Unformatted text preview: Homework 5 DueTuesday October 27 http: nsgsun.aae.uiuc.edu AAE250 1. Problem 44 from Greenwood: Particles m1 = m and m2 = 2m are connected by a massless string and undergo free planar motion. Initially they rotate about each other with a constant separation l0 and an angular velocity !0: Then a device in m1 reels in the connecting string until its length is 1 l0 and is again constant. At 2 this later time, nd a the angular velocity ! b the tension in the string 2. Consider the following system of particles ^ m1 = 2 kg; r1 = ^ + j + k m; v1 = 7^ m=s; i i m2 = 1 kg; r2 = 4^ + 3k m; v2 = ,6^ m=s j ^ j m3 = 1:5 kg; r3 = 2^ + 2k m; v3 = ,3^ m=s i ^ i ^ m4 = :5 kg; r4 = 4k m; v4 = 12^ + 5k m=s i ^ a Find the mass center of the system of particles at the instant shown. b Find the linear momentum of the system of particles at the instant shown. c Find the angular momentum of the system of particles about point O at the instant shown. d Find the angular momentum of the system of particles about the center of mass at the instant shown. e If the only forces acting are the internal forces of attraction between the particles, what can be said about the subsequent motion of the system? What quantities will be conserved? 3. A mass particle m is launched in a uniform gravitational eld with launch speed v0 and launch angle 0 as shown. The value of m is constant. AAE 250 1 y 1 0 1 0g 1 0 1 0 1 0
v0 11 00m
0 x a Determine the Lagrangian function L for the system and obtain the equations of motion1 b Find all available constants of the motion and evaluate them in terms of initial launch conditions. c Solve the equations of motion for the given initial conditions. 4. Problem 67 from Greenwood: A double pendulum consists of two massless rods of length l and two particles of mass m which can move in a given vertical plane as shown see gure in text. a Assuming frictionless joints, and using and as coordinates, obtain the di erential equations of motion. b What are the linearized equations2 for small and ? Note: and v0 are initial conditions on the solutions to the equations of motion, but do not appear in the equations of motion themselves 2To linearize, approximate sin by and cos by 1 why?, and neglect terms containing products of coordinates and or velocities for example, neglect 2 or _:
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 Spring '09
 Mass, 1 kg, 2k, 2 kg, 1 0 1 0g

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