems of electic flu. lectic Flu Φ though an aea ~ Numbe of Field Lines
More informationOn the Relativistic Forms of Newton's Second Law and Gravitation
On the Relativistic Foms of Newton's Second Law and avitation Mohammad Bahami,*, Mehdi Zaeie 3 and Davood Hashemian Depatment of physics, College of Science, Univesity of Tehan,Tehan, Islamic Republic
More informationNuclear models: FermiGas Model Shell Model
Lectue Nuclea mode: FemiGas Model Shell Model WS/: Intoduction to Nuclea and Paticle Physics The basic concept of the Femigas model The theoetical concept of a Femigas may be applied fo systems of weakly
More informationChapter 4. Electric Potential
Chapte 4 Electic Potential 4.1 Potential and Potential Enegy... 43 4.2 Electic Potential in a Unifom Field... 47 4.3 Electic Potential due to Point Chages... 48 4.3.1 Potential Enegy in a System of
More informationELECTRIC CHARGES AND FIELDS
Chapte One ELECTRIC CHARGES AND FIELDS 1.1 INTRODUCTION All of us have the expeience of seeing a spak o heaing a cackle when we take off ou synthetic clothes o sweate, paticulaly in dy weathe. This is
More informationNUCLEAR MAGNETIC RESONANCE
19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic
More informationTheory and measurement
Gavity: Theoy and measuement Reading: Today: p11  Theoy of gavity Use two of Newton s laws: 1) Univesal law of gavitation: ) Second law of motion: Gm1m F = F = mg We can combine them to obtain the gavitational
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationPhys 2101 Gabriela González. cos. sin. sin
1 Phys 101 Gabiela González a m t t ma ma m m T α φ ω φ sin cos α τ α φ τ sin m m α τ I We know all of that aleady!! 3 The figue shows the massive shield doo at a neuton test facility at Lawence Livemoe
More information2. SCALARS, VECTORS, TENSORS, AND DYADS
2. SCALARS, VECTORS, TENSORS, AND DYADS This section is a eview of the popeties of scalas, vectos, and tensos. We also intoduce the concept of a dyad, which is useful in MHD. A scala is a quantity that
More informationStructure and evolution of circumstellar disks during the early phase of accretion from a parent cloud
Cente fo Tubulence Reseach Annual Reseach Biefs 2001 209 Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The
More informationSolutions for Physics 1301 Course Review (Problems 10 through 18)
Solutions fo Physics 1301 Couse Review (Poblems 10 though 18) 10) a) When the bicycle wheel comes into contact with the step, thee ae fou foces acting on it at that moment: its own weight, Mg ; the nomal
More informationChapter 4: Fluid Kinematics
Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian
More informationFluids Lecture 15 Notes
Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2D, this velocit
More informationLecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3
Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each
More informationdz + η 1 r r 2 + c 1 ln r + c 2 subject to the boundary conditions of noslip side walls and finite force over the fluid length u z at r = 0
Poiseuille Flow Jean Louis Maie Poiseuille, a Fench physicist and physiologist, was inteested in human blood flow and aound 1840 he expeimentally deived a law fo flow though cylindical pipes. It s extemely
More informationPhysics 505 Homework No. 5 Solutions S51. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.
Physics 55 Homewok No. 5 s S5. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm
More informationTransformations in Homogeneous Coordinates
Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) YanBin Jia Aug, 6 Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a b c u au + bv +
More informationOrbital Motion & Gravity
Astonomy: Planetay Motion 1 Obital Motion D. Bill Pezzaglia A. Galileo & Fee Fall Obital Motion & Gavity B. Obits C. Newton s Laws Updated: 013Ma05 D. Einstein A. Galileo & Fee Fall 3 1. Pojectile Motion
More informationProblems of the 2 nd and 9 th International Physics Olympiads (Budapest, Hungary, 1968 and 1976)
Poblems of the nd and 9 th Intenational Physics Olympiads (Budapest Hungay 968 and 976) Péte Vankó Institute of Physics Budapest Univesity of Technology and Economics Budapest Hungay Abstact Afte a shot
More informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuentcaying conducto is basic to evey electic moto  tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More informationMagnetic Bearing with Radial Magnetized Permanent Magnets
Wold Applied Sciences Jounal 23 (4): 495499, 2013 ISSN 18184952 IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.23.04.23080 Magnetic eaing with Radial Magnetized Pemanent Magnets Vyacheslav Evgenevich
More informationRelativistic Quantum Mechanics
Chapte Relativistic Quantum Mechanics In this Chapte we will addess the issue that the laws of physics must be fomulated in a fom which is Loentz invaiant, i.e., the desciption should not allow one to
More informationEXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH S MAGNETIC FIELD
260 161. THEORY EXPERMENT 16 THE MAGNETC MOMENT OF A BAR MAGNET AND THE HORZONTAL COMPONENT OF THE EARTH S MAGNETC FELD The uose of this exeiment is to measue the magnetic moment μ of a ba magnet and
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationChapter 30: Magnetic Fields Due to Currents
d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.11 T) i to ue a lage cuent flowing though a wie.
More informationESCAPE VELOCITY EXAMPLES
ESCAPE VELOCITY EXAMPLES 1. Escape velocity is the speed that an object needs to be taveling to beak fee of planet o moon's gavity and ente obit. Fo example, a spacecaft leaving the suface of Eath needs
More informationPAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII  SPETO  1995. pod patronatem. Summary
PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8  TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More informationLab M4: The Torsional Pendulum and Moment of Inertia
M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disklike mass suspended fom a thin od o wie. When the mass is twisted about the
More informationGravitational Mechanics of the MarsPhobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning
Gavitational Mechanics of the MasPhobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More information