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Spring 2002 Qualifying ExaminationClassical Mechanics

IMPORTANT: Please answer all three questions.________________________________________________________________________

1. [40 points total] A platform of mass M sits on a frictionless table. Two identicalblocks of mass m are attached with identical springs to a post fixed to the platform. The

springs are massless and have force constant k. The blocks move on the frictionlesssurface of the platform and are constrained to move along the x axis (parallel to the table

surface and in the plane of this paper).

m m

X M

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A. [10 points] Give the Lagrangian of the system of masses and springs.

B. [25 points] Calculate the normal frequencies of the system.

C. [5 points] Describe the normal modes of vibration corresponding to these frequencies.

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2. [30points total] A particle of mass m moves in a central potential U(r).

A. [4 points] Show the angular momentum, L, about the center force, r=0, is

conserved.

B. [2 points] Show that the motion of the particle must lie in a plane perpendicular

to L.

C. [5 points] Show that the total energy may be written

E=1/2 m 2rD + Ueff , where Ueff (r) =L2/(2 m r2) + U(r)

and where L is the magnitude ofL.

D. [5 points] Assume a circular orbit of radius R exists; then show

E = Ueff (R) and

d Ueff /dr|r=R = 0

E. [4 points] State the condition for which the circular orbits specified in D arestable.

F. [10 points] Let the central force be F = - (b/ r2 c/ r4) r, where b>0 and c>0 andr is the unit vector in the radial direction. Calculate the values of R that give rise

to stable orbits as a function of L, b and c.

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3. [30points total] A wedge of mass M sits on a frictionless table. A block of mass mslides on the frictionless slope of the wedge. The angle of the wedge with respect to the

table is . Take x positive to the right. Let the coordinates for m be (x1,y1) and for thepoint of the wedge x2.

y m

M

x

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A. [8 points] Write the constraint equation for the block sliding on the wedge and the

Lagrangian for the block and wedge.

B. [22 points] Calculate the Lagrange equations of motion for the block and wedge usingthe method of Lagrange Multipliers to incorporate the constraint..

M