分析力学第一次作业解答.pdf

1 1 Spherical coordinates Many of the problems which we will solve will possess spherical symmetry so that the most natural coordinates are spherical coordinates We ll need to know how to express the position velocity acceleration and kinetic energy of a particle in this coordinate system The spherical coordinates r are related to the rectangular coordinates x y z throughsincosxr sinsinyr coszr The unit vectors are a Draw a careful figure which illustrates the meaning of the spherical coordinates and their relation to the rectangular coordinates b Show that this is indeed on orthogonal coordinate system In other words show that 0 0 0 rr eeeeee Also show that each of the vectors r e e e has a length of one Finally show that the coordinate system is right handed so that rrr eee eee eee c Use these relations to show that d In spherical coordinates the position vector is r r re Use your results above to calculate the velocity v r i the kinetic energy 2 2Tmv and the acceleration a r i of a particle in spherical coordinates Solution b 22 sincossinsincos coscoscossinsin sincoscossincossinsincos0 rxyzxyz e eeeeeee ii sincossinsincossincos coscoscossinsin xyz rxy eee eeeee The others are straightforward c coscossinsin cossinsincos sin coscoscossinsin sin sincos sin rxyz xyzxy eeee eeeee ee i iiiii ii ii The others are straightforward d This closely follows the derivation in class for cylindrical coordinates The key point to remember is that the unit vectors are time dependent so you need to use Eqs above and the product rule when carrying out the di erentiations The results are 1 2 Central forces A central force is a force directed along the line connecting the two interacting particles which only depends on the separation between the particles Assuming that the center of the force is at the origin r 0 then the central force can be written as F r f r er with r the distance from the origin Important examples of central forces are the gravitational force the electrostatic force Coulomb s law and the force exerted by a spring on a mass a Assume that a particle of mass m is acted upon by a central force The natural coordinates to describe the motion of the particle are spherical coordinates r Separate Newton s Second Law in spherical coordinates and write down the three equations of motion one for each component b Show that for any f r a solution of the and components of these equations has 2 and 2 mrl i with l a constant which is independent of time What is the physical significance of these results c Use the conditions above to simplify the equation for the radial coordinate r Show that Solution a Use the results from Problem 1 using ma F in spherical coordinates we have b Taking independent of time then substituting these results into Eqs above we find Multiply both sides by r we can then write this as So that with l a constant independent of time If we recognize as the tangential component of the particle momentum then is the angular momentum about the center of force Therefore the condition is simply a statement of conservation of angular momentum about the center of force The fact that means that the motion occurs in a plane c Substitute and into we can get the result 1 4 Spherical pendulum A mass m is attached to the ceiling by an inextensible i e fixed length massless string of length b as shown in the figure a Draw a free body diagram for the mass showing all forces acting upon it b Use F ma in spherical coordinates to derive the equations of motion for and c Use your equations of motion to show that the quantity d Use your result from part c to eliminate the dependence upon in your equation of motion for You should find that Solution a The figure is below b Resolving the forces into components along the unit vectors we have Use F ma and the expression for acceleration in spherical coordinates along with the constraint that r b and to obtain the equations of motion And As a check note that when the equation for is which is the equation of motion for the plane pendulum c Differentiate Factoring sin from this expression we see that it is indeed zero The quantity is the angular momentum about the z axis which is conserved d The angular momentum is a constant whose value is determined from initial conditions Using we immediately obtain 3 1 Lagrangian vs Newtonian mechanics Construct the Lagrangians and derive the equations of motion for a the spherical pendulum and b the bead on the hoop In both cases compare your results with the results which you obtained in previous assignments using Newton s Second Law Solution a Let s begin with the spherical pendulum In spherical coordinates the kinetic energy is The pendulum string has a constant length R so r R and0r i There are two degrees of freedom corresponding to the two generalized coordinates and The kinetic energy is then Setting the zero of the potential energy to be when the string is horizontal we have The Lagrangian is then Lagrange s equation for is which after a little rearranging yields For we see that L is independent of so that is a cyclic coordinate Thus where is a constant b For the bead on the hoop we ll use the angle the angle that the line connecting the bead to the center of the hoop makes with the vertical as our generalized coordinate The constraint is that r R so that 0r i and that the hoop rotates with a constant angular velocity so that There is now only one degree of freedom so there is one generalized coordinate which we will choose to be From Equation above we see that the kinetic energy is If we choose the potential energy to be zero at then The Lagrangian is Lagrange s equation is Taking the derivatives and rearranging a bit we obtain the equation of motion Note that in both cases the equations of motion are the same as we obtained using Newton s Second Law However using the Lagrangian method we do not obtain the forces of constraint 3 2 Moving pendulum A plane pendulum of length l has a bob of mass m which is attached to a cart of mass M which can move on a frictionless horizontal table a Express the rectangular coordinates of the pendulum bob in terms of the coordinates b Using X and as your generalized coordinates construct the Lagrangian for this system c Using Lagrange s equations derive the equations of motion for the coupled pen dulum and cart system d Find all first integrals of the motion i e find the conserved quantities Solution a The coordinates of the pendulum bob are and cos b y the components of the velocity are then b The kinetic energy of the bob is then and the kinetic energy of the cart is The potential energy of the bob is Pulling all of this together we have for the Lagrangian c The equations of motion are obtained from Lagrange s equations For we have which after some algebra becomes For X we have The Lagrangian is independent of X X is a cyclic coordinate so that is a constant This is a statement of the conservation of the x component of the momentum of the system The value of Px is determined from the initial conditions We can differentiate this equation again to obtain We could use this equation to eliminate the from to get a single nonlinear second order di erential equation for d In addition to the momentum Px the total energy of the system is conserved 3 3 Pendulum A pendulum consisting of a rigid massless rod of l