Table Of Contents - Casaxps
Copyright 2011 Casa Software Ltd. www.casaxps.comTable of ContentsVariable Forces and Differential Equations . 2Differential Equations . 3Second Order Linear Differential Equations with Constant Coefficients . 6Reduction of Differential Equations to Standard Forms by Substitution . 18Simple Harmonic Motion. 21The Simple Pendulum . 23Solving Problems using Simple Harmonic Motion . 24Circular Motion . 28Mathematical Background . 28Polar Coordinate System . 28Polar Coordinates and Motion . 31Examples of Circular Motion . 351
Copyright 2011 Casa Software Ltd. www.casaxps.comVariable Forces and Differential EquationsA spring balance measures the weight for a rangeof items by exerting an equal and opposite force tothe gravitational force acting on a mass attached tothe hook. The spring balance is therefore capableof applying a variable force, the source of which isthe material properties of a spring. When inequilibrium, the spring balance and mass attachedto the hook causes the spring to extend from aninitial position until the resultant force is zero.Provided the structure of the spring is unaltered bythese forces, the tension in the spring isproportional to the extension of the spring fromthe natural length of the spring.The tension due to the spring is an example of aforce which is a function of displacement:Hooke’s Law, empirically determined (determined by experiment), states for a spring of naturallengthwhen extendedbeyond the natural length exerts a tension proportional to theextension . Introducing the constantknown as modulus of elasticity for a particular spring (orextensible string), the tension due to the extension of the spring is given by:The term natural length means the length of a spring before any external forces act to stretch orcompress the spring.If a particle is attached to a light spring and the spring is stretched to produce a displacementfrom the natural length of the spring, then the force acting upon the particle due to the spring isgiven byApplying Newton’s second law of motionterms ofand derivatives of, wherethe equation can be written inas follows.Equation (1) is a second order linear differential equation, the solution of which provides thedisplacement as a function of timein the form. Differential equations are often2
Copyright 2011 Casa Software Ltd. www.casaxps.comencountered when studying dynamics, therefore before returning to problems relating to themotion of particles attached to elastic strings and springs the technical aspects of differentialequations will be considered.Differential EquationsOrdinary differential equations involve a function and derivative of the function with respect to anindependent variable. For example the displacement from an origin of a particle travelling in astraight line might be expressed in the form of a differential equation for the displacementin the formA differential equation is a prescription for how a functionand functions obtained bydifferentiating the function can be combined to produce a specific function, in this case.Whenever the derivative of a function is involved, a certain amount of information is lost. Theintegral of a derivative of a function is the function plus an arbitrary constant. The arbitrary constantrepresents the lost information resulting from when the derivative is calculated. For example,The two functions andboth have the same derivativetherefore if presented with thederivative alone, the precise nature of the function is unknown; hence the use of a constant ofintegration whenever a function is integrated.Combining derivatives to form a differential equation for a function also means information aboutthe function is missing within the definition and for this reason the solution to a differentialequation must be expressed as a family of solutions corresponding to constants introduced toaccommodate the potential loss of information associated with the derivatives. A general solutionto Equation (2) isandare constants yet to be determined. Bothandare solutions to the differential equation as are any number of other choicesfor the values of and . For a given problem, if at a given time the position and the derivative ofposition are known, then a specific solution from the set of solutions represented by Equation (3)can be obtained. The method used to establish solutions to equations of the standard form, ofwhich Equation (2) is an example, will be discussed in detail later.Solving general differential equations is a large subject, so for sixth form mechanics the types ofdifferential equations considered are limited to a subset of equations which fit standard forms.Equations (1) and (2) are linear second order differential equations with constant coefficients. Tobegin with, solutions for certain standard forms of first order differential equations will beconsidered.3
Copyright 2011 Casa Software Ltd. www.casaxps.comThe differential equations used to model the vertical motion of a particle with air resistanceprescribe the rate of change of velocity in terms of velocity:Or depending on the model used for the resistance force,Equations (3) and (4) are first order differential equations specifying the velocity as a function oftime. Equation (3) is a linear first order differential equation sincewithout products such as,orandappear in the equation. Equation (4) is nonlinear becauseappears inthe equation. These first order differential equations (3) and (4) are also in a standard form, namely,The key point being the derivative can be expressed as the product of two function where onefunction expresses a relationship between the dependent variable while the other only involvesthe independent variable . For equation (4)and. The solution for thestandard form (5) is obtained by assumingThe solution relies on the separation of the variables. For Equation (3),and, therefore the solution can be obtained as follows:If the is particle initially released from rest, thenwhen, therefore, henceThe same procedure could be used to find a solution for the nonlinear differential equation (4).Equation (3) represents a first order linear differential equation for which two standard forms canapply. In addition to being open to direct integration using (5) and (6), Equation (3) is of the form:Differential equations of the form (7) can be solved by determining a, so called, integrating factorsuch that the differential equation can be reduced to an equivalent equation:4
Copyright 2011 Casa Software Ltd. www.casaxps.comIf, then by the rule for differentiating productsIf Equation (7) is multiplied throughout by the integrating factorEquation (9) will reduce to Equation (8) providedAndEquation (10) is valid providedOrApplying the solution based on separation of variables yieldsEquation (9) can now be written in the formTherefore ifis used in Equation (8), an equivalent differential equation to Equation(11) is obtained as followsSince Equation (3) can be written in the standard form defined by Equation (7), namely,5
Copyright 2011 Casa Software Ltd. www.casaxps.comWe can therefore identify the following functionsrequires an integrating factor ofand, therefore the solution, thereforeApplying the same initial conditions as before, namely,whenyieldsresultingin the same answer as beforeTwo different methods applied to a single problem leading to the same conclusion provide a senseof reassurance. An alternative to explicitly solving a differential equation is to calculate the solutionusing numerical methods. It is important to realise, however, that even when an expression or anumerical solution is produced, there is the possibility an assumption used in the solution is invalidand therefore the solution is only valid for a limited range of the independent variable. An equationof the formrequires the conditionsince the solution involves . The importance of such restrictions canbe nicely illustrated by the follow sequence of algebraic steps applied to any numbercontradiction.So far so good, but attempting to divide byleading to aleads toIn terms of manipulation of numbers, these steps appear fine but for the step in whichiseliminated. Dividing by zero is clearly shown to produce an incorrect answer. Differential equationsmay have conditions leading to similar issues, but for now it is sufficient to understand the solutiontechniques for differential equations and defer these problematic considerations for those studyingmathematics at a higher level than this text.Second Order Linear Differential Equations with Constant CoefficientsDynamics problems involving Newton’s second law of motion often involve second order lineardifferential equations as illustrated in the derivation of Equation (1) for a particle attached to a lightspring. For an understanding of simple harmonic motion it is sufficient to investigate the solution ofdifferential equations with constant coefficients:6
Copyright 2011 Casa Software Ltd. www.casaxps.comThat is, equations of the form (12) for which , and are all constant.The equation of motion for a particle attached to a light spring is of the form (12)where,,and.Apart from being important mathematical methods for mechanics in their own right, solutions offirst order differential equations play a role in solving equations of the form (12). Before writingdown the solution for Equation (12), first the solution for the equationmust be established.Whileis a function obtained from the function, the act of differentiatingcould bedefined in the sense thatSimilarly, the second derivative ofmight be expressed asUsing these alternative forms for the first and second derivative ofexpressed asEquation (14) could beIt might seem reasonable to think of these operations expressed by Equation (15) in an equivalentform using the analogy for factorising a quadratic equationasIf Equations (15) and (16) are equivalent, then the solutionobtained from the first order differential equation7might reasonably be expected to be
Copyright 2011 Casa Software Ltd. www.casaxps.comApplying separation of variablesThus a solution to Equation (16) obtain from the methods above is.Since the roots for the quadratic polynomial are also interchangeable when Equation (17) waschosen, it might also be reasonable to assumeis also a function which satisfiesEquation (14).SinceTherefore substituting into Equation (14)And since the value is a root ofis a solution of the differential equation (14). Similarly,,, hencemust be a solution and sinceis also a solution of the differential equation (14).Equation (18) is consistent with the previous discussion about potential loss of information resultingfrom differentiating a function, namely, the second derivative of a function potentially needs twoconstants of integration to allow for a class of functions all of which have the same secondderivative. The introduction of two constants in the solution serves to introduce the necessarygenerality needed to accommodate the range of functionssatisfying Equation (14).Repeated Root forThe generality of the solution (18) runs into problems if the quadratic equationhas repeated roots , in which case Equation (18) reduces toNamely only a single constant and function appear in the solution. It becomes necessary to look fora further solution before all the possible solutions to the differential equation are obtained. It canbe shown that ifis a solution of (14), thenis also a solution of (14). The8
Copyright 2011 Casa Software Ltd. www.casaxps.comfact that a second solution is required and the method for constructing the second solution are bothconsequences of theory beyond the scope of this text, so simply showing thatis asolution of (14) will suffice.Substituting into the left-hand side of (14)Ifis a repeated root ofthensoandFor repeated roots of the auxiliary equation, the general solution of (14) isComplex Roots forThe motivation for considering differential equations was the equation of motion for a particleattached to a light spring. The resulting differential equation is written in the form of a second orderdifferential equation with constant coefficients:The auxiliary equation is thereforeThis quadratic equation has no real roots, however the complex roots are, whereUsing Euler’s formulaandand. The solution (18) still applies in the sense, the complex solution to Equation (19) isWhile expressed as a complex valued function of a real variable, the Equation (20) suggestsandare solutions of Equation (19).9
Copyright 2011 Casa Software Ltd. www.casaxps.comFirst considerThereforeis indeed a solution of (19). Similarlyreal valued general solution of (19) is therefore of the formDefining the alternative constantsis another solution. Theand as follows:The solution to equation (19) can be written as follows:Usingthe solution can be expressed in the formThe solution (22) is an alternative formulation of solution (21), in which the constants and canbe interpreted as the amplitude or maximum displacement from the centre for the oscillation of aparticle attached to a spring and as defining the initial displacement of the particle at the time. Equation (22) is the more common form used when analysing dynamics problems described assimple harmonic motion, of which a particle on a spring is one example of this type of motion.More generally, the auxiliary equationhas complex roots of the formandwhenever theand. Under these circumstancesthe solution as prescribed by Equation (18) takes the form:Following a similar analysis used to obtain Equation (20) the complex valued solution is of the form10
Copyright 2011 Casa Software Ltd. www.casaxps.comSince the differential equation (14) has real coefficients and equates to zero, it might be reasonableto assume both real and imaginary part of the complex solution must be solutions of the differentialequation (14). A solution of the formis therefore a nature first choice to test bysubstitution into the differential equation.Substituting intoTherefore,Since the complex roots of the auxiliary equationit follows thatandare obtained from, thereforeSimilarlyThus,is a solution ofwhenever the auxiliary equationhas complex rootsand11, with.
Copyright 2011 Casa Software Ltd. www.casaxps.comA similar argument showsis also a solution of (14) and therefore a real valuedsolution ofcan be expressed in the form:To solve a second-order homogeneous linear differential equation of the form:Determine the rootandof the auxiliary equationThe general solution for the differential equation is then one of the following three options:Roots of Auxiliary EquationReal roots:andGeneral SolutionReal roots:Complex Roots:There are problems in mechanics for which the homogeneous differential equation is replaced by anequation of the form:For example, the motion of a particle of mass attached to a spring with a constant additionalforce in the direction of the x-axis given byresults in the equation of motion:OrThe general solution for these types of problems reduces to three steps:1. Find the general solution2. Find any functionfor the complementary homogenous equationnot part of the complementary solution which satisfies12
Copyright 2011 Casa Software Ltd. www.casaxps.com3. Add the two solutions together to form the general solution for (24)The function referred to asis known as a particular integral, whileis thecomplementary function. Since the complementary function is established using the tableabove, the problem is therefore to construct a particular integral for equations of the form (24).Consider the equation of motion:Ifthen the solution forhas already been established to beThe problem is to find the simplest functionsuch thatSince the functionappears on the left-hand-side ifis used as a solution whereis a constant, the first and second derivatives ofare zero and thereforeThe general solution is thereforeFor these specific types of second order differential equations it is possible to find manydifferent particular integrals, however it can be shown that the general solution constructedfrom the complementary function and any one of these particular integrals result in the sameanswer when boundary conditions are applied of the formand.Methods for determining particular integral for differential equations of the form (24) are againbeyond the scope of this text, so a limited number of special cases will be tabulated with theiruse illustrated by example.13
Copyright 2011 Casa Software Ltd. www.casaxps.comForm for f(t)Form for Particular IntegralTo illustrate the use of these particular integrals, consider the problem of a particle attached toa spring, where instead of a constant forcethe disturbing force varies with time accordingto. Newton’s second law of motion yields the equationOr ifSolving Equation (25) involves determining the complementary functionequationand finding an appropriate particular integralfor the functionmatchesmust be of the formsubstituting into the identityfor the functionwhere, where both andGiven the formAndThereforeEquating coefficients for sine and cosine yields14for the homogeneous. Since the form, the particular integralmust be determined by
Copyright 2011 Casa Software Ltd. www.casaxps.comTherefore forthe particular integral isAnd the general solution isExample:Given the boundary conditionsFindandatand the differential equationas a function of .Solution:The first step is to calculate the general solution to the homogenous differential equationIt is important to determine the complementary function first since there is always the possibilitythe standard option for the particular integral corresponding tois included in the twofunctions used to construct the complementary function. Obtaining the complement functionallows an informed decision to be made when selected the form for the particular integral.The auxiliary equation corresponding toisThe complementary function is therefore constructed using the form for two distinct real roots:Sincecannot be constructed fromthe particular integral can be chosen to beSubstituting into15
Copyright 2011 Casa Software Ltd. www.casaxps.comThe particular integral must satisfy the differential equationThereforeThe general solution for the differential equation isApplying the boundary conditionsThereforeatandatresults in the equation for the constantsSolving the simultaneous equations (a) and (b) yieldsresult is the particular solutionandand. The boundary conditionsExample:A particle of massattached to a spring is subject to three forces:i.A tension force –ii.A damping force proportional to the velocity –iii.A disturbing forceBy applying Newton’s second law of motion, express the displacement in terms of time as adifferential equation and solve the differential equation for the general solution.Solution:Newton’s second law of motionallows these three forces to be combined in the form16
Copyright 2011 Casa Software Ltd. www.casaxps.comEquation (a) is a second order linear differential equation with constant coefficients. The solution istherefore of the form.The complementary function is obtained from the general solution for the correspondinghomogeneous equationThe auxiliary equation for Equation (b) isSincethe roots for the auxiliary equation are complex andtherefore the complementary function is of the formWhereThe particular integralmatchesthe forminto the identityand.for the function. Since the form for the functionwhereand, the particular integral must be of, where both and must be determined by substitutingGiven the formAndThereforeCollecting coefficients ofand:17
Copyright 2011 Casa Software Ltd. www.casaxps.comCoefficient of:Coefficient of:The particular integral for Equation (a) is thereforeWith general solutionReduction of Differential Equations to Standard Forms by SubstitutionThe discussions above are concerned with finding solutions to a select group of differentialequations appearing in standard forms, namely,1. first order differential equations where the variables can be separated to allow directintegration,2. first order linear differential equations by determining an integrating factor and3. second order linear differential equations with constant coefficients.These standard forms can also be useful for problems where a change of variable transforms adifferential equation from one form to a standard form for which a solution can be found.By way of example, consider the second order differential equation for a particle attached to a lightspring.Rather than treating the problem as a second order differential equation, usingThereforeis can be expressed as a first order differential equation involving velocitydisplacement :18and
Copyright 2011 Casa Software Ltd. www.casaxps.comSubstitutingimplies, therefore the equation is transformed to a first order lineardifferential equationUsing direct integrationSinceIf the constant of integration is rewritten in the forrequire by(imposingEquation (26) relates velocity to displacement and sinceis also a first order differentialequation for.Using separation of variablesMaking the substitutionHenceSince,19which is
Copyright 2011 Casa Software Ltd. www.casaxps.comThese two solutions are of the formThat is, the same solution for the original differential equation is recovered by direct integration asby applying the theory for the second order linear differential equation to the displacement as afunction of time. Equation (22) and Equation (27) are identical.20
Copyright 2011 Casa Software Ltd. www.casaxps.comSimple Harmonic MotionWhile the introductory problems in mechanics involving the motion of a particle are oftenconcerned with moving a particle from one place to another, there is an important class ofproblems where a particle goes through a motion, but at some point in the trajectory the particlereturns to the initial position. An obvious example of repetitious motion is a Formula 1 race carwhich must execute a sequence of laps of a race circuit. Other examples might be the hands of ananalogue clock or the vibrations in a tuning fork. The key characteristic for all these motions is thatafter a time period, the particle or particles retraces over ground previously encountered.While periodic motion is often complex in nature, many problems can be reduced by approximationto a more simple form known as Simple Harmonic Motion (SHM). An example of such anapproximation is a simple pendulum, where for small oscillations the motion can be approximatedto simple harmonic motion.Simple Harmonic Motion is an oscillation of a particle in a straight line. The motion is characterisedby a centre of oscillation, acceleration for the particle which is always directed towards the centreof oscillation, and the acceleration is proportional to the displacement of the particle from thecentre of oscillation. These statements are encapsulated in the differential equation.Simple Harmonic MotionFor constantmotionthe equation ofhas solutionSuch a linear motion is precisely the motion of a particle of mass attached to a spring of naturallength when moving on a smooth horizontal surface after being displaced a distance from thenatural length of the spring before being released.When approximating a motion as simple harmonic, the problem is reduced to that of a straight linetrajectory for a particle corresponding to the x-coordinate of an equivalent particle moving in acircle of radius with a constant speed. This motion of a particle in a circle provides a geometricperspective for simple harmonic motion expressed in the solution.21
Copyright 2011 Casa Software Ltd. www.casaxps.comThe trajectory for a particle undergoing Simple Harmonic Motion is described by acosine functions in terms of time, hence the name for the motion.sine orSimple Harmonic behaviour:More complex oscillation can be analysed in terms of combinations of sine and cosine functions, sounderstanding the more fundamental problem of simple harmonic motion provides the basis forunderstanding these more general problems.More complicated periodic motion can be recreated by combining these three sinusoidalmotions representing simple harmonic oscillations. For example the motion of a piano stringand be synthetically modelled from a number of sinusoidal motions.A technologically significant problem is that of interpreting infrared spectra, which can beunderstood in terms of oscillations associated with molecular bonds. A molecular bond is modelledas springs connecting two masses, hence the relevance of SHM. Infrared spectra are used to22
Copyright 2011 Casa Software Ltd. www.casaxps.comcharacterise materials for medical science and other key areas of technology. The following FourierTransform Infrared (FTIR) spectrum illustrates numerous oscillations in intensity which can betraced back to vibrations associated with carbon-hydrogen bonds in polystyrene (PS).FTIR Spectra from polystyrene.The Simple PendulumA simple pendulum consists of a particle of mass attached to one end of a light inextensible stringof length where the other end of the string is attached to a fixed point. The particle when at resthangs vertically below the fixed point. The particle and string when displaced from the equilibriumposition oscillate in a circular arc in the same vertical plane. This physical description suggests theparticle moves in 2D and therefore the motion will not behave like simple harmonic motion. Thevalue in studying the simple pendulum lies in observing the types of approximation and restrictionsto the motion of the particle that allow a description in terms of simple harmonic motion.A diagram for the simple pendulum showing theforces acting on the particle of mass helps towrite down the equations of motionusingthe unit vectors and , to express the displacementfrom the origin in terms of the tension exertedby the inextensible string and weight:andIn general, these equations for the simple pendulum do not match the equation for simpleharmonic motion, however if the length of the string is large compared with thevertical displacementthen, therefore23
Copyright 2011 Casa Software Ltd. www.casaxps.comThe assumption that is large compared with also suggests that the acceleration in theis small too, therefore the component for the equations of motion yieldsApplying these approximation to the equation of motion for thethen becomes on replacingdirectiondirectionandwhereThus, the motion of a simple pendulum for which the length of the string is large compared to thevertical displacement of the mass reduces under approximation to simple harmonic motion.Note the condition that is large compared with is equivalent to stating the maximum angle forthe oscillations is small. Also, the assertion thatis geometrically equivalent to observing forlarge and small the trajectory of the particle for small angles is almost without curve, that is, canbe approximated by a straight line.The reason for analysing a mechanical system such as the simple pendulum is to extract usefulinformation. Historically, a pendulum offered a means of measuring time, the method being tocount the number of complete oscillations. Once the motion of a pendulum is characterised interms of simple harmonic motion, the mathematics of the solutionprovidesthe means of calculating such useful parameters.Solving Problems using Simple Harmonic MotionSimple harmonic motion is referred to a periodic because after a time interval or period, the sametrajectory for the particle begins afresh. This statement is mathematically described by thedisplacementfor the particle must be the same at two times and :The shortest timeis called the periodand is thereforeFor a simple pendulum of length , the time period is determined by24.and is therefore
Copyright 2011 Casa Software Ltd. www.casaxps.comIn general, if a problem can be expressed in the form of simple harmonic motion, that is, theequation of motion is of the formthen the time for one complete oscillation is given byComparing solution toImpliesNote: the time period for a particle moving under simple harmonic motion is independent of themaximum displacement from the centre of oscillation known as the amplitude. The velocity forthe particle does depend on the amplitude.Given the displacement for SHM,Eliminating from these two equations by multiplying Equation (1) bythe resulting equations yieldsUsing25before squaring and adding
Copyright 2011 Casa Software Ltd. www.casaxps.comEquation (3) shows the velocity for a particle moving in SHM is a maximum whenand zerowhen the displacement of the particle is at either of the extreme positions from the centre ofoscillation, namely,.ExampleThe port at Teignmouth is in the Teignestuary. A sand bar at the mouth of theriver Teign prevents ships from enteringthe port apart from when
Solving general differential equations is a large subject, so for sixth form mechanics the types of differential equations considered are limited to a subset of equations which fit standard forms. Equations (1) and (2) are linear second order differential equations with constant coefficients. To
Creating a table of contents The Insert Index/Table window (Figure 1) has five tabs. All of them can be used when creating a table of contents: Use the Index/Table tab to set the attributes of the table of contents. Use the Entries and Styles tabs to format the entries in the table of contents. Use the Background tab to add color or a graphic to the background of the table of
Creating a table of contents The Insert/Index Table window has five tabs. Four of them are used when creating a table of contents: Use the Index/Table tab to set the table's attributes. Use the Entries and Styles tabs to format the table entries. Use the Background tab to add color or a graphic to the table background. The next four sections of this chapter tell you how to use each . /p div class "b_factrow b_twofr" div class "b_vlist2col" ul li div strong File Size: /strong 554KB /div /li /ul ul li div strong Page Count: /strong 15 /div /li /ul /div /div /div
Word. Modifying the appearance To change how the table of contents looks – font type, size, indentation etc. – click in the table and on Table of Contents on the References tab, then choose Custom Table of Contents again. In the Table of Contents dialog box, click the Modify button to
The TABLE OF CONTENTS will have shifted. If you need to re-insert the TABLE OF CONTENTS, this margin fix will not stay in place. You will have to follow the temporary fix again OR make a permanent fix to the TABLE OF CONTENTS. Permanent fix: Place your cursor at the first entry in the TABLE OF CONTENTS.
quantized, the resulting energy spectra exhibit resonance peaks characteristic of the electronic structure for atoms at the sample surface. While the x-rays may penetrate deep into the sample, the . by an electron from the L shell coupled with the ejection of an electron from an L shell. Unlike the photoelectric peaks, the kinetic energy of .
single Experiment Frame. Other times, the data is best maintained as individual files. In either case the files can be opened in one action. Both the Open and Merge and Open options on the File menu (Figure 1) offer a File Dialog window and by selecting a range of files, the action associated with the window will be applied to the selected .
Figure 6 is a scatter plot for the normalized peak areas of the non-PAA peaks against the saturated C 1s peak from the pure PAA model. The peaks at 288.4 eV and 285 eV have areas that are anti-correlated with the pure PAA model (all the peak areas within the PAA model are constrained to one
Python is a general-purpose, interpreted high-level programming language. Its syntax is clear and emphasize readability. Python has a large and comprehensive standard library. Python supports multiple programming paradigms, primarily but not limited to object-oriented, imperative and, to a lesser extent, functional programming .
