p605hw3f08wsoln - Phys 605. Homework 2 Due 5pm, Monday,...

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Unformatted text preview: Phys 605. Homework 2 Due 5pm, Monday, September 22, 2008 Problem 2-]: [10 pts.] Goldstein, Poole, and Sofko, (3rd. ed), Problem 8, Chapter 1, pg. 30. Problem 2-2: [10 pts.] Goldstein, Poole, and Saflco, (31rd. cal), Problem 9, Chapter 1, pg. 31. Strong hint: Apply the results of GPS Problem 8, Chapter 1. Problem 2-3: [15 pts] A solid sphere of mass m and radius 'r is placed at the center of the top horizontal surface of a wedge of mass M, which slides down a frictionless surface as shown in the diagram below. The sphere rolls without slipping. Initially the sphere and the wedge are at rest at the top of the incline surface. ' (a) Find the Lagrangian for the system using the generalized coordinates x and 3 shown. (b) Find the equations of motion for m and M and solve for a and (c) Check your answer for some special case(s) for which you know the correct answer. ((1) Describe the initial motion (t small) of the sphere with respect to the wedge; that is, do they initially move together, or do they accel- erate relative to each other and, if they accel— erate relative to each other, in which direction does the sphere move relative to the wedge. Problem 2—4: [15 pts.] Consider (again!) a thin disk of radius a rolling without slipping on a plane. (See pg. 15 in the Goldstein text and/or my class notes Section 2, pgs. 2-6 and 2—7.) For this new problem there is a vertical gravitational field 9 and the plane is inclined by angle ,8 relative to the horizontal so the disk will tend to roll down the incline. The disk is still constrained such that the plane of the disk always remains perpendicular to the inclined plane. Take the four generalized coordinates describing the configuration of the disk at any time t (as defined in the class notes and text) to be (m, y, 6, :15). Now a: and y are the cartesian coordinates of the point of contact of the disk with the incline plane and the a: and y axes lie in the incline plane with the y-axis pointing directly up the plane. The angle o is is the angle of rotation of the disk about its axis and 3 is the angle a line perpendicular to the plane of the disk makes with the m—axis. The discussion in the text and class notes shows that the rolling without friction constraint is expressed by two differential constraint equations: d2: — a sin 0 do = 0, and dy + (1 cos 3 do = 0. You showed in last weeks homework that these are nonholonomic constraints. This problem is to use the Lagrange multiplier method described in class to include the constraint forces associated with these constraints into the Lagrange formulation to find the correct equations of motion. Take the moment of inertia of the disk about its rotational axis1 I¢ = mag/2 and the moment of inertia for rotation of the disk about a diameter line to be In = mag/4. (a) Find the Lagrangian for this system. (b) Use Lagrange multipliers and the differential constraint equations to find expressions for the generalized constraint forces Q§,Q;, Q3, Q3 (See class notes section 2 pp 2-25 through 2—29.) Write down the Lagrange equations including the constraint forces. Show all the equations that you would need to solve for the motion of the disk. Clearly list the unknown functions of ti [You should have 6 rather simple ordinary differential equations and 6 unknown functions of it. These could be solved completely for general initial conditions (270,310, 00, (to, do, do) but, I do not ask you to do that] (c) However, it is easy to consider the special case when the disk is not initially spinning about its diameter perpendicular to the incline. That is, assume 60 2 0. 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This note was uploaded on 02/09/2012 for the course PHYS 605 taught by Professor Staff during the Fall '09 term at Ohio University- Athens.

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p605hw3f08wsoln - Phys 605. Homework 2 Due 5pm, Monday,...

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