{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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). ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern