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3. Damped SHO – A damped simple harmonic oscillator has an oscillation frequency ω’ that does not match the natural frequency ωo of the undamped oscillator.

a.) If the frequency shift if relatively small say ωo2 -ω’2 = 10-6 ωo2 , find the oscillator’s quality factor Q. and its logarithmic decrementδ.

b.) If ωo = 105 and the mass of the system is m = 10-5 kg, find the stiffness s of the system, and the damage coefficient r.

c.) If the maximum displacement of the oscillator is at t = 0, and has a value of 10-3 m, find the energy of the system, and find the timeτfor the amplitude to drop to 1/e of this maximum value.

4. Consider s pendulum on a string og length L. Add to this two parallel frictionless rails, which lie horizontally with the string passing between them, at a changeable position up or down the string between the anchor and the pendulum bob. The string is free to slide along the rails, but when the pendulum oscillates along the perpendicular direction, the stationary rails limit the string’s motion; Take the gravity is along the z-direction, so that the pendulum bob will swing in an x-y plane. Begin by assuming there is not friction or dissipation/damping.

a.) Describe and/or sketch, for an arbitrary position of the rails, the result this new addition has on the 2D motion of the pendulum over the surface, considering these initial conditions: i) start from rest at (x,y) = (1,0); ii) i) start from rest at (x,y) = (0,1); iii) start from rest at (x,y) = (1,1); iv)start at (x,y) = (1,1), but with an initial velocity (1,-1), ( ie, a tangential velocity, one directed a t right angles to the displacement)

b.) Where must these rails be located in order that the pendulum be able to swing in a circle? Where must it be so that the pendulum can swing in a figure-eight shape?

c.) Describe and sketch the trajectory of the bob in (a) part iv), if the rails are located on the string at position z = L (1-π2 ) ?

d.) Assume now that there is a damage force proportional to the velocity of the pendulum. Sketch the x-y trajectory of the pendulum bob under initial conditions of (a) part iv) when the parallel rails are located at z = 0. Sketch the x-y trajectory of the pendulum bob when the parallel rails are located at z = L/2

5. Seismograph – A simple seismograph is constructed with a mass hung from a spring suspended on a rigid frame attached to the earth. The spring force and the damping force depend on the displacement and velocity of the mass relative to the earth’s surface, but the important acceleration, inertia and energy of the mass is “absolute” (depending on its position in space – relative to the “fixed stars”, not relative to the frame as the frame moves).

a.) Using y to denote the displacement of M relative to the earth, and η to denote the displacement of the earth’s surface itself, show the equation of motion is:

(d2y/dt2) + γ(dy/dt) + ωo2y = - (d2η/ dt2)

b.) Solve for y in steady-state vibration if the earth oscillates as η = C cosωt

c.) Sketch a graph of the amplitude A of the displacement y as a function of ω (supposing C to be the same for all ω).

d.) A typical long-period seismometer has a period of about 30 s, and a Q of about 2. As a result of a violent earthquake, the earth’s surface may oscillate with a period of about 20 minutes and with amplitude such that the maximum acceleration is about 10-9 m s-2. How small alvalue of A must be observable if this is ti be detected?

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