PHYS 1441 – Section 004 Lecture #23AnnouncementsSinusoidal Behavior of SHMThe Period and Sinusoidal Nature of SHMExample 11-5Example 11-6The SHM Equation of MotionSlide 8The Simple PendulumExample 11-8Damped OscillationForced Oscillation; ResonanceWednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu1PHYS 1441 – Section 004Lecture #23Wednesday, Apr. 28, 2004Dr. Jaehoon Yu•Period and Sinusoidal Behavior of SHM•Pendulum•Damped Oscillation•Forced Vibrations, Resonance•WavesToday’s homework is #13 and is due 1pm, next Wednesday !Final Exam Monday, May. 10!Wednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu2Announcements•Final exam Monday, May 10–Time: 11:00am – 12:30pm in SH101–Chapter 8 – whatever we cover next Monday–Mixture of multiple choices and numeric problems–Will give you exercise test problems next Monday•Review next Wednesday, May 5.Wednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu3Sinusoidal Behavior of SHMWednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu4The Period and Sinusoidal Nature of SHMv0Consider an object moving on a circle with a constant angular speed If you look at it from the side, it looks as though it is doing a SHMsin q201xv vA� �= -� �� �0v02 ATvp=2012mv2mTkp=1 12kfT mp= =Since it takes T to complete one full circular motionFrom an energy relationship in a spring SHM0kv Am=Thus, T isNatural Frequency0vv=2 2A xA-=21xA� �= -� �� �2 ATp=2 Afp=212kA=Wednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu5Example 11-5Car springs. When a family of four people with a total mass of 200kg step into their 1200kg car, the car’s springs compress 3.0cm. The spring constant of the spring is 6.5x104N/m. What is the frequency of the car after hitting the bump? Assume that the shock absorber is poor, so the car really oscillates up and down. fT2mkp=414002 0.926.5 10sp= =�1T=12kmp=41 6.5 101.092 1400Hzp�= =Wednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu6Example 11-6Spider Web. A small insect of mass 0.30 g is caught in a spider web of negligible mass. The web vibrates predominantly with a frequency of 15Hz. (a) Estimate the value of the spring constant k for the web.f(b) At what frequency would you expect the web to vibrate if an insect of mass 0.10g were trapped?fkSolve for k12kmp=15Hz=2 24 mfp=( )22 44 3 10 15 2.7 /N mp-=�״=12kmp=41 2.7262 1 10Hzp-= =�Wednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu7The SHM Equation of Motionv0The object is moving on a circle with a constant angular speed How is x, its position at any given time expressed with the known quantities?cosx A q=tq v=cosx A tv=sincecos 2A ftp=How about its velocity v at any given time? 2 fv p=andv0sinv q=-( )0sinv tv=-( )0sin 2v ftp=-How about its acceleration a at any given time? 0kv Am=aFm=kxm-=( )cos 2kAftmp� �=-� �� �( )0cos 2a ftp=-From Newton’s 2nd law0kAam=Wednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu8Sinusoidal Behavior of SHM( )cos 2x A ftp=( )0sin 2v v ftp=-( )0cos 2a a ftp=-Wednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu9The Simple PendulumA simple pendulum also performs periodic motion.The net force exerted on the bob is rFx Lq=Satisfies conditions for simple harmonic motion!It’s almost like Hooke’s law with.Since the arc length, x, is tF F= =�mgkL=The period for this motion isTThe period only depends on the length of the string and the gravitational accelerationAmgTcos0tFAmgsinmgq�-mgxL-2mkp=2mLmgp=2Lgp=Wednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu10Example 11-8Grandfather clock. (a) Estimate the length of the pendulum in a grandfather clock that ticks once per second. Since the period of a simple pendulum motion isTThe length of the pendulum in terms of T is 224gTL Thus the length of the pendulum when T=1s is 224T gLp=gL2(b) What would be the period of the clock with a 1m long pendulum?gL2T1.02 2.09.8sp= =21 9.80.254mp�= =Wednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu11Damped OscillationMore realistic oscillation where an oscillating object loses its mechanical energy in time by a retarding force such as friction or air resistance.How do you think the motion would look?Amplitude gets smaller as time goes on since its energy is spent.Types of dampingA: OverdampedB: Critically dampedC: UnderdampedWednesday, Apr. 28, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu12Forced Oscillation; ResonanceWhen a vibrating system is set into motion, it oscillates with its natural frequency f0.However a system may have an external force applied to it that has its own particular frequency (f), causing forced vibration.For a forced vibration, the amplitude of vibration is found to be dependent on the different between f and f0. and is maximum when f=f0.A: light dampingB: Heavy dampingThe amplitude can be large when f=f0, as long as damping is small.This is called resonance. The natural frequency f0 is also called resonant
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