Question

From Classical Mechanics by Taylor

3. (30 pt +10 bonus pt) The mass shown in the figure below is resting on a frictionless horizontal table. Each of the two identical springs has force constant and un-stretched length o The mass rests at the origin of an r-y coordinate system, and the two springs are on theヱaxis as shown in the figure. The distances a are not necessarily equal to lo- (That is, the spring may already be stretched or compressed.) The two outer ends of the springs are fixed in space, but the cnds attached to the mass are free to move as the mass moves Figure 0.2: A mass attached to two springs a, (10 pt) when the mass is displaced a distance from the origin to the point (,), knowing that the potential energy of a spring is write down the potential energy U(z,v) (in terms of k, α· b. (5 pt) When oth , and y are small, show that the potential energy has the form for an anisotropie oscillator. What is the efective spring constant & (expressed with k, a, and o) aong the r direction, and what is k, along the y direction? Show that if a 4-the equilibrium at the origin is unstable and explain why c. (10 pt) Set α = 1, k = 1, lo = 1.5: Use your favorite plotting software to plot the potential energy U(r, ) (from part a.) over the range of -1,1 and y2,2. Where are the points (o.o) that give the miniotential energy? [HINT #1 : If you could not get your plotting software to work, don't despair two springs in its un-stretched length, lo. How would you move the mass attached to the two springs so that the springs can be in its natural unstretched length?] [HINT #2: If you You can reason that the is3 energy point must have the use mathematica, you might want to try Plot 3D. d. (5 pt) Contine with part e, expand the potential energy around the global mininnim to yield (0.2) where ór and óy are small displacements along and , respectively, around the global minim ( What is the frequency of (small amplitude) oscillations around the global minimum?

Answer 1