From a13e302260d604ca791a468bea0358628b30ed1a Mon Sep 17 00:00:00 2001 From: Jack Halford Date: Tue, 4 Jun 2024 23:02:21 +0200 Subject: [PATCH] textract all of book 1 after page 40 --- .gitignore | 1 - 1/10-mechanical-similarity.md | 8 + 1/10_problems.md | 10 - 1/11-motion-in-one-dimension.md | 35 +- 1/11_problems.md | 31 - 1/13-the-reduced-mass.md | 12 +- 1/13_problem.md | 11 - 1/14-motion-in-a-central-field.md | 159 + 1/15-keplers-problem.md | 216 + 1/16-disintegration-of-particles.md | 408 ++ 1/17-elastic-collisions.md | 140 + 1/18-scattering.md | 117 + 1/19-rutherfords-formula.md | 0 ...he-lagrangian-for-a-system-of-particles.md | 76 +- 1/5_problems.md | 72 - 1/7-momentum.md | 12 + 1/7_problem.md | 15 - 1/8-centre-of-mass.md | 9 + 1/8_problem.md | 11 - 1/9-angular-momentum.md | 28 + 1/9_problems.md | 30 - 1/index.md | 72 +- pdfs/1-mechanics.pdf | Bin 0 -> 8088918 bytes textract/1-mechanics-15-keplers-problem.txt | 591 ++ ...chanics-16-disintegration-of-particles.txt | 127 + 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5ea1836..1fcb152 100644 --- a/.gitignore +++ b/.gitignore @@ -1,2 +1 @@ out -*.pdf diff --git a/1/10-mechanical-similarity.md b/1/10-mechanical-similarity.md index c138683..e385adc 100644 --- a/1/10-mechanical-similarity.md +++ b/1/10-mechanical-similarity.md @@ -82,3 +82,11 @@ which express $\bar{U}$ and $\bar{T}$ in terms of the total energy of the system In particular, for small oscillations ($k=2$) we have $\bar{T} = \bar{U}$, i.e. the mean values of the kinetic and potential energies are equal. For a Newtonian interaction $(k = - 1)$, $2\bar{T} = - \bar{U}$, and $E=-\bar{T}$, in accordance with the fact that, in such an interaction, the motion takes place in a finite region of space only if the total energy is negative (see `1/15`). [^1]: The expression on the right of `1/10.5` is sometimes called the virial of the system. + + + + + + + + diff --git a/1/10_problems.md b/1/10_problems.md deleted file mode 100644 index 5c68f33..0000000 --- a/1/10_problems.md +++ /dev/null @@ -1,10 +0,0 @@ ---- ---- - -PROBLEMS -PROBLEM 1. Find the ratio of the times in the same path for particles having different -masses but the same potential energy. -SOLUTION. t'/t = (m'/m). -PROBLEM 2. Find the ratio of the times in the same path for particles having the same mass -but potential energies differing by a constant factor. -SOLUTION. I'/t diff --git a/1/11-motion-in-one-dimension.md b/1/11-motion-in-one-dimension.md index 02ab3dd..9d46f74 100644 --- a/1/11-motion-in-one-dimension.md +++ b/1/11-motion-in-one-dimension.md @@ -22,7 +22,8 @@ The equations of motion corresponding to these Lagrangians can be integrated in ``` The two arbitrary constants in the solution of the equations of motion are here represented by the total energy E and the constant of integration. -Since the kinetic energy is essentially positive, the total energy always exceeds the potential energy, i.e. the motion can take place only in those regions of space where U(x) < E. For example, let the function U(x) be of the form shown in 'TODO Fig. 6 (p. 26)'. If we draw in the figure a horizontal line corresponding to a given value of the total energy, we immediately find the possible regions of motion. In the example of Fig. 6, the motion can occur only in the range AB or in the range to the right of C. + +Since the kinetic energy is essentially positive, the total energy always exceeds the potential energy, i.e. the motion can take place only in those regions of space where $U(x) \lt E$. For example, let the function U(x) be of the form shown in 'TODO Fig. 6 (p. 26)'. If we draw in the figure a horizontal line corresponding to a given value of the total energy, we immediately find the possible regions of motion. In the example of Fig. 6, the motion can occur only in the range AB or in the range to the right of C. The points at which the potential energy equals the total energy, ```load @@ -40,3 +41,35 @@ from X1 to X2 (because of the reversibility property, `1/5`) or, by `1/11.3`), ``` where $x_1$ and $x_2$ are roots of equation `1/11.4` for the given value of $E$. This formula gives the period of the motion as a function of the total energy of the particle. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/1/11_problems.md b/1/11_problems.md deleted file mode 100644 index 4f26575..0000000 --- a/1/11_problems.md +++ /dev/null @@ -1,31 +0,0 @@ -PROBLEMS -PROBLEM 1. Determine the period of oscillations of a simple pendulum (a particle of mass -m suspended by a string of length l in a gravitational field) as a function of the amplitude of -the oscillations. -SOLUTION. The energy of the pendulum is E = 1ml2j2-mgl cos = -mgl cos to, where -o is the angle between the string and the vertical, and to the maximum value of . Calculating -the period as the time required to go from = 0 to = Do, multiplied by four, we find --cos -po) -The substitution sin $ = sin 10/sin 100 converts this to T = /(l/g)K(sin 100), where -1/75 -- -is the complete elliptic integral of the first kind. For sin 100 22 100 < 1 (small oscillations), -an expansion of the function K gives -T = -§12 -Determination of the potential energy -27 -The first term corresponds to the familiar formula. -PROBLEM 2. Determine the period of oscillation, as a function of the energy, when a -particle of mass m moves in fields for which the potential energy is -(a) U = Alx -(b) U = Uo/cosh2ax, -U0 0, (c) U = Uotan2ax. -SOLUTION. (a): -T = -By the substitution yn = u the integral is reduced to a beta function, which can be expressed -in terms of gamma functions: -The dependence of T on E is in accordance with the law of mechanical similarity (10.2), -(10.3). -(b) T = (7/a)V(2m/E). -(c) T =(t/a)v[2m/(E+U0)] diff --git a/1/13-the-reduced-mass.md b/1/13-the-reduced-mass.md index 42fa046..6bef1c8 100644 --- a/1/13-the-reduced-mass.md +++ b/1/13-the-reduced-mass.md @@ -34,4 +34,14 @@ where is called the *reduced mass*. The function `1/13.3` is formally identical with the Lagrangian of a particle of mass $m$ moving in an external field $U(\v{r})$ which is symmetrical about a fixed origin. -Thus the problem of the motion of two interacting particles is equivalent to that of the motion of one particle in a given external field $U(\v{r})$. From the solution $\v{r} = \v{r}(t)$ of this problem, the paths $\v{r}_1 = \v{r}_1(t)$ and $\v{r}_2 = \v{r}_2(t)$ of the two particles separately, relative to their common centre of mass, are obtained by means of formulae `1/13.2`. + + + + + + + + + + + diff --git a/1/13_problem.md b/1/13_problem.md deleted file mode 100644 index 242613a..0000000 --- a/1/13_problem.md +++ /dev/null @@ -1,11 +0,0 @@ - -PROBLEM -A system consists of one particle of mass M and n particles with equal masses m. Eliminate -the motion of the centre of mass and so reduce the problem to one involving n particles. -SOLUTION. Let R be the radius vector of the particle of mass M, and Ra (a = 1, 2, ..., n) -those of the particles of mass m. We put ra = Ra-R and take the origin to be at the centre -of mass: MR+mER = 0. Hence where =M + nm; Ra = R + ra. -Substitution in the Lagrangian L = gives -ra. -The potential energy depends only on the distances between the particles, and so can be -written as a function of the ra. diff --git a/1/14-motion-in-a-central-field.md b/1/14-motion-in-a-central-field.md index 5dfe0bc..5f45e33 100644 --- a/1/14-motion-in-a-central-field.md +++ b/1/14-motion-in-a-central-field.md @@ -21,3 +21,162 @@ In the present case, the generalised momentum $p_\phi=mr^2\dot{\phi}$ is the sam ``` This law has a simple geometrical interpretation in the plane motion of a single particle in a central field. The expression $\mfrac{1}{2}r\cdot r\dd{\phi}$ is the area of the sector bounded by two neighbouring radius vectors and an element of the path `1/fig8` + +§14 +Motion in a central field +31 +(Fig. 8). Calling this area df, we can write the angular momentum of the par- +ticle as +M = 2mf, +(14.3) +where the derivative f is called the sectorial velocity. Hence the conservation +of angular momentum implies the constancy of the sectorial velocity: in equal +times the radius vector of the particle sweeps out equal areas (Kepler's second +law).t +rdd +dd +0 +FIG. 8 +The complete solution of the problem of the motion of a particle in a central +field is most simply obtained by starting from the laws of conservation of +energy and angular momentum, without writing out the equations of motion +themselves. Expressing in terms of M from (14.2) and substituting in the +expression for the energy, we obtain +E = = +(14.4) +Hence +(14.5) +or, integrating, +constant. +(14.6) +Writing (14.2) as do = M dt/mr2, substituting dt from (14.5) and integrating, +we find +constant. +(14.7) +Formulae (14.6) and (14.7) give the general solution of the problem. The +latter formula gives the relation between r and , i.e. the equation of the path. +Formula (14.6) gives the distance r from the centre as an implicit function of +time. The angle o, it should be noted, always varies monotonically with time, +since (14.2) shows that & can never change sign. +t The law of conservation of angular momentum for a particle moving in a central field +is sometimes called the area integral. +32 +Integration of the Equations of Motion +§14 +The expression (14.4) shows that the radial part of the motion can be re- +garded as taking place in one dimension in a field where the "effective poten- +tial energy" is +(14.8) +The quantity M2/2mr2 is called the centrifugal energy. The values of r for which +U(r) +(14.9) +determine the limits of the motion as regards distance from the centre. +When equation (14.9) is satisfied, the radial velocity j is zero. This does not +mean that the particle comes to rest as in true one-dimensional motion, since +the angular velocity o is not zero. The value j = 0 indicates a turning point +of the path, where r(t) begins to decrease instead of increasing, or vice versa. +If the range in which r may vary is limited only by the condition r > rmin, +the motion is infinite: the particle comes from, and returns to, infinity. +If the range of r has two limits rmin and rmax, the motion is finite and the +path lies entirely within the annulus bounded by the circles r = rmax and +r = rmin- This does not mean, however, that the path must be a closed curve. +During the time in which r varies from rmax to rmin and back, the radius +vector turns through an angle Ao which, according to (14.7), is given by +Mdr/r2 +(14.10) +The condition for the path to be closed is that this angle should be a rational +fraction of 2n, i.e. that Ao = 2nm/n, where m and n are integers. In that case, +after n periods, the radius vector of the particle will have made m complete +revolutions and will occupy its original position, so that the path is closed. +Such cases are exceptional, however, and when the form of U(r) is arbitrary +the angle is not a rational fraction of 2nr. In general, therefore, the path +of a particle executing a finite motion is not closed. It passes through the +minimum and maximum distances an infinity of times, and after infinite time +it covers the entire annulus between the two bounding circles. The path +shown in Fig. 9 is an example. +There are only two types of central field in which all finite motions take +place in closed paths. They are those in which the potential energy of the +particle varies as 1/r or as r2. The former case is discussed in §15; the latter +is that of the space oscillator (see §23, Problem 3). +At a turning point the square root in (14.5), and therefore the integrands +in (14.6) and (14.7), change sign. If the angle is measured from the direc- +tion of the radius vector to the turning point, the parts of the path on each +side of that point differ only in the sign of for each value of r, i.e. the path +is symmetrical about the line = 0. Starting, say, from a point where = rmax +the particle traverses a segment of the path as far as a point with r rmin, +§14 +Motion in a central field +33 +then follows a symmetrically placed segment to the next point where r = rmax, +and so on. Thus the entire path is obtained by repeating identical segments +forwards and backwards. This applies also to infinite paths, which consist of +two symmetrical branches extending from the turning point (r = rmin) to +infinity. +'max +min +so +FIG. 9 +The presence of the centrifugal energy when M # 0, which becomes +infinite as 1/22 when r -> 0, generally renders it impossible for the particle to +reach the centre of the field, even if the field is an attractive one. A "fall" of +the particle to the centre is possible only if the potential energy tends suffi- +ciently rapidly to -00 as r 0. From the inequality +1mr2 = E- U(r) - M2/2mr2 +or r2U(Y)+M2/2m < Er2, it follows that r can take values tending to zero +only if +(14.11) +i.e. U(r) must tend to - 8 either as - a/r2 with a > M2/2m, or proportionally +to - 1/rn with n > 2. +PROBLEMS +PROBLEM 1. Integrate the equations of motion for a spherical pendulum (a particle of mass +m moving on the surface of a sphere of radius l in a gravitational field). +SOLUTION. In spherical co-ordinates, with the origin at the centre of the sphere and the +polar axis vertically downwards, the Lagrangian of the pendulum is +1ml2(02 + 62 sin20) +mgl cos 0. +2* +34 +Integration of the Equations of Motion +§14 +The co-ordinate is cyclic, and hence the generalised momentum Po, which is the same as the +z-component of angular momentum, is conserved: +(1) +The energy is +E = cos 0 +(2) += 0. +Hence +(3) +where the "effective potential energy" is +Ueff(0) = COS 0. +For the angle o we find, using (1), +do +(4) +The integrals (3) and (4) lead to elliptic integrals of the first and third kinds respectively. +The range of 0 in which the motion takes place is that where E > Ueff, and its limits +are given by the equation E = Uell. This is a cubic equation for cos 0, having two roots +between -1 and +1; these define two circles of latitude on the sphere, between which the +path lies. +PROBLEM 2. Integrate the equations of motion for a particle moving on the surface of a +cone (of vertical angle 2x) placed vertically and with vertex downwards in a gravitational +field. +SOLUTION. In spherical co-ordinates, with the origin at the vertex of the cone and the +polar axis vertically upwards, the Lagrangian is sin2x) -mgr cos a. The co- +ordinate is cyclic, and Mz = mr2 sin²a is again conserved. The energy is += a. +By the same method as in Problem 1, we find +== +The condition E = Ueff(r) is (if M + 0) a cubic equation for r, having two positive roots; +these define two horizontal circles on the cone, between which the path lies. +PROBLEM 3. Integrate the equations of motion for a pendulum of mass M2, with a mass M1 +at the point of support which can move on a horizontal line lying in the plane in which m2 +moves (Fig. 2, §5). +SOLUTION. In the Lagrangian derived in §5, Problem 2, the co-ordinate x is cyclic. The +generalised momentum Px, which is the horizontal component of the total momentum of the +system, is therefore conserved +Px = cos = constant. +(1) +The system may always be taken to be at rest as a whole. Then the constant in (1) is zero +and integration gives +(m1+m2)x+m2) sin = constant, +(2) +which expresses the fact that the centre of mass of the system does not move horizontally. diff --git a/1/15-keplers-problem.md b/1/15-keplers-problem.md new file mode 100644 index 0000000..9adda1a --- /dev/null +++ b/1/15-keplers-problem.md @@ -0,0 +1,216 @@ +§15 +Kepler's problem +35 +Using (1), we find the energy in the form +E +(3) +Hence +Expressing the co-ordinates X2 sin o, y = l cos of the particle m2 in terms of +by means of (2), we find that its path is an arc of an ellipse with horizontal semi- +axis lm1/(m1+m2) and vertical semi-axis l. As M1 8 we return to the familiar simple pen- +dulum, which moves in an arc of a circle. +$15. Kepler's problem +An important class of central fields is formed by those in which the poten- +tial energy is inversely proportional to r, and the force accordingly inversely +proportional to r2. They include the fields of Newtonian gravitational attrac- +tion and of Coulomb electrostatic interaction; the latter may be either attrac- +tive or repulsive. +Let us first consider an attractive field, where +U=-a/r +(15.1) +with a a positive constant. The "effective" potential energy +(15.2) +is +of the form shown in Fig. 10. As r 0, Ueff tends to + 00, and as +r +8 +it tends to zero from negative values ; for r = M2/ma it has a minimum value +Ueff, min = -mx2/2M2. +(15.3) +Ueff +FIG. 10 +It is seen at once from Fig. 10 that the motion is finite for E <0 and infinite +for E > 0. +36 +Integration of the Equations of Motion +§15 +The shape of the path is obtained from the general formula (14.7). Substi- +tuting there U = - a/r and effecting the elementary integration, we have +o = +- constant. +Taking the origin of such that the constant is zero, and putting +P = M2/ma, e= [1 1+(2EM2/mo2)] +(15.4) +we can write the equation of the path as +p/r = 1+e coso. +(15.5) +This is the equation of a conic section with one focus at the origin; 2p is called +the latus rectum of the orbit and e the eccentricity. Our choice of the origin of +is seen from (15.5) to be such that the point where = 0 is the point nearest +to the origin (called the perihelion). +In the equivalent problem of two particles interacting according to the law +(15.1), the orbit of each particle is a conic section, with one focus at the centre +of mass of the two particles. +It is seen from (15.4) that, if E < 0, then the eccentricity e < 1, i.e. the +orbit is an ellipse (Fig. 11) and the motion is finite, in accordance with what +has been said earlier in this section. According to the formulae of analytical +geometry, the major and minor semi-axes of the ellipse are +a = p/(1-e2) = a2E, b =p/v(1-e2)Mv(2mE) = +(15.6) +y +X +2b +ae +2a +FIG. 11 +The least possible value of the energy is (15.3), and then e = 0, i.e. the ellipse +becomes a circle. It may be noted that the major axis of the ellipse depends +only on the energy of the particle, and not on its angular momentum. The +least and greatest distances from the centre of the field (the focus of the +ellipse) are +rmin = =p/(1+e)=a(1-e), +max=p(1-e)=a(1+e). = = (15.7) +These expressions, with a and e given by (15.6) and (15.4), can, of course, +also be obtained directly as the roots of the equation Ueff(r) = E. +§15 +Kepler's problem +37 +The period T of revolution in an elliptical orbit is conveniently found by +using the law of conservation of angular momentum in the form of the area +integral (14.3). Integrating this equation with respect to time from zero to +T, we have 2mf = TM, where f is the area of the orbit. For an ellipse +f = nab, and by using the formulae (15.6) we find +T = 2ma3/2-(m/a) += ma((m2E3). +(15.8) +The proportionality between the square of the period and the cube of the +linear dimension of the orbit has already been demonstrated in §10. It may +also be noted that the period depends only on the energy of the particle. +For E > 0 the motion is infinite. If E > 0, the eccentricity e > 1, i.e. the +the path is a hyperbola with the origin as internal focus (Fig. 12). The dis- +tance of the perihelion from the focus is +rmin ==pl(e+1)=a(e-1), = = +(15.9) +where a = p/(e2-1) = a/2E is the "semi-axis" of the hyperbola. +y +p +ale-1) +FIG. 12 +If E = 0, the eccentricity e = 1, and the particle moves in a parabola with +perihelion distance rmin = 1p. This case occurs if the particle starts from rest +at infinity. +The co-ordinates of the particle as functions of time in the orbit may be +found by means of the general formula (14.6). They may be represented in a +convenient parametric form as follows. +Let us first consider elliptical orbits. With a and e given by (15.6) and (15.4) +we can write the integral (14.6) for the time as +t += += +38 +Integration of the Equations of Motion +§15 +The obvious substitution r-a = - ae cos $ converts the integral to +sioant +If time is measured in such a way that the constant is zero, we have the +following parametric dependence of r on t: +r = a(1-e cos ), t = +(15.10) +the particle being at perihelion at t = 0. The Cartesian co-ordinates +x = r cos o, y = r sin (the x and y axes being respectively parallel to the +major and minor axes of the ellipse) can likewise be expressed in terms of +the parameter $. From (15.5) and (15.10) we have +ex = = = +y is equal to W(r2-x2). Thus +x = a(cos & - e), +y = =av(1-e2) $. +(15.11) +A complete passage round the ellipse corresponds to an increase of $ from 0 +to 2nr. +Entirely similar calculations for the hyperbolic orbits give +r = a(e cosh & - 1), t = V(ma3/a)(esinh - - $), +(15.12) +x = a(e-cosh ) y = a1/(e2-1)sinh & +where the parameter $ varies from - 00 to + 00. +Let us now consider motion in a repulsive field, where +U +(a>0). +(15.13) +Here the effective potential energy is +Utt +and decreases monotonically from + 00 to zero as r varies from zero to +infinity. The energy of the particle must be positive, and the motion is always +infinite. The calculations are exactly similar to those for the attractive field. +The path is a hyperbola (or, if E = 0, a parabola): +pr r = =1-e coso, = +(15.14) +where P and e are again given by (15.4). The path passes the centre of the +field in the manner shown in Fig. 13. The perihelion distance is +rmin =p(e-1)=a(e+1). = +(15.15) +The time dependence is given by the parametric equations += =(ma3/a)(esinh+) = +(15.16) +x += a(cosh & e , +y = av((e2-1) sinh & +§15 +Kepler's problem +39 +To conclude this section, we shall show that there is an integral of the mo- +tion which exists only in fields U = a/r (with either sign of a). It is easy to +verify by direct calculation that the quantity +vxM+ar/r +(15.17) +is constant. For its total time derivative is v +since M = mr xv, +Putting mv = ar/r3 from the equation of motion, we find that this expression +vanishes. +y +0 +(I+e) +FIG. 13 +The direction of the conserved vector (15.17) is along the major axis from +the focus to the perihelion, and its magnitude is ae. This is most simply +seen by considering its value at perihelion. +It should be emphasised that the integral (15.17) of the motion, like M and +E, is a one-valued function of the state (position and velocity) of the particle. +We shall see in §50 that the existence of such a further one-valued integral +is due to the degeneracy of the motion. +PROBLEMS +PROBLEM 1. Find the time dependence of the co-ordinates of a particle with energy E = 0 +moving in a parabola in a field U = -a/r. +SOLUTION. In the integral +we substitute r = M2(1+n2)/2ma = 1p(1+n2), obtaining the following parametric form of +the required dependence: +r=1p(1+n2), +t= +y=pn. +40 +Integration of the Equations of Motion +§15 +The parameter n varies from - 00 to +00. +PROBLEM 2. Integrate the equations of motion for a particle in a central field +U = - a/r2 (a > 0). +SOLUTION. From formulae (14.6) and (14.7) we have, if and t are appropriately measured, +(a) for E > andM 0 and +(b) for E>0 0nd and M 2/2m a, +(c) for E <0 and Ms1 +In all three cases +In cases (b) and(c) the particle"falls"to the centre along a path which approaches the +origin as +00. The fall from a given value of r takes place in a finite time, namely +PROBLEM 3. When a small correction SU(r) is added to the potential energy U = -a/r, +the paths of finite motion are no longer closed, and at each revolution the perihelion is dis- +placed through a small angle so. Find 80 when (a) SU = B/r2, (b) SU = y/r3. +SOLUTION. When r varies from rmin to rmax and back, the angle varies by an amount +(14.10), which we write as +in order to avoid the occurrence of spurious divergences. We put U= - -a/r+8U, and +expand the integrand in powers of SU; the zero-order term in the expansion gives 2nr, and +the first-order term gives the required change so: +(1) +where we have changed from the integration over r to one over , along the path of the "un- +perturbed" motion. +In case (a), the integration in (1) is trivial: 80 = -2nBm/M2 = -2nB/ap, where 2p (15.4) +is the latus rectum of the unperturbed ellipse. In case (b) r2SU = y/r and, with 1/r/given by +(15.5), we have 80 = -6naym2/M4 = -6ny/ap2. diff --git a/1/16-disintegration-of-particles.md b/1/16-disintegration-of-particles.md new file mode 100644 index 0000000..83fe807 --- /dev/null +++ b/1/16-disintegration-of-particles.md @@ -0,0 +1,408 @@ +§16. Disintegration of particles +IN many cases the laws of conservation of momentum and energy alone can +be used to obtain important results concerning the properties of various mech- +anical processes. It should be noted that these properties are independent of +the particular type of interaction between the particles involved. +Let us consider a "spontaneous" disintegration (that is, one not due to +external forces) of a particle into two "constituent parts", i.e. into two other +particles which move independently after the disintegration. +This process is most simply described in a frame of reference in which the +particle is at rest before the disintegration. The law of conservation of momen- +tum shows that the sum of the momenta of the two particles formed in the +disintegration is then zero; that is, the particles move apart with equal and +opposite momenta. The magnitude Po of either momentum is given by the +law of conservation of energy: +here M1 and m2 are the masses of the particles, E1t and E2i their internal +energies, and E the internal energy of the original particle. If € is the "dis- +integration energy", i.e. the difference +E= E:-E11-E2i, +(16.1) +which must obviously be positive, then +(16.2) +which determines Po; here m is the reduced mass of the two particles. The +velocities are V10 = Po/m1, V20 = Po/m2. +Let us now change to a frame of reference in which the primary particle +moves with velocity V before the break-up. This frame is usually called the +laboratory system, or L system, in contradistinction to the centre-of-mass +system, or C system, in which the total momentum is zero. Let us consider +one of the resulting particles, and let V and V0 be its velocities in the L and +the C system respectively. Evidently V = V+vo, or V -V = V0, and SO +(16.3) +where 0 is the angle at which this particle moves relative to the direction of +the velocity V. This equation gives the velocity of the particle as a function +41 +42 +Collisions Between Particles +§16 +of its direction of motion in the L system. In Fig. 14 the velocity V is repre- +sented by a vector drawn to any point on a circle+ of radius vo from a point +A at a distance V from the centre. The cases V < vo and V>00 are shown +in Figs. 14a, b respectively. In the former case 0 can have any value, but in +the latter case the particle can move only forwards, at an angle 0 which does +not exceed Omax, given by +(16.4) +this is the direction of the tangent from the point A to the circle. +C +V +V +VO +VO +max +oo +oo +A +V +A +V +(a) VVo +FIG. 14 +The relation between the angles 0 and Oo in the L and C systems is evi- +dently (Fig. 14) +tan +(16.5) +If this equation is solved for cos Oo, we obtain +V +0 +(16.6) +For vo > V the relation between Oo and 0 is one-to-one (Fig. 14a). The plus +sign must be taken in (16.6), so that Oo = 0 when 0 = 0. If vo < V, however, +the relation is not one-to-one: for each value of 0 there are two values of Oo, +which correspond to vectors V0 drawn from the centre of the circle to the +points B and C (Fig. 14b), and are given by the two signs in (16.6). +In physical applications we are usually concerned with the disintegration +of not one but many similar particles, and this raises the problem of the +distribution of the resulting particles in direction, energy, etc. We shall +assume that the primary particles are randomly oriented in space, i.e. iso- +tropically on average. +t More precisely, to any point on a sphere of radius vo, of which Fig. 14 shows a diametral +section. +§16 +Disintegration of particles +43 +In the C system, this problem is very easily solved: every resulting particle +(of a given kind) has the same energy, and their directions of motion are +isotropically distributed. The latter fact depends on the assumption that the +primary particles are randomly oriented, and can be expressed by saying +that the fraction of particles entering a solid angle element doo is proportional +to doo, i.e. equal to doo/4rr. The distribution with respect to the angle Oo is +obtained by putting doo = 2m sin Oo doo, i.e. the corresponding fraction is +(16.7) +The corresponding distributions in the L system are obtained by an +appropriate transformation. For example, let us calculate the kinetic energy +distribution in the L system. Squaring the equation V = V0 + V, we have +2.2 = V02 + +200 cos Oo, whence d(cos 00) = d(v2)/200V. Using the +kinetic energy T = 1mv2, where m is m1 or M2 depending on which kind of +particle is under consideration, and substituting in (16.7), we find the re- +quired distribution: +(1/2mvov) dT. +(16.8) +The kinetic energy can take values between Tmin = 3m(e0-V)2 and +Tmax = 1m(vo+V)2. The particles are, according to (16.8), distributed +uniformly over this range. +When a particle disintegrates into more than two parts, the laws of con- +servation of energy and momentum naturally allow considerably more free- +dom as regards the velocities and directions of motion of the resulting particles. +In particular, the energies of these particles in the C system do not have +determinate values. There is, however, an upper limit to the kinetic energy +of any one of the resulting particles. To determine the limit, we consider +the system formed by all these particles except the one concerned (whose +mass is M1, say), and denote the "internal energy" of that system by Ei'. +Then the kinetic energy of the particle M1 is, by (16.1) and (16.2), +T10 = 2/2m1 = (M-m1)(E:-E1-EiM where M is the mass of the +primary particle. It is evident that T10 has its greatest possible value +when E/' is least. For this to be so, all the resulting particles except M1 +must be moving with the same velocity. Then Ei is simply the sum of their +internal energies, and the difference E;-E-E; is the disintegration +energy E. Thus +T10,max = (M - M1) E M. +(16.9) +PROBLEMS +PROBLEM 1. Find the relation between the angles 01, O2 (in the L system) after a disintegra- +tion into two particles. +SOLUTION. In the C system, the corresponding angles are related by 010 = n-020. Calling +010 simply Oo and using formula (16.5) for each of the two particles, we can put +7+10 cos Oo = V10 sin Oo cot 01, V-V20 cos Oo = V20 sin Oo cot O2. From these two +equations we must eliminate Oo. To do so, we first solve for cos Oo and sin Oo, and then +44 +Collisions Between Particles +§17 +form the sum of their squares, which is unity. Since V10/2.20 = m2/m1, we have finally, using +(16.2), +(m2/m1) sin202+(m1/m2) sin201-2 sin O1 sin O2 cos(A1+6 +PROBLEM 2. Find the angular distribution of the resulting particles in the L system. +SOLUTION. When vo > V, we substitute (16.6), with the plus sign of the radical, in (16.7), +obtaining +(0 . +When vo < V, both possible relations between Oo and 0 must be taken into account. Since, +when 0 increases, one value of Oo increases and the other decreases, the difference (not the +sum) of the expressions d cos Oo with the two signs of the radical in (16.6) must be taken. +The result is +(0 max). +PROBLEM 3. Determine the range of possible values of the angle 0 between the directions +of motion of the two resulting particles in the L system. +SOLUTION. The angle 0 = 01+02, where 01 and O2 are the angles defined by formula (16.5) +(see Problem 1), and it is simplest to calculate the tangent of 0. A consideration of the extrema +of the resulting expression gives the following ranges of 0, depending on the relative magni- +tudes of V, V10 and V20 (for definiteness, we assume V20 > V10): 0 0 77 if V10 +< 0 < TT if V V V10, 0 < Oo if V > U20. The value of Oo is given by +sin = +§17. Elastic collisions +A collision between two particles is said to be elastic if it involves no change +in their internal state. Accordingly, when the law of conservation of energy +is applied to such a collision, the internal energy of the particles may be +neglected. +The collision is most simply described in a frame of reference in which the +centre of mass of the two particles is at rest (the C system). As in $16, we +distinguish by the suffix 0 the values of quantities in that system. The velo- +cities of the particles before the collision are related to their velocities V1 and +V2 in the laboratory system by V10 = M2V/(m1+m2), V20 = -m1V/(m1+m2), +where V = V1-V2; see (13.2). +Because of the law of conservation of momentum, the momenta of the two +particles remain equal and opposite after the collision, and are also unchanged +in magnitude, by the law of conservation of energy. Thus, in the C system +the collision simply rotates the velocities, which remain opposite in direction +and unchanged in magnitude. If we denote by no a unit vector in the direc- +tion of the velocity of the particle M1 after the collision, then the velocities +of the two particles after the collision (distinguished by primes) are +V10' m20120/(m1+m2), V20' = -mjono/(m1+m2). +(17.1) +§17 +Elastic collisions +45 +In order to return to the L system, we must add to these expressions the +velocity V of the centre of mass. The velocities in the L system after the +collision are therefore +V1' = +(17.2) +V2' = +No further information about the collision can be obtained from the laws +of conservation of momentum and energy. The direction of the vector no +depends on the law of interaction of the particles and on their relative position +during the collision. +The results obtained above may be interpreted geometrically. Here it is +more convenient to use momenta instead of velocities. Multiplying equations +(17.2) by M1 and M2 respectively, we obtain +(17.3) +P2' muno+m2(p1+p2)/(m1+m2) +where m = m1m2/(m1+m2) is the reduced mass. We draw a circle of radius +mv and use the construction shown in Fig. 15. If the unit vector no is along +OC, the vectors AC and CB give the momenta P1' and P2' respectively. +When p1 and P2 are given, the radius of the circle and the points A and B +are fixed, but the point C may be anywhere on the circle. +C +p' +no +P'2 +B +A +FIG. 15 +Let us consider in more detail the case where one of the particles (m2, say) is +at rest before the collision. In that case the distance OB = m2p1/(m1+m2) = mv +is equal to the radius, i.e. B lies on the circle. The vector AB is equal to the +momentum P1 of the particle M1 before the collision. The point A lies inside +or outside the circle, according as M1 < M2 or M1 > M2. The corresponding +46 +Collisions Between Particles +§17 +diagrams are shown in Figs. 16a, b. The angles 01 and O2 in these diagrams +are the angles between the directions of motion after the collision and the +direction of impact (i.e. of P1). The angle at the centre, denoted by X, which +gives the direction of no, is the angle through which the direction of motion +of m1 is turned in the C system. It is evident from the figure that 01 and O2 +can be expressed in terms of X by +(17.4) +C +p' +P2 +pi +P2 +0 +max +10, +X +O2 +O2 +B +B +A +0 +A +Q +0 +(a) m < m2 +(b) m, m m m +AB=p : AO/OB= m/m2 +FIG. 16 +We may give also the formulae for the magnitudes of the velocities of the +two particles after the collision, likewise expressed in terms of X: +ib +(17.5) +The sum A1 + O2 is the angle between the directions of motion of the +particles after the collision. Evidently 01 + O2 > 1/1 if M1 < M2, and 01+O2 < 1st +if M1 > M2. +When the two particles are moving afterwards in the same or in opposite +directions (head-on collision), we have X=TT, i.e. the point C lies on the +diameter through A, and is on OA (Fig. 16b ; P1' and P2' in the same direc- +tion) or on OA produced (Fig. 16a; P1' and P2' in opposite directions). +In this case the velocities after the collision are +(17.6) +This value of V2' has the greatest possible magnitude, and the maximum +§17 +Elastic collisions +47 +energy which can be acquired in the collision by a particle originally at rest +is therefore +(17.7) +where E1 = 1M1U12 is the initial energy of the incident particle. +If M1 < M2, the velocity of M1 after the collision can have any direction. +If M1 > M2, however, this particle can be deflected only through an angle +not exceeding Omax from its original direction; this maximum value of A1 +corresponds to the position of C for which AC is a tangent to the circle +(Fig. 16b). Evidently +sin Omax = OC|OA = M2/M1. +(17.8) +The collision of two particles of equal mass, of which one is initially at +rest, is especially simple. In this case both B and A lie on the circle (Fig. 17). +C +p' +P2 +Q2 +B +A +0 +FIG. 17 +Then +01=1x, +A2 = 1(-x), +(17.9) +12 += +(17.10) +After the collision the particles move at right angles to each other. +PROBLEM +Express the velocity of each particle after a collision between a moving particle (m1) and +another at rest (m2) in terms of their directions of motion in the L system. +SOLUTION. From Fig. 16 we have P2' = 20B cos O2 or V2' = 2v(m/m2) cos O2. The momen- +tum P1' = AC is given by OC2 = AO2tp12-2AO.p cos or +Hence +for m1 > M2 the radical may have either sign, but for M2 > M1 it must be taken positive. +48 +Collisions Between Particles +§18 +§18. Scattering +As already mentioned in §17, a complete calculation of the result of a +collision between two particles (i.e. the determination of the angle x) requires +the solution of the equations of motion for the particular law of interaction +involved. +We shall first consider the equivalent problem of the deflection of a single +particle of mass m moving in a field U(r) whose centre is at rest (and is at +the centre of mass of the two particles in the original problem). +As has been shown in $14, the path of a particle in a central field is sym- +metrical about a line from the centre to the nearest point in the orbit (OA +in Fig. 18). Hence the two asymptotes to the orbit make equal angles (o, +say) with this line. The angle X through which the particle is deflected as it +passes the centre is seen from Fig. 18 to be +X = -200. +(18.1) +A +X +to +FIG. 18 +The angle do itself is given, according to (14.7), by +(M/r2) dr +(18.2) +taken between the nearest approach to the centre and infinity. It should be +recalled that rmin is a zero of the radicand. +For an infinite motion, such as that considered here, it is convenient to +use instead of the constants E and M the velocity Voo of the particle at infinity +and the impact parameter p. The latter is the length of the perpendicular +from the centre O to the direction of Voo, i.e. the distance at which the particle +would pass the centre if there were no field of force (Fig. 18). The energy +and the angular momentum are given in terms of these quantities by +E = 1mvoo², +M = mpVoo, +(18.3) +§18 +Scattering +49 +and formula (18.2) becomes +dr +(18.4) +Together with (18.1), this gives X as a function of p. +In physical applications we are usually concerned not with the deflection +of a single particle but with the scattering of a beam of identical particles +incident with uniform velocity Voo on the scattering centre. The different +particles in the beam have different impact parameters and are therefore +scattered through different angles X. Let dN be the number of particles +scattered per unit time through angles between X and X + dx. This number +itself is not suitable for describing the scattering process, since it is propor- +tional to the density of the incident beam. We therefore use the ratio +do = dN/n, +(18.5) +where n is the number of particles passing in unit time through unit area of +the beam cross-section (the beam being assumed uniform over its cross- +section). This ratio has the dimensions of area and is called the effective +scattering cross-section. It is entirely determined by the form of the scattering +field and is the most important characteristic of the scattering process. +We shall suppose that the relation between X and P is one-to-one; this is +so if the angle of scattering is a monotonically decreasing function of the +impact parameter. In that case, only those particles whose impact parameters +lie between p(x) and p(x) + dp(x) are scattered at angles between X and ++ dx. The number of such particles is equal to the product of n and the +area between two circles of radii P and p+dp, i.e. dN = 2mp dp n. The +effective cross-section is therefore +do = 2mp dp. +(18.6) +In order to find the dependence of do on the angle of scattering, we need +only rewrite (18.6) as +do = 2(x)|dp(x)/dx|dx +(18.7) +Here we use the modulus of the derivative dp/dx, since the derivative may +be (and usually is) negative. t Often do is referred to the solid angle element +do instead of the plane angle element dx. The solid angle between cones +with vertical angles X and x+dx is do = 2n sin x dx. Hence we have from +(18.7) +do. +(18.8) +t If the function p(x) is many-valued, we must obviously take the sum of such expressions +as (18.7) over all the branches of this function. +50 +Collisions Between Particles +§18 +Returning now to the problem of the scattering of a beam of particles, not +by a fixed centre of force, but by other particles initially at rest, we can say +that (18.7) gives the effective cross-section as a function of the angle of +scattering in the centre-of-mass system. To find the corresponding expression +as a function of the scattering angle 0 in the laboratory system, we must +express X in (18.7) in terms of 0 by means of formulae (17.4). This gives +expressions for both the scattering cross-section for the incident beam of +particles (x in terms of 01) and that for the particles initially at rest (x in terms +of O2). +PROBLEMS +PROBLEM 1. Determine the effective cross-section for scattering of particles from a perfectly +rigid sphere of radius a (i.e. when the interaction is such that U = 8 for r < a and U = 0 +for r>a). +SOLUTION. Since a particle moves freely outside the sphere and cannot penetrate into it, +the path consists of two straight lines symmetrical about the radius to the point where the +particle strikes the sphere (Fig. 19). It is evident from Fig. 19 that +a sin to = a sin 1(-x) = a cos 1x. +A +to +p +& +FIG. 19 +Substituting in (18.7) or (18.8), we have +do = 1ma2 sin X do, +(1) +i.e. the scattering is isotropic in the C system. On integrating do over all angles, we find that +the total cross-section o = na2, in accordance with the fact that the "impact area" which the +particle must strike in order to be scattered is simply the cross-sectional area of the sphere. +In order to change to the L system, X must be expressed in terms of 01 by (17.4). The +calculations are entirely similar to those of $16, Problem 2, on account of the formal resemb- +lance between formulae (17.4) and (16.5). For M1 < m2 (where M1 is the mass of the particle +and m2 that of the sphere) we have +do1, +where do1 = 2n sin 01 d01. If, on the other hand, M2 < M1, then +For m1 = M2, we have do = a²|cos 01| do1, which can also be obtained directly by sub- +stituting X = 201 from (17.9) in (1). diff --git a/1/17-elastic-collisions.md b/1/17-elastic-collisions.md new file mode 100644 index 0000000..cae51e6 --- /dev/null +++ b/1/17-elastic-collisions.md @@ -0,0 +1,140 @@ +§17. Elastic collisions +A collision between two particles is said to be elastic if it involves no change +in their internal state. Accordingly, when the law of conservation of energy +is applied to such a collision, the internal energy of the particles may be +neglected. +The collision is most simply described in a frame of reference in which the +centre of mass of the two particles is at rest (the C system). As in $16, we +distinguish by the suffix 0 the values of quantities in that system. The velo- +cities of the particles before the collision are related to their velocities V1 and +V2 in the laboratory system by V10 = M2V/(m1+m2), V20 = -m1V/(m1+m2), +where V = V1-V2; see (13.2). +Because of the law of conservation of momentum, the momenta of the two +particles remain equal and opposite after the collision, and are also unchanged +in magnitude, by the law of conservation of energy. Thus, in the C system +the collision simply rotates the velocities, which remain opposite in direction +and unchanged in magnitude. If we denote by no a unit vector in the direc- +tion of the velocity of the particle M1 after the collision, then the velocities +of the two particles after the collision (distinguished by primes) are +V10' m20120/(m1+m2), V20' = -mjono/(m1+m2). +(17.1) +§17 +Elastic collisions +45 +In order to return to the L system, we must add to these expressions the +velocity V of the centre of mass. The velocities in the L system after the +collision are therefore +V1' = +(17.2) +V2' = +No further information about the collision can be obtained from the laws +of conservation of momentum and energy. The direction of the vector no +depends on the law of interaction of the particles and on their relative position +during the collision. +The results obtained above may be interpreted geometrically. Here it is +more convenient to use momenta instead of velocities. Multiplying equations +(17.2) by M1 and M2 respectively, we obtain +(17.3) +P2' muno+m2(p1+p2)/(m1+m2) +where m = m1m2/(m1+m2) is the reduced mass. We draw a circle of radius +mv and use the construction shown in Fig. 15. If the unit vector no is along +OC, the vectors AC and CB give the momenta P1' and P2' respectively. +When p1 and P2 are given, the radius of the circle and the points A and B +are fixed, but the point C may be anywhere on the circle. +C +p' +no +P'2 +B +A +FIG. 15 +Let us consider in more detail the case where one of the particles (m2, say) is +at rest before the collision. In that case the distance OB = m2p1/(m1+m2) = mv +is equal to the radius, i.e. B lies on the circle. The vector AB is equal to the +momentum P1 of the particle M1 before the collision. The point A lies inside +or outside the circle, according as M1 < M2 or M1 > M2. The corresponding +46 +Collisions Between Particles +§17 +diagrams are shown in Figs. 16a, b. The angles 01 and O2 in these diagrams +are the angles between the directions of motion after the collision and the +direction of impact (i.e. of P1). The angle at the centre, denoted by X, which +gives the direction of no, is the angle through which the direction of motion +of m1 is turned in the C system. It is evident from the figure that 01 and O2 +can be expressed in terms of X by +(17.4) +C +p' +P2 +pi +P2 +0 +max +10, +X +O2 +O2 +B +B +A +0 +A +Q +0 +(a) m < m2 +(b) m, m m m +AB=p : AO/OB= m/m2 +FIG. 16 +We may give also the formulae for the magnitudes of the velocities of the +two particles after the collision, likewise expressed in terms of X: +ib +(17.5) +The sum A1 + O2 is the angle between the directions of motion of the +particles after the collision. Evidently 01 + O2 > 1/1 if M1 < M2, and 01+O2 < 1st +if M1 > M2. +When the two particles are moving afterwards in the same or in opposite +directions (head-on collision), we have X=TT, i.e. the point C lies on the +diameter through A, and is on OA (Fig. 16b ; P1' and P2' in the same direc- +tion) or on OA produced (Fig. 16a; P1' and P2' in opposite directions). +In this case the velocities after the collision are +(17.6) +This value of V2' has the greatest possible magnitude, and the maximum +§17 +Elastic collisions +47 +energy which can be acquired in the collision by a particle originally at rest +is therefore +(17.7) +where E1 = 1M1U12 is the initial energy of the incident particle. +If M1 < M2, the velocity of M1 after the collision can have any direction. +If M1 > M2, however, this particle can be deflected only through an angle +not exceeding Omax from its original direction; this maximum value of A1 +corresponds to the position of C for which AC is a tangent to the circle +(Fig. 16b). Evidently +sin Omax = OC|OA = M2/M1. +(17.8) +The collision of two particles of equal mass, of which one is initially at +rest, is especially simple. In this case both B and A lie on the circle (Fig. 17). +C +p' +P2 +Q2 +B +A +0 +FIG. 17 +Then +01=1x, +A2 = 1(-x), +(17.9) +12 += +(17.10) +After the collision the particles move at right angles to each other. +PROBLEM +Express the velocity of each particle after a collision between a moving particle (m1) and +another at rest (m2) in terms of their directions of motion in the L system. +SOLUTION. From Fig. 16 we have P2' = 20B cos O2 or V2' = 2v(m/m2) cos O2. The momen- +tum P1' = AC is given by OC2 = AO2tp12-2AO.p cos or +Hence +for m1 > M2 the radical may have either sign, but for M2 > M1 it must be taken positive. diff --git a/1/18-scattering.md b/1/18-scattering.md new file mode 100644 index 0000000..76df87a --- /dev/null +++ b/1/18-scattering.md @@ -0,0 +1,117 @@ +§18 +§18. Scattering +As already mentioned in §17, a complete calculation of the result of a +collision between two particles (i.e. the determination of the angle x) requires +the solution of the equations of motion for the particular law of interaction +involved. +We shall first consider the equivalent problem of the deflection of a single +particle of mass m moving in a field U(r) whose centre is at rest (and is at +the centre of mass of the two particles in the original problem). +As has been shown in $14, the path of a particle in a central field is sym- +metrical about a line from the centre to the nearest point in the orbit (OA +in Fig. 18). Hence the two asymptotes to the orbit make equal angles (o, +say) with this line. The angle X through which the particle is deflected as it +passes the centre is seen from Fig. 18 to be +X = -200. +(18.1) +A +X +to +FIG. 18 +The angle do itself is given, according to (14.7), by +(M/r2) dr +(18.2) +taken between the nearest approach to the centre and infinity. It should be +recalled that rmin is a zero of the radicand. +For an infinite motion, such as that considered here, it is convenient to +use instead of the constants E and M the velocity Voo of the particle at infinity +and the impact parameter p. The latter is the length of the perpendicular +from the centre O to the direction of Voo, i.e. the distance at which the particle +would pass the centre if there were no field of force (Fig. 18). The energy +and the angular momentum are given in terms of these quantities by +E = 1mvoo², +M = mpVoo, +(18.3) +§18 +Scattering +49 +and formula (18.2) becomes +dr +(18.4) +Together with (18.1), this gives X as a function of p. +In physical applications we are usually concerned not with the deflection +of a single particle but with the scattering of a beam of identical particles +incident with uniform velocity Voo on the scattering centre. The different +particles in the beam have different impact parameters and are therefore +scattered through different angles X. Let dN be the number of particles +scattered per unit time through angles between X and X + dx. This number +itself is not suitable for describing the scattering process, since it is propor- +tional to the density of the incident beam. We therefore use the ratio +do = dN/n, +(18.5) +where n is the number of particles passing in unit time through unit area of +the beam cross-section (the beam being assumed uniform over its cross- +section). This ratio has the dimensions of area and is called the effective +scattering cross-section. It is entirely determined by the form of the scattering +field and is the most important characteristic of the scattering process. +We shall suppose that the relation between X and P is one-to-one; this is +so if the angle of scattering is a monotonically decreasing function of the +impact parameter. In that case, only those particles whose impact parameters +lie between p(x) and p(x) + dp(x) are scattered at angles between X and ++ dx. The number of such particles is equal to the product of n and the +area between two circles of radii P and p+dp, i.e. dN = 2mp dp n. The +effective cross-section is therefore +do = 2mp dp. +(18.6) +In order to find the dependence of do on the angle of scattering, we need +only rewrite (18.6) as +do = 2(x)|dp(x)/dx|dx +(18.7) +Here we use the modulus of the derivative dp/dx, since the derivative may +be (and usually is) negative. t Often do is referred to the solid angle element +do instead of the plane angle element dx. The solid angle between cones +with vertical angles X and x+dx is do = 2n sin x dx. Hence we have from +(18.7) +do. +(18.8) +t If the function p(x) is many-valued, we must obviously take the sum of such expressions +as (18.7) over all the branches of this function. +50 +Collisions Between Particles +§18 +Returning now to the problem of the scattering of a beam of particles, not +by a fixed centre of force, but by other particles initially at rest, we can say +that (18.7) gives the effective cross-section as a function of the angle of +scattering in the centre-of-mass system. To find the corresponding expression +as a function of the scattering angle 0 in the laboratory system, we must +express X in (18.7) in terms of 0 by means of formulae (17.4). This gives +expressions for both the scattering cross-section for the incident beam of +particles (x in terms of 01) and that for the particles initially at rest (x in terms +of O2). +PROBLEMS +PROBLEM 1. Determine the effective cross-section for scattering of particles from a perfectly +rigid sphere of radius a (i.e. when the interaction is such that U = 8 for r < a and U = 0 +for r>a). +SOLUTION. Since a particle moves freely outside the sphere and cannot penetrate into it, +the path consists of two straight lines symmetrical about the radius to the point where the +particle strikes the sphere (Fig. 19). It is evident from Fig. 19 that +a sin to = a sin 1(-x) = a cos 1x. +A +to +p +& +FIG. 19 +Substituting in (18.7) or (18.8), we have +do = 1ma2 sin X do, +(1) +i.e. the scattering is isotropic in the C system. On integrating do over all angles, we find that +the total cross-section o = na2, in accordance with the fact that the "impact area" which the +particle must strike in order to be scattered is simply the cross-sectional area of the sphere. +In order to change to the L system, X must be expressed in terms of 01 by (17.4). The +calculations are entirely similar to those of $16, Problem 2, on account of the formal resemb- +lance between formulae (17.4) and (16.5). For M1 < m2 (where M1 is the mass of the particle +and m2 that of the sphere) we have +do1, +where do1 = 2n sin 01 d01. If, on the other hand, M2 < M1, then +For m1 = M2, we have do = a²|cos 01| do1, which can also be obtained directly by sub- +stituting X = 201 from (17.9) in (1). diff --git a/1/19-rutherfords-formula.md b/1/19-rutherfords-formula.md new file mode 100644 index 0000000..e69de29 diff --git a/1/5-the-lagrangian-for-a-system-of-particles.md b/1/5-the-lagrangian-for-a-system-of-particles.md index afb4ec4..24cd30a 100644 --- a/1/5-the-lagrangian-for-a-system-of-particles.md +++ b/1/5-the-lagrangian-for-a-system-of-particles.md @@ -86,11 +86,79 @@ is said to be uniform. The potential energy in such a field is evidently To conclude this section, we may make the following remarks concerning the application of Lagrange's equations to various problems. It is often necessary to deal with mechanical systems in which the interaction between different bodies (or particles) takes the form of constraints, i.e. restrictions on their relative position. In practice, such constraints are effected by means of rods, strings, hinges and so on. This introduces a new factor into the problem, in that the motion of the bodies results in friction at their points of contact, and the problem in general ceases to be one of pure mechanics `1/25`. In many cases, however, the friction in the system is so slight that its effect on the motion is entirely negligible. If the masses of the constraining elements of the system are also negligible, the effect of the constraints is simply to reduce the number of degrees of freedom $S$ of the system to a value less than $3N$. To determine the motion of the system, the Lagrangian `1/5.5` can again be used, with a set of independent generalised co-ordinates equal in number to the actual degrees of freedom. -## Problems + -Find the Lagrangian for each of the following systems when placed in a uniform gravitational field (acceleration $g$). + -#### PROBLEM 1. A coplanar double pendulum + -`1/fig1` + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/1/5_problems.md b/1/5_problems.md deleted file mode 100644 index 8d22ffb..0000000 --- a/1/5_problems.md +++ /dev/null @@ -1,72 +0,0 @@ ---- -section: The Equations of Motion ---- - -Find the Lagrangian for each of the following systems when placed in a uniform gravitational field (acceleration g). - -TODO figures - -§5 -The Lagrangian for a system of particles -11 -PROBLEM 1. A coplanar double pendulum (Fig. 1). -'!!!!!!! -pl4 -m, -2 -L2 -M2 -FIG. 1 -SOLUTION. We take as co-ordinates the angles 01 and O2 which the strings l1 and l2 make -with the vertical. Then we have, for the particle M1, T1 = U = -migh cos 01. In -order to find the kinetic energy of the second particle, we express its Cartesian co-ordinates -x2, y2 (with the origin at the point of support and the y-axis vertically downwards) in terms -of the angles 01 and O2: X2 = l1 sin 01 sin O2, y2 = l1 cos 1+12 cos O2. Then we find -T2 = tm2(x22++22) -= -Finally -L = +m2) m2/1/20102 cos(01-02)+(m1+m2)gl cos 01 + m2gl2 cos O2. -PROBLEM 2. A simple pendulum of mass M2, with a mass M1 at the point of support which -can move on a horizontal line lying in the plane in which m2 moves (Fig. 2). -x -m, -l -M2 -FIG. 2 -SOLUTION. Using the co-ordinate x of M1 and the angle between the string and the -vertical, we have -L = cos b) +m2gl cos p. -PROBLEM 3. A simple pendulum of mass m whose point of support (a) moves uniformly -on a vertical circle with constant frequency y (Fig. 3), (b) oscillates horizontally in the plane -of motion of the pendulum according to the law x = a cos yt, (c) oscillates vertically accord- -ing to the law y = a cos yt. -SOLUTION. (a) The co-ordinates of m are x = a cos rt+l sin , y = -a sin rt+l cos o. -The Lagrangian is -L = sin(-t) +mgl cos ; -here terms depending only on time have been omitted, together with the total time derivative -of mlay cos(-t). -12 -The Equations of Motion -§5 -(b) The co-ordinates of m are x = a cos yt+l sin , y = l cos p. The Lagrangian is (omit- -ting total derivatives) -L = st sin +mgl cos . -(c) Similarly -L = cos st cos +mgl cos . -ly -m -FIG. 3 -PROBLEM 4. The system shown in Fig. 4. The particle M2 moves on a vertical axis and the -whole system rotates about this axis with a constant angular velocity S2. -A -a -m -m, -a -m2 -FIG. 4 -SOLUTION. Let 0 be the angle between one of the segments a and the vertical, and $ the -angle of rotation of the system about the axis; ; b = S. For each particle M1, the infinitesimal -displacement is given by dl-2 = a2 sin² 0 do2. The distance of M2 from the point -of support A is 2a cos 0, and so dl2 = -2a sin 0 d0. The Lagrangian is -L = m1a2(j2 + Q2 sin20) +2m2a202 sin20+2(m1 +m2)ga cos 0. diff --git a/1/7-momentum.md b/1/7-momentum.md index 3a23d83..f04ef65 100644 --- a/1/7-momentum.md +++ b/1/7-momentum.md @@ -73,3 +73,15 @@ are called generalised forces. In this notation, Lagrange's equations are ``` In Cartesian co-ordinates the generalised momenta are the components of the vectors $\v{p}_a$. In general, however, the $p_i$ are linear homogeneous functions of the generalised velocities $\dot{q}_i$, and do not reduce to products of mass and velocity. + + + + + + + + + + + + diff --git a/1/7_problem.md b/1/7_problem.md deleted file mode 100644 index 1ee1d59..0000000 --- a/1/7_problem.md +++ /dev/null @@ -1,15 +0,0 @@ ---- -section: Conservation laws ---- - -PROBLEM -A particle of mass m moving with velocity V1 leaves a half-space in which its potential energy -is a constant U1 and enters another in which its potential energy is a different constant U2. -Determine the change in the direction of motion of the particle. -SOLUTION. The potential energy is independent of the co-ordinates whose axes are parallel -to the plane separating the half-spaces. The component of momentum in that plane is -therefore conserved. Denoting by 01 and O2 the angles between the normal to the plane and -the velocities V1 and V2 of the particle before and after passing the plane, we have V1 sin 01 -= U2 sin O2. The relation between V1 and V2 is given by the law of conservation of energy, -and the result is -sin diff --git a/1/8-centre-of-mass.md b/1/8-centre-of-mass.md index b2a8fc5..e252a87 100644 --- a/1/8-centre-of-mass.md +++ b/1/8-centre-of-mass.md @@ -62,3 +62,12 @@ E ``` This formula gives the law of transformation of energy from one frame to another, corresponding to formula `1/8.1` for momentum. If the centre of mass is at rest in $K'$, then $\v{P}' = 0$, $E' = E_i$, and we have `1/8.4`. + + + + + + + + + diff --git a/1/8_problem.md b/1/8_problem.md deleted file mode 100644 index 32a8fea..0000000 --- a/1/8_problem.md +++ /dev/null @@ -1,11 +0,0 @@ ---- ---- - -PROBLEM -Find the law of transformation of the action S from one inertial frame to another. -SOLUTION. The Lagrangian is equal to the difference of the kinetic and potential energies, -and is evidently transformed in accordance with a formula analogous to (8.5): -Integrating this with respect to time, we obtain the required law of transformation of the -action: -S=S'tuV.R'+1uV2t, -where R' is the radius vector of the centre of mass in the frame K'. diff --git a/1/9-angular-momentum.md b/1/9-angular-momentum.md index 007482c..5c587f2 100644 --- a/1/9-angular-momentum.md +++ b/1/9-angular-momentum.md @@ -128,3 +128,31 @@ $$ The Lagrangian is, in terms of these co-ordinates, and substitution of this in `1/9.7` gives `1/9.8`. + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/1/9_problems.md b/1/9_problems.md deleted file mode 100644 index f4e77b1..0000000 --- a/1/9_problems.md +++ /dev/null @@ -1,30 +0,0 @@ ---- ---- - -PROBLEMS -PROBLEM 1. Obtain expressions for the Cartesian components and the magnitude of the -angular momentum of a particle in cylindrical co-ordinates r, , Z. -SOLUTION. Mx = m(rz-zi) sin - -mrzo cos , -My = -m(rz-zi) cos -mrzo sin , -Mz = mr2 -M2 = -PROBLEM 2. The same as Problem 1, but in spherical co-ordinates r, 0, o. -SOLUTION. Mx = -mr2(8 sin + sin 0 cos 0 cos ), -My = mr2(j cos - sin 0 cos 0 sin b), -Mz = mr2sin20, -M2 = -PROBLEM 3. Which components of momentum P and angular momentum M are conserved -in motion in the following fields? -(a) the field of an infinite homogeneous plane, (b) that of an infinite homogeneous cylinder, -(c) that of an infinite homogeneous prism, (d) that of two points, (e) that of an infinite homo- -geneous half-plane, (f) that of a homogeneous cone, (g) that of a homogeneous circular torus, -(h) that of an infinite homogeneous cylindrical helix. - -SOLUTION. (a) Px, Py, Mz (if the plane is the xy-plane), (b) M, Pz (if the axis of the -cylinder is the z-axis), (c) P (if the edges of the prism are parallel to the z-axis), -(d) Mz (if the line joining the points is the z-axis), (e) Py (if the edge of the half- -plane is the y-axis), (f) Mz (if the axis of the cone is the z-axis), (g) Mz (if the axis -of the torus is the z-axis), (h) the Lagrangian is unchanged by a rotation through an angle -so about the axis of the helix (let this be the z-axis) together with a translation through a -distance h86/2m along the axis (h being the pitch of the helix). Hence SL = 8z aL/dz+ -+80 0L/26 = = 0, so that I+hPz/2n = constant. diff --git a/1/index.md b/1/index.md index 8ef3016..0b1897e 100644 --- a/1/index.md +++ b/1/index.md @@ -28,43 +28,43 @@ III. INTEGRATION OF THE EQUATIONS OF MOTION 14. [Motion in a central field](14-motion-in-a-central-field.html) -15. [Kepler's problem]() +15. [Kepler's problem](15-keplers-problem.html) IV. COLLISION BETWEEN PARTICLES -16. Disintegration of particles -17. Elastic collisions -18. Scattering -19. Rutherford's formula -20. Small-angle scattering +16. [Disintegration of particles](16-disintegration-of-particles.html) +17. [Elastic collisions](17-elastic-collisions.html) +18. [Scattering](18-scattering.html) +19. [Rutherford's formula](19-rutherfords-formula.html) +20. [Small-angle scattering](20-small-angle-scattering.html) V. SMALL OSCILLATIONS -21. Free oscillations in one dimension -22. Forced oscillations -23. Oscillations of systems with more than one degree of freedom -24. Vibrations of molecules -25. Damped oscillations -26. Forced oscillations under friction -27. Parametric resonance -28. Anharmonic oscillations -29. Resonance in non-linear oscillations -30. Motion in a rapidly oscillating field +21. [Free oscillations in one dimension](21-free-oscillations-in-one-dimension.html) +22. [Forced oscillations](22-forced-oscillations.html) +23. [Oscillations of systems with more than one degree of freedom](23-oscillations-of-systems-with-more-than-one-degree-of-freedom.html) +24. [Vibrations of molecules](24-vibrations-of-molecules.html) +25. [Damped oscillations](25-damped-oscillations.html) +26. [Forced oscillations under friction](26-forced-oscillations-under-friction.html) +27. [Parametric resonance](27-parametric-resonance.html) +28. [Anharmonic oscillations](28-anharmonic-oscillations.html) +29. [Resonance in non-linear oscillations](29-resonance-in-non-linear-oscillations.html) +30. [Motion in a rapidly oscillating field](30-motion-in-a-rapidly-oscillating-field.html) VI. MOTION OF A RIGID BODY -31. Angular velocity -32. The inertia tensor -33. Angular momentum of a rigid body -34. The equations of motion of a rigid body -35. Eulerian angles -36. Euler's equations -37. The asymmetrical top -38. Rigid bodies in contact -39. Motion in a non-inertial frame of reference +31. [Angular velocity](31-angular-velocity.html) +32. [The inertia tensor](32-the-inertia-tensor.html) +33. [Angular momentum of a rigid body](33-angular-momentum-of-a-rigid-body.html) +34. [The equations of motion of a rigid body](34-the-equations-of-motion-of-a-rigid-body.html) +35. [Eulerian angles](35-eulerian-angles.html) +36. [Euler's equations](36-eulers-equations.html) +37. [The asymmetrical top](37-the-asymmetrical-top.html) +38. [Rigid bodies in contact](38-rigid-bodies-in-contact.html) +39. [Motion in a non-inertial frame of reference](39-motion-in-a-non-inertial-frame-of-reference.html) VII. THE CANONICAL EQUATIONS -40. Hamilton's equations -41. The Routhian -42. Poisson brackets -43. The action as a function of the co-ordinates -44. Maupertuis' principle -45. Canonical transformations -46. Liouville's theorem -47. The Hamilton-Jacobi equation -48. Separation of the variables -49. Adiabatic invariants -50. General properties of motion in `s` dimensions +40. [Hamilton's equations](40-hamiltons-equations.html) +41. [The Routhian](41-the-routhian.html) +42. [Poisson brackets](42-poisson-brackets.html) +43. [The action as a function of the co-ordinates](43-the-actions-as-a-function-of-the-co-ordinates.html) +44. [Maupertuis' principle](44-maupertuis-principle.html) +45. [Canonical transformations](45-canonical-transformations.html) +46. [Liouville's theorem](46-louivilles-theorem.html) +47. [The Hamilton-Jacobi equation](47-the-hamilton-jacobi-equations.html) +48. [Separation of the variables](48-separation-of-the-variables.html) +49. [Adiabatic invariants](49-adiabatic-invariants.html) +50. 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