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Ph.D. Qualifying Exam - Classical Mechanics Sep. 2004
Complete 3 out of4
1. A mass-spring ring system as shown at right consists of N
identical masses, m, and/Vsprings with spring constant &.
(a) Show the equation of motion of this system when the
rnasses move alo.g- the circle of this mass_spring system.
fli?f H*, *S"de
angular frequencv orthemode in
Consider two pendula of equal length b and equal rloss zt1 =
ttt2= ot connected by a spring of force constant fr and both conshained to move in the same
ql*q. The spring is unstetched when the system is in its static equilibrium configuration, and
the pivot points are separated by a disAnce.[.
(a) Write down the Lagrangian for the system. Do not assume snall-amplitude motion. How
would you determine the equations ofmotion.
(b) Noq in the limit of small-amplitude motion, the equations ofmotion simpliff to:
#ru,r*(f . L)u,-*r,: o
#*u.(f . }u,-Lr,: o
Using normal coordinates 0r+02 and 01-02, derive the corresponding eigenfreque,trcies o1 and
(D2.
4 fluid o{dgsrty p and viscosity n now#constant rate between two plates separated by a
distance L. Tnc total fluid flow per unit length between the walls perpendiculario the
direction of current is ,L Assume that fte ptEssute varies only in the direction of flow.
G) ,Find the component of velocity parallel to the walls as a function of the distance from the
midpoint between the plates and the pressure gradient parallel to the walls
(b) Find a relationship between &e pressure gradient and the fluid flow per rmit length.
A cable is hung at equilibrirm with the end points at (x,4, y1) nd @r, yn). The cable is
supporting a load that is uniformly dishibuted inthe horizontal direction where w is the
weight per unit length. The tension of the cable at the lowest point is ?g. Find an expression
for the span.L : trB - xAas a finction of w, To, lA nd ye.
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