PHYS 1443 – Section 501 Lecture #25AnnouncementsSimple Block-Spring SystemMore Simple Block-Spring SystemExample for Spring Block SystemSlide 6Energy of the Simple Harmonic OscillatorExample for Energy of Simple Harmonic OscillatorThe PendulumExample for PendulumTorsion PendulumDamped OscillationMore on Damped OscillationWednesday April 28, 2004 1 PHYS 1443-501, Spring 2004Dr. Andrew BrandtPHYS 1443 – Section 501Lecture #25Wednesday, April 28, 2004Dr. Andrew Brandt1. Simple Block-Spring System2. Energy of the Simple Harmonic Oscillator3. Pendulums4. Damped oscillatorWednesday April 28, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt2Announcements•The last HW on Ch. 12+13 is due tonight•No optional sections (*) in Ch 11-15 Explicitly, covered 11.1-5,7 12.1-6 13.1-8 14.1-5,7 •Last lecture will be Monday May 3 on Ch. 15•On May 5, I will answer any questions that you may have on any of the material. If you have none it will be a short class—if you can use the time better with independent study that’s fine•Final exam will be Monday May 10 @5:30. It will be multiple choice only, so bring a Scantron form. It will be comprehensive, perhaps with a little more emphasis on ch 10-14.Wednesday April 28, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt3Simple Block-Spring SystemDoes this solution satisfy the differential equation?A block attached at the end of a spring on a frictionless surface experiences acceleration when the spring is displaced from an equilibrium position.This becomes a second order differential equation tAx cosLet’s take derivatives with respect to timexmka xmkdtxd22If we denotemk2The resulting differential equation becomesxdtxd22Since this satisfies condition for simple harmonic motion, we can take the solutiondtdxNow the second order derivative becomes22dtxdWhenever the force acting on a particle is linearly proportional to the displacement from some equilibrium position and is in the opposite direction, the particle moves in simple harmonic motion. tdtdA cos tsin tdtdsin tcos2x2see boardWednesday April 28, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt4More Simple Block-Spring SystemHow do the period and frequency of this harmonic motion look?Since the angular frequency isThe period, T, becomesSo the frequency is •Frequency and period do not depend on amplitude•Period is inversely proportional to spring constant and proportional to massTfWhat can we learn from these?mk2km2T12mk21Wednesday April 28, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt5Example for Spring Block SystemA car with a mass of 1300kg is constructed so that its frame is supported by four springs. Each spring has a force constant of 20,000N/m. If two people riding in the car have a combined mass of 160kg, find the frequency of vibration of the car after it is driven over a pothole in the road.Let’s assume that mass is evenly distributed on all four springs. Thus the frequency of vibration of each spring is The total mass of the system is 1460kg.Therefore each spring supports 365kg each.From the frequency relationship based on Hooke’s law fHzsmkf 18.118.13652000021211How long does it take for the car to complete two full vibrations?The period is skmfT 849.021For two cycles sT 70.12 T12mk21Wednesday April 28, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt6Example for Spring Block SystemA block with a mass of 200g is connected to a light spring for which the force constant is 5.00 N/m and is free to oscillate on a horizontal, frictionless surface. The block is displaced 5.00 cm from equilibrium and released from rest. Find the period of its motion.From Hooke’s law, we obtain From the general expression of the simple harmonic motion, the speed is As we know, period does not depend on the amplitude or phase constant of the oscillation, therefore the period, T, is simplyTDetermine the maximum speed of the block.maxvdtdx tAsinAsm /25.005.000.5 2 s26.100.52mk100.520.000.5 sWednesday April 28, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt7Energy of the Simple Harmonic OscillatorHow do you think the mechanical energy of the harmonic oscillator look without friction?Kinetic energy of a harmonic oscillator isThe elastic potential energy stored in the springTherefore the total mechanical energy ofthe harmonic oscillator is KEPEEETotal mechanical energy of a simple harmonic oscillator is a constant for a particular motion and is proportional to the square of the amplitude Maximum KE is when PE=0maxKESincemkOne can obtain speedE221mv tm222sin21221kx tk22cos21PEKE tktm22222cossin21PEKE tktk2222cossin21221kA2max21mv tm222sin212221m221 kPEKE 222121kxmv 221 kv 22xAmk 22xA xA-AKE/PEE=KE+PE=kA2/2Wednesday April 28, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt8Example for Energy of Simple Harmonic OscillatorA 0.500kg cube connected to a light spring with a force constant of 20.0 N/m oscillates on a horizontal, frictionless track. a) Calculate the total energy of the system and the maximum speed of the cube if the amplitude of the motion is 3.00 cm.The total energy of the cube isEFrom the problem statement, A and k aremNk /0.20Maximum speed occurs when kinetic energy is the same as the total energy2maxmax21mvKE b) What is the velocity of the cube when the displacement is 2.00 cm.velocity at any given displacement isvc) Compute the kinetic and potential energies of the system when the displacement is 2.00 cm.Kinetic energy, KEKEPotential energy, PEPEmcmA 03.000.3 PEKE 221kA J321000.903.00.2021E221kAmaxvmkAsm /190.0500.00.2003.0 22xAmk sm /141.0500.0/02.003.00.2022221mv J321097.4141.0500.021221kx J321000.402.00.2021Wednesday April 28, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt9The PendulumA simple
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