分析力学第三次作业解答.pdfVIP

5.1 Block on a cylinder II. Lets return to the block on a cylinder problem of Problem Set 5. A block of mass m starts at rest at the top of a cylinder of mass M and radius R which is placed on the floor, as shown in the figure. At time t = 0 the block is given a gentle tap and it begins sliding down the surface of the cylinder. Ignore the friction between the block and the cylinder and the cylinder and the floor. Use the method of Lagrange multipliers to nd the normal force which the cylinder exerts upon the block. Determine the angle (as a function of the mass ratio ) at which the block loses contact with the cylinder. Check your result in the limit . Solution: Start by introducing the generalized coordinates , such that . At this point we haven’t enforced the constraint, which is , since we want to determine the force of constraint and find the angle at which it vanishes. In terms of these coordinates, the Lagrangian is Lagrange’s equation for is After a litter algebra, this becomes By enforcing the constraint, , this becomes By drawing a free body diagram, you can convince yourself that is the normal force which the cylinder exerts upon the block, which will vanish when the block leaves the surface of the cylinder. The equations for are like Where . We also have Using the equations above Solving the equation for Expressing the result completely in terms of ; after lots of algebra, To find the critical angle at which the block loses contact with the cylinder, set ; this results in the following cubic equation for For , we recover the result which we had obtained for the immobile cylinder. If so th