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Unformatted text preview: U) sci/45' Phys 605. Homework 2 Due 5pm, Monday, September 22, 2008 Problem 2—1: [10 pts.] Consider a system of interacting particles, each with mass mi. Show
that the magnitude ROM of the position vector for the center of mass is given by 2 2 2 1 2
M RCM = MZmi'ri — EZWiijﬁ,
i is where n; is the magnitude of the position vector of the i—th particle and nj is the distance
between particles 2' and 3'. Note: take n, E 0. Problem 22: [10 pts.] Consider a thin vertical disk of radius a rolling without slipping on a
horizontal plane (see pg. 15 in the Goldstein text and/or my class notes Section 2, pg. 2—6).
Four generalized coordinates describing the conﬁguration of the disk at any time t (as deﬁned
in the class notes and text) are (.13, y, 6, 915). The discussion in the text and class notes shows
that the rolling without friction constraint is expressed by two differential constraint equations: (in: — asin qu5 = O,
dy+acos€d¢ = 0. This problem is to show that this constraint is nonholonornic. If it was a holonomic constraint, then two functions would exist such that G1(w,y,6,¢) = 0 and G2(s:,y,6,¢) = 0 and the diﬂ’erential constraint equations above would be the exact differentials: dG1 = 0 and £2 = 0. Thus1 we need to Show that it is impossible to ﬁnd two such functions. (a) Show that neither of these two diﬂ’erential constraint equations is an exact differential as
written. Hint: see class notes pg 2—73.. (h) Fin‘ther, show that no integrating factor function exists (see class notes pg 2—7b) for either
differential constraint equation that will turn it into an exact differential. Completing (b) we have shown neither differential constraint equation could be an exact dif—
ferential. All we really needed to do was show one of the two differential constraint equations
could not be an exact differential to show that a disk rolling without slipping on a plane is a
nonholonomic constraint. Problem 23: [20 pts.] Two point masses, each of mass m, are joined by a rigid weightless rod
of length 8, the center of which is constrained to remain on a circle of radius a. The circle of
radius 0: lies in a vertical plane. {Only the center of the rod is constrained to the plane of the
circle; the masses at the ends of the rod are thus constrained to a Sphere (of radius 12/ 2) and the
center of the sphere remains on the circle] (a) Deﬁne a set of generalized coordinates for this system. (b) Express the total kinetic energy of the masses in terms of your generalized coordinates. (c) Assume the whole system is in a uniform gravitational ﬁeld 9 directed downward. Write an expression for the potential energy of the system.
(d) Find the Lagrange equations of motion for this system. . 3
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