ANALYTISK MEKANIK I HT 2016

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1 Karlstads Universitet Fysik ANALYTISK MEKANIK I HT 2016 Kursens kod: FYGB08 Kursansvarig lärare: Jürgen Fuchs rum 21F 316 tel el.mail: juerfuch@kau.se FYGB08 HT 2016 Exercises

2 1. [ utdelning: inlämning: ] 4p. a) For each of the following force fields determine whether the force is conservative or not, and if it is, give a corresponding potential function: Constant force. Spring force satisfying Hooke s law. Gravitational force resulting from a point mass. Lorentz force resulting from a magnetic field. Van der Waals force between molecules. Rayleigh-type fluid friction. ( 1 p.) b) Consider a system of N point masses m i (i=1,2,...,n, with N arbitrary), with a total force F i tot acting on the ith mass. Show that the center of mass R c.m. :=( N i=1 m i r i )/( N i=1 m i) moves like a single particle of mass N m i acted upon by a force F c.m. = i=1 N i=1 F tot i. Is one allowed to ignore the internal forces when calculating F c.m.? c) Determine, for arbitrary initial conditions, the equation of motion for a point particleonwhichboththegravitationalforce F g andafrictionforceofmagnitude F fr = k v act. Show that, for arbitrary initial conditions, the speed v of the particle cannot exceed mg/k. (If you cannot solve this problem for general initial conditions, assume instead that the particle starts at rest.) d) Compute the differential scattering cross section σ(ϑ) for scattering in the central potential V = V(r) = k r 2. FYGB08 HT 2016 Exercises

3 2. [ utdelning: inlämning: ] 6p. a) 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.) b) 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]. c) Show that the Euler--Lagrange equations do not change when a total time derivative d F(q,t) is added to the Lagrangian. dt FYGB08 HT 2016 Exercises

4 d) 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? e) Considerasystemconsistingofasimplependulumofmass m andlength l whose upper end is attached to a massless block, which in turn is attached to a wall by a spring of spring constant κ. The block is constrained to move horizontally, and the pendulum is only allowed to move in a plane that contains the line of motion of the block. (The usual idealizations are made, i.e. the block moves without friction, the spring and the rod of the pendulum are massless, etc.) Determine the number of degrees of freedom. Find a Lagrangian for the system and obtain the equations of motion as Euler-Lagrange equations for this Lagrangian. Discuss solutions with the block being at rest and solutions with the pendulum hanging vertically. ( 1 p.) f) Derive the Euler--Lagrange equation for the time-dependent Lagrangian L(q, q) = 1 2 eγt (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). Discuss the possible physical meaning of the resulting differential equations. ( 1 p.) FYGB08 HT 2016 Exercises

5 3. [ utdelning: inlämning: ] 5p. a) Consider a pendulum of length l and mass m that originally is in equilibrium in a constant vertical gravitational field of strength g. (As usual, assume that m describes a point mass and neglect the mass of the rod 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. Thereby translate the effect of the acceleration into a modification of the gravitational force. b) The orbit of the comet Hyakutake around the sun has an eccentricity e = , and its perihelion is.2302 AU. The last time the comet appeared on the sky was the period from March to May When will it appear next time? Compute the aphelion and compare it to the one of the (former) planet Pluto. (Some astronomical data: Astronomical unit: 1AU= meter; solar mass: M = kg; semimajor axis of Pluto s orbit: 39.37AU; eccentricity of Pluto s orbit:.249. The comet s mass is negligible compared to M.) c) Study the motion of a point particle in a central potential of the form V(r) = a r + b r 2 Show that the orbits have the form d r(θ) = 1+ε cos(αθ) with suitable constants d, ε and α. with a>0. 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.) FYGB08 HT 2016 Exercises

6 d) 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). e) 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. FYGB08 HT 2016 Exercises

7 4. [ utdelning: inlämning: ] 5p. a) 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. ( 3 p.) b) 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 ). (2p.) FYGB08 HT 2016 Exercises

8 5. [ utdelning: inlämning: ] 4p. a) A rigid body of the form of a homogeneous straight stick (of mass M, length l and infinitesimal thickness) is placed in a homogeneous gravitational field. The stick is held fixed at one of its ends, but otherwise allowed to move freely. Give the Lagrangian of the system in spherical coordinates for a coordinate system with origin at the fixed point. Obtain the Euler--Lagrange equations. Solve them partially in the form of an orbit equation ϕ = ϕ(θ). Discuss the solution for the particular initial condition that at time t = 0 the stick is rotating in the plane perpendicular to the gravitational force. Determine the turning points for the motion obtained in the previous point. b) Consider a pendulum consisting of two parts: a uniform rod of mass m, length l, negligible thickness, and with one end fixed; and a uniform disk of mass µ and radius ρ. Therodismovinginaplane, andthediskisattachedatapointp onitsboundary to the non-fixed end of the rod, in such a way that it can freely rotate about P in the plane in which the rod is moving. Obtain the Lagrangian and the equations of motion. Give the kinetic and potential energy matrices in the limit of small deviations from stable equilibrium. ( 1 p.) c) A particle moving with initial velocity v in a one-dimensional space is subjected to a frictional force in direction opposite to the motion whose magnitude is proportional to the square of the speed, F = κv 2. Assuming that the particle is relativistic, so that the momentum is given by p=γm v, find a relation between its velocity v and time. At what time does v become zero? Discuss the non-relativistic limit of the motion. FYGB08 HT 2016 Exercises

9 d) 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. FYGB08 HT 2016 Exercises

10 6. [ utdelning: inlämning: ] 6p. a) Consider a one-dimensional particle of mass m moving in a potential of the form V(x) = 1 2 k(x2 x 2 )e x/x. Determine all (stable or unstable) equilibrium positions. Describe the possible types of (bounded or unbounded) motion, clearly stating the energy range in which they exist. Draw a curve, or several curves, in phase space for each type of motion. Make sure to indicate any separatrix that may exist. Compute the frequency of small oscillations about any stable equilibrium. ( 2 p.) b) 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. (2p.) 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.) FYGB08 HT 2016 Exercises

ANALYTISK MEKANIK I HT 2014

Karlstads Universitet Fysik ANALYTISK MEKANIK I HT 2014 Kursens kod: FYGB08 Undervisande lärare: Jürgen Fuchs rum 21F 316 tel. 054-700 1817 el.mail: jfuchs@fuchs.tekn.kau.se FYGB08 HT 2014 Exercises 1