By Derek F. Lawden
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Extra info for A Course in Applied Mathematics, Vols 1 & 2
49): qR 1 D qR 2 D m1 m2 gI m1 C m2 D 2g m1 m2 : m1 C m2 (2) Rolling Barrel on an Inclined Plane The ‘barrel’ is a hollow cylinder of mass M whose moment of inertia J has been calculated in Sect. 3 of Vol. r/ is the mass density of the hollow cylinder. 114). It is here again about a holonomic problem. We consider as generalized coordinates (see Fig. 21) q1 D x I q2 D # . j D 2/ with the rolling off condition R d# D dx as constraint. This is of course integrable and therewith holonomic. But intentionally, that shall not be exploited here.
28 Bead on a rotating wire-ring under the influence of the gravitational force Fig. 29 On a table frictionlessly rotating mass m being connected by a thread to another mass M which experiences the gravitational force 1. Introduce suitable generalized coordinates and find the Lagrangian! 2. Solve the Lagrange equations of motion! 9 A rod of the length 2L with circular cross-section R2 is slipping down a wall (y axis) because of the gravitational force (Fig. 27). The rod possesses a homogeneous mass distribution (mass M, homogeneous density 0 ).
5 Let the position of a particle be described by cylindrical coordinates . ; '; z/. The potential energy of the particle is given as V. / D V0 ln 0 ; V0 D const ; 1. Write down the Lagrangian! 2. Formulate the Lagrange equations of motion! 3. Find and interpret at least two conservation laws! 2 The d’Alembert’s Principle 51 Fig. 24 Cylinder rolling on the inner surface of the side wall of another (‘larger’) cylinder Fig. 25 Point mass m on the inner surface of a circular cone in the earth’s gravitational field Fig.
A Course in Applied Mathematics, Vols 1 & 2 by Derek F. Lawden