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210 chapter 9Here N is the normal force on the object under consideration, µ and b are parameters,and v is the velocity. This model for static friction is appropriate for objects at rest,while the model for kinetic friction is appropriate for an object sliding on a drysurface. If the surface is lubricated, or if the object is moving through a viscousmedium, then a frictional force proportional to velocity is a better model. 41. Extend your harmonic oscillator code to include the three types of friction in(9.43) and observe how the motion differs for each.2. Hint: For the simulation with static plus kinetic friction, each time the oscillatorhas v =0you need to check that the restoring force exceeds the static forceof friction. If not, the oscillation must end at that instant. Check that yoursimulation terminates at nonzero x values.3. For your simulations with viscous friction, investigate the qualitative changesthat occur for increasing b values:Underdamped: b2mω 0Oscillation within a decaying envelopeNonoscillatory, finite time decayNonoscillatory, infinite time decay9.7.2 Resonances and Beats: Model and ImplementationAll stable physical systems will oscillate if displaced slightly from their rest positions.The frequency ω 0 with which such a system executes small oscillations aboutits rest positions is called its natural frequency. If an external sinusoidal force isapplied to this system, and if the frequency of the external force equals the naturalfrequency ω 0 , then a resonance may occur in which the oscillator absorbs energy fromthe external force and the amplitude of oscillation increases with time. If the oscillatorand the driving force remain in phase over time, the amplitude will increasecontinuously unless there is some mechanism, such as friction or nonlinearities, tolimit the growth.If the frequency of the driving force is close to the natural frequency of the oscillator,then a related phenomena, known as beating, may occur. In this situation thereis interference between the natural amplitude that is independent of the drivingforce plus an amplitude due to the external force. If the frequency of the driver isvery close to the natural frequency, then the resulting motion,(x ≃ x 0 sin ωt + x 0 sin ω 0 t = 2x 0 cos ω − ω 0t2)sin ω + ω 0t, (9.44)2looks like the natural vibration of the oscillator at the average frequency ω + ω02, yetwith an amplitude 2x 0 cos ω − ω02t that varies with the slow beat frequency ω − ω02.4 The effect of air resistance on projectile motion is studied in Unit III of this chapter.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 210