ANALYTISK MEKANIK I HT 2014

Transcription

1 Karlstads Universitet Fysik ANALYTISK MEKANIK I HT 2014 Kursens kod: FYGB08 Undervisande lärare: Jürgen Fuchs rum 21F 316 tel el.mail: jfuchs@fuchs.tekn.kau.se FYGB08 HT 2014 Exercises

2 Analytisk Mekanik I(FYGB08) HT [ utdelning: inlämning: ] 5p. a) A double pendulum consists of an ordinary pendulum of length l 1 and mass m 1 attached to a fixed point in space and a second pendulum of length l 2 and mass m 2 attached to the end point of the first pendulum. Explain why, with the usual idealizations (i.e., each pendulum consisting of an inextensible rod with a point mass at the end, no friction, etc.) the number of degrees of freedom is 2 when the motion is confined to a vertical plane. How many degrees of freedom are there if the motion is not confined to a plane? Assuming that the motion is in a plane, find the equations of motion in terms of appropriate angles θ 1 and θ 2 (e.g. those used in figure 1.4 on p. 14 of [Goldstein--Poole--Safko]. b) Show that the Euler--Lagrange equations do not change when a total time derivative d F(q,t) is added to the Lagrangian. dt c) Consider the Atwood machine, as e.g. described in chapter 1.6 of [Goldstein--Poole--Safko]. Assume that one of the two bodies is yourself and that the other body has the same mass as you. Further assume that at some initial time both you and the other mass are at rest, while from this time on you are climbing up the rope, at some (possibly time-dependent) speed u. Solve this system with the help of the Lagrangian formalism. In particular, obtain the motion of the other mass as compared to your own motion. Discuss whether the system conservative. ( 1 p.) FYGB08 HT 2014 Exercises

3 d) Consider a pendulum of length l and mass m that originally is in equilibrium in a constant vertcal gravitational field of strength g. (As usual, assume that m describes a point mass and neglect the mass of the string etc.) Until some time t the point of suspension of the pendulum is at rest, while from this time on it is subjected to a constant upwards acceleration b at an angle γ with respect to the vertical. Describe the resulting motion of the pendulum. Hint: Study the motion in a suitable accelerated reference frame. Therevy translate the effect of the acceleration into a modification of the gravitational force. e) The pressure p(r) and density ρ(r) of a self-gravitating fluid that is distributed in a spherically symmetric way obey the differential equation ( ) d r 2 dp = 4πGr 2 ρ, where G is the gravitational constant. dr ρ dr Solve this equation subject to the boundary condition that p=ρ=0 for all values of r larger than some value R, separately for the following two cases: ρ = ρ 0 = const (for r R) and p = cρ 2 with c=const. To what physical systems might these solutions be approximations? FYGB08 HT 2014 Exercises

4 Analytisk Mekanik I(FYGB08) HT [ utdelning: inlämning: ] 7p. a) A point mass of mass m is connected by an inextensible massless wire with a spring of spring constant k, which in turn is attached to a vertical wall. Between the point mass and the spring, the wire partly runs along a cylindrical wheel, in such a way that the point mass is hanging vertically, while the spring is horizontal. When the point mass moves vertically, the wheel rotates correspondingly around its cylinder axis. Determine the extension l of the spring at equilibrium, compared to its extension l in the absence of the mass. Find the motion of the system, assuming that at time t = 0 the extension of the spring is l(t=0) = l. Hint: 1) A convenient generalized coordinate is d := l l. 2) The system is conservative. ( 1 p.) b) Derive the Euler--Lagrange equation for the time-dependent Lagrangian L(q, q) = e γt (m q 2 kq 2 ) (with γ, m, k constants) in terms of the generalized coordinate q. Repeat the calculation, using the new generalized coordinate q := e γt/2 q instead of q. Solve the resulting equation of motion for q(t). Interpret the result. FYGB08 HT 2014 Exercises

5 c) A uniform circular inextensible hoop can roll frictionless and without slipping on a horizontal plane along a fixed direction. A point particle is constrained to move along the inside of the hoop. The particle and the hoop are acted upon by a constant vertical gravitational force. Consider the situation that at some initial time t=0 the hoop is at rest and the particle is located at the top of the hoop and has horizontal velocity v top. Using the Lagrangian formalism, compute the horizontal velocity v bot of the particle at the time when it first passes the bottom of the hoop. Consider the result specifically in the limits that the mass of the particle is either much smaller or much larger than the one of the hoop. d) Consider two point masses, of masses m 1 and m 2, which can move in one direction (with coordinate to be called x) under the influence of a potential V =V(x)= 1 2 kx2 with constant k>0, and which are connected by a massless inextensible rod of length l. The positions x 1 and x 2 of the point masses are such that x 2 >x 1. Obtain the constraint equation. Is the constraint holonomic? Implement the constraint by a Lagrange multiplier λ, and write down the Lagrangian of the system in terms of λ and of the positions x 1 and x 2 and their time derivatives. Use the Euler--Lagrange equations to obtain an expression for λ in terms of x 1 and x 2. Deduce from the result whether, for the special case m 1 =m 2, the effect of the constraint force on the rod is compression or tension. e) A particle is moving in a central potential V = V(r) in such a way that its orbit is a circle that passes through the origin r=0. Describe the orbit in polar coordinates as a suitable function r = r(θ). With the help of the orbit equation, reconstruct the potential from this function r(θ). FYGB08 HT 2014 Exercises

6 f) Study the motion of a point particle in a central potential of the form V(r) = a r + b with a>0. r 2 Show that the orbits have the form d r(θ) = 1+ε cos(αθ) with suitable constants d, ε and α. What do the orbits look like when α=1+η with η 1? (Discuss separately the cases when ε>1, ε=1 and ε<1.) g) Consider the Kepler problem (i.e. a point mass in a 1/r-potential) in a viscous medium, described by an isotropic (meaning that k x =k y =k z ) Rayleigh dissipation function. Show that, just like in the absence of dissipation, the motion is confined to a plane. Determine the time-dependence of the magnitude l of the angular momentum and use your result to describe the orbits qualitatively. Write down the equations of motion in terms of planar polar coordinates. Discuss the solution of the equations of motion. Thinking of the Earth as a ball satisfying Stokes law of friction, estimate an upper bound for the viscosity of the interplanetary medium based on the stability of Earth s orbit. FYGB08 HT 2014 Exercises

7 Analytisk Mekanik I(FYGB08) HT [ utdelning: inlämning: ] 6p. a) Compute the differential scattering cross section σ(ϑ) dϑ for scattering in the central potential V = V(r) = k r 2. b) Assume that someone stops the Earth on its orbit around the Sun. Once the Earth is released again, it will fall towards the Sun. Deduce a formula for the time that it takes until the Earth hits the Sun, and compare it with the period of the original orbit. (Treat the Earth and Sun as point-like and assume that the original orbit is a circle.) Hint: To obtain the motion in time for the fall-in, use energy conservation. du u 1 You will then need the integral u 2 u 1 = +arctan( u 1). u c) A ladder is leaning on a horizontal floor against a vertical wall. Due to its weight, the ladder will slide down the wall and along the floor. Aftersometime t c theladderwilllosecontactwiththewall. Beforethishappens, the system has one degree of freedom, say the angle θ between the ladder and the floor. Describe the motion for t t c, assuming that the wall and floor are frictionless: Express the kinetic energy T through θ and θ. Introduce a constraint force N and write down the equations of motion for the center of mass position and for the angle θ. Then eliminate (the components of) the constraint force from the equation of motion for θ to obtain an uncoupled differential equation for θ. Use energy conservation to determine the angle θ c =θ(t c ) and thereby obtain an expression for the time t c. Hint: To determine T, either regard the ladder as a rigid body, or proceed as follows: divide the ladder into infinitesimal pieces of length dl and of mass dm=(m/l)dl (with L the length and M the mass of the ladder), and obtain T by summing up the contributions from all these pieces. ( 2 p.) FYGB08 HT 2014 Exercises

8 d) Compute the moment of inertia tensor I with respect to the origin for a cuboid of constant mass density whose edges (of lengths a,b,c) are along the x,y,zaxes, with one corner at the origin. For a=b=c, i.e. a cube, find the principal axes and the principal moments of inertia. Determine the angular momentum L and the angle between L and ω when thecube is rotatingwith angular velocity ω about oneof the coordinateaxes. e) A rigid body of the form of a homogeneous ball is initially at rest. At some time t the body is hit instantaneously, in such a manner that directly afterwards it is sliding uniformly, with initial speed v, in a fixed direction. Due to the sliding friction, the ball will in addition start to roll in the same direction. Assume that the force F slide describing the sliding friction is directed oppositely to the motion and that its magnitude is proportional to the weight of the ball and independent of the speed, i.e. F slide = γmg with m the mass of the ball and γ a coefficient of sliding friction. Assume further that the rolling motion is frictionless (which is a good approximation), Write down the equations of motion for the translational coordinate and for the angle of rotation. Show that after a certain time t 1 the motion of the ball is a pure rolling. Determine the time t 1 and the fraction of the initial kinetic energy T(t ) that is transferred to the rolling motion. FYGB08 HT 2014 Exercises

9 Analytisk Mekanik I(FYGB08) HT [ utdelning: inlämning: ] 6p. a) A homogeneous cone of total mass M, height h and base radius R is rolling without slipping on a horizontal plane, with angular velocity of constant magnitude ω. Compute the kinetic energy. Hints: Since the line along which the cone touches the plane is temporarily at rest, the direction of the angular velocity vector is given by this line. The center of mass of the cone is located along its symmetry axis at distance 3h/4 from the top. In the body system, with origin at the center of mass and x -direction along the symmetry axis, the moment of inertia tensor is diagonal, with diagonal 3 entries 10 MR2 3, 80 M (4R2 +h 2 3 ), 80 M (4R2 +h 2 ). b) Consider a system of two equal-mass particles with ordinary kinetic energy T = 1 2 m( q2 1 + q2 2 ) and with conservative forces described by a potential ( ) V = 1 V1 2 q V q with V = W with constants V 1, V 2 and W. W V 2 Write down the equations of motion and solve them with the help of diagonalizing the mtrix V. Assuming that V 1 V 2, expand the eigenvalues and eigenvectors of V in the variable γ := W V 1 V 2. Assuming further that γ 1, truncate the so obtained expressions to the lowest non-trivial order in γ. FYGB08 HT 2014 Exercises

10 c) A bead slides without friction in a uniform gravitational field on a vertical circular hoop. The hoop is rotating at constant angular velocity about its vertical diameter. Denote the mass of the bead by m, the radius of the hoop by R, the angular velocity of the hoop by ω, and the angle between the vertical and the line that connects the bead with the origin of the hoop by ϑ (in such a way that ϑ=0 when the bead is at the bottom of the hoop). Write down the Lagrangian L as a function of ϑ and ϑ. Determine the equilibrium values of ϑ as a function of ω and discuss their stability. Find the frequencies of small vibrations about the stable equilibrium positions. Discuss in particular the case that the rotation frequency has the special value ω= g/r, with g the gravitational acceleration. (2p.) d) Consider a particle of mass m, sliding (without friction) under the influence of gravity (with acceleration g) along a straight wire in a vertical plane. The wire is moving in such a way that, using Cartesian coordinates with x in horizontal and z in vertical direction in the plane, its position in the plane is given by the formula z = x tanθ+ 1 2 αt2. Eliminate the coordinate z through the constraint and thereby obtain the Lagrangian as a function of the coordinate x alone. Then obtain the Euler--Lagrange equation of motion for x. Determine the Hamiltonian H as a function of x and ẋ. Is H conserved? Does it coincide with the total energy? Obtain the Lagrangian by using both x and z as variables, without using the constraint, and implement the constraint by a Lagrange multiplier λ. Find the resulting equations of motion for x and z, as well as an expression for λ. Show that the power resulting from the constraint force does not vanish. e) Consider the time-dependent Hamiltonian H(q,p;t) = with constant m and ω. p 2 2m sin 2 (ωt) ωpq cot(ωt) m 2 ω2 sin 2 (ωt)q 2 Find a corresponding Lagrangian L=L(q, q;t). Then obtain, by choosing a suitable new coordinate q, an equivalent time-independent Lagrangian L. Determine the new Hamiltonian H that corresponds to this new Lagrangian. Discuss similarities and differences between H and H. FYGB08 HT 2014 Exercises

11 Analytisk Mekanik I(FYGB08) HT [ utdelning: inlämning: ] 6 p. a) Usingsphericalcoordinates (r,ϑ,ϕ), obtainthehamiltonianandthehamilton equations of motion for a particle in a central potential V(r). Study how the Hamilton equations of motion simplify when one imposes the initial conditions p ϕ (0)=0 and ϕ(0)=0. b) A point mass m experiences a force F( r) = k r e 2 r mg e z (r= x 2 +y 2 +z 2 ). Determine the potential and suggest a physical interpretation for it. Introduce coordinates (u, v, ϑ) that are determined by x = uv cosϑ, y = uv sinϑ, z = 1 2 (u v). Obtain the Lagrangian and the Hamiltonian in these coordinates, and find at least two constants of motion. c) The suspension point of a pendulum (wich, as usual, is assuemd to consist of a mass point at the end of a massless inextensible rod) is placed on a horizontal wire, on which it can slide without friction. Determine the Lagrangian as a function of the position of the suspension point along the wire and of the angle of the pendulum with respect to the vertical. Obtain the conjugate momenta to the latter variables. Obtain the Hamiltonian by the usual Legendre transformation and write down the Hamilton equations of motion. Give all cyclic coordinates and the corresponding conserved quantities. ( 1 p.) FYGB08 HT 2014 Exercises

12 d) Consider a particle moving on a sphere (say, on the surface of the Earth) under the influence of a conservative force with potential V. Write down the Hamiltonian using spherical coordinates ϑ, ϕ on the sphere. Assume that the sphere is rotating about the z-axis with angular velocity ω. Then besides the previously used coordinates ϑ, ϕ which describe the particle in a non-rotating coordinate system, also a rotating coordinate system ϑ,ϕ is of interest, where ϑ = ϑ and ϕ = ϕ+ωt. Determine the generating function that affords the canonical transformation (ϑ, p ϑ, ϕ, p ϕ ) (ϑ, P ϑ, ϕ, P ϕ ) and obtain the Hamiltonian in rotating coordinates. ( 1 p.) e) Consider a canonical transformation with generating function F 2 (q,p) = qp +εg 2 (q,p), where ε is a small parameter. Write down the explicit form of the transformation. Neglecting terms of order ε 2 and higher, find a relation between this transformation and Hamilton s equations of motion, by setting G 2 =H (why is this allowed?) and ε=dt. f) Starting from the Lagrangian of the heavy symmetric top, expressed in terms of Euler angles in the body system, determine the conjugate momenta and obtain the Hamiltonian by a Legendre transformation. Then perform a time-dependent canonical transformation corresponding to the choice F 3 F 3 (p φ,p θ,p ψ ;Φ,Θ,Ψ;t) = 1 ( 1 2 I 1 ) 3 I p 2 1 ψ t p φφ p θ Θ p ψ Ψ of generating function. Write out the relation between old and new coordinates and momenta and determine the new Hamiltonian. Finally write the Hamiltonian equations of motion for the new Hamiltonian, using instead of Θ the more convenient variable z:= cos Θ, and discuss the resulting differential equation for z as the equation of motion for a mechanical system with one degree of freedom. FYGB08 HT 2014 Exercises