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CM_2006a - were Wit T%7j‘ ‘H‘ [email protected] 6 F36” Classical...

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Unformatted text preview: were Wit T%7j‘; ‘H‘ 4: @oo 6. F36”) Classical Mechanics Problem 1 A solid of homogeneous sphere of radiUS r and of mass m rolls without slipping on the inside of a stationary larger cylinder of radius R as shown in Fig. 1. (a) 15% Using variables given in Fig. l and assuming that the potential energy zero at the lowest position of the sphere, determine the Lagrangian, the equation of constraint. and Lagrangian equation of motion including the Lagrangian multipliert (b) 10% Find the Lagrangian multiplier for at the particular angles, 6 : o: and d: = 6. and the the frequency for small oscillations. (c‘) 10% After eliminating (15, find the Hamiltonian in terms of l9 and its corresponding generalized momentum. Problem 2 Consider a coplanar double pendulum shown in Fig, 2, in which the masses of rods that connect m1 and m2 can be neglected. (a) 10% Find the Lagrangian of the system using the variables given in Fig. 2. (b) 7% Derive the equation of motion. (08% Assuming that :51 = (2 2 l and m1 3 m2 = m. find the normal-mode frequencies when all angles are small. If 61(0) = 60 and 62(0) 2 0, find 510:) and 62(13). (d) 15% Suppose that the pendulum is attached to a. cart of mass 2m that moves without friction on a horizontal surface (see Fig. 3). Assuming that 11 = [2 = l and m1 = m; = To, find the normalimod-e-rlirequencies for oscillations of small angles. Problem 3 10% Find the total cross section for the elastic scattering of small marbles of mass m and radius a from a fixed solid sphere of radius R and mass M. Problem 4 A particle of mass m moves in two dimension under the influence of the potential V(x, y) = ax/zt2 -l- 3,12. (3.) 8 ‘70 By using the polar coordinates, r and it} find the Hamiltonian in terms of r and 45 and their corresponding generalized momentum. (b) 'T % Suppose that the initial conditions of the particle at t = O are: r(D) = r0, (35(0) = O . %(0) = O, and 5%(0) = we. By using the Hamilton's equation of motion, find r(t) and 4::(75). ...
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