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ROCHESTER PHY 121 - PHY 121 Lecture 20 Notes

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Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121, April 3, 2008.Equilibrium and Simple Harmonic Motion.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.April 3, 2008.• Course Information• Topics to be discussed today:• Requirements for Equilibrium (a brief review)• Stress and Strain• Introduction to Harmonic MotionFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.April 3, 2008.• Homework set # 7 is due on Saturday morning, April 5, at8.30 am. This assignment has two components:• WeBWorK (75%)• Video analysis (25%): you can calculate the angular momentumquickly by using the expression for the vector product in terms of thecomponents of the individual vectors (the x and y components of theposition and velocity of the puck).• Homework set # 8 is now available. This assignmentcontains only WeBWorK questions and will be due onSaturday morning, April 12, at 8.30 am.• Exam # 3 will take place on Tuesday April 22.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterComments on Homework # 7.Rolling motion causes much confusion!Two views of rolling motion: 1) Pure rotation aroundthe instantaneous axis or 2) rotation and translation.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterComments on Homework # 7.Rolling motion causes much confusion!Note: friction provides thetorque with respect to the center-of-mass.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterHomework # 8.Due: Saturday April 12, 2008.All problems in this assignment are relatedto equilibrium. In all cases you need toidentify all forces andtorques that act on thesystem.Remember to choose thereference point in a smartway!Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.Quiz lecture 20.• The quiz today will have 3 questions!Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterEquilibrium (a quick review).• An object is in equilibrium is thefollowing conditions are met:Net force = 0 N (first conditionfor equilibrium) . This implies p= constant.andNet torque = 0 Nm (secondcondition for equilibrium). Thisimplies L = constant.• Conditions for static equilibrium:• p = 0 kg m/s• L = 0 kg m2/sFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterEquilibrium.Summary of conditions (a quick review).• Equilibrium in 3D:• Equilibrium in 2D: Fx!= 0"x!= 0Fy!= 0 and"y!= 0Fz!= 0"z!= 0 Fx!= 0Fy!= 0"z!= 0Torque condition must be satisfiedwith respect to any reference point.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterStress and strain.The effect of applied forces.• When we apply a force to anobject that is kept fixed at oneend, its dimensions can change.• If the force is below a maximumvalue, the change in dimension isproportional to the applied force.This is called Hooke’s law:F = k ΔL• This force region is called theelastic region.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterStress and strain.The effect of applied forces.• When the applied force increasesbeyond the elastic limit, the materialenters the plastic region.• The elongation of the materialdepends not only on the applied forceF, but also on the type of material, itslength, and its cross-sectional area.• In the plastic region, the materialdoes not return to its original shape(length) when the applied force isremoved.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterStress and strain.The effect of applied forces.• The elongation ΔL can bespecified as follows:whereL0 = original lengthA = cross sectional areaE = Young’s modulus• Stress is defined as the force perunit area (= F/A).• Strain is defined as the fractionalchange in length (ΔL0/L0).Note: the ratio of stress to strain is equal to the Young’s Modulus.!L =1EFAL0Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterStress and strain.Direction matters.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterStress and strain. A simple calculation couldhave prevented the death of 114 people.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterStress and strain. A simple calculation couldhave prevented the death of 114 people.Initial DesignActual DesignCredit: http://www.glendale-h.schools.nsw.edu.au/faculty_pages/ind_arts_web/bridgeweb/Hyatt_page.htmFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterAnd now something completely different!Harmonic motion.• We will continue our discussion of mechanics with thediscussion of harmonic motion (simple and complex). Thismaterial is covered in Chapter 14 of our text book.• Chapter 14 will be the last Chapter included in the materialcovered on Exam # 3 (which will cover Chapters 10, 11, 12,and 14).• Note: We will not discuss the material discussed in Chapter13 of the book, dealing with fluids, and this material will notbe covered on our exams.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterHarmonic motion.Motion that repeats itself at regular intervals.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple harmonic motion.AmplitudePhase ConstantAngular Frequency x(t) = Acos!t +"( )Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple harmonic motion.• Instead of the angular frequency ω the motion can also bedescribed in terms of its period T or its frequency ν.• The period T is the time required to complete oneoscillation:x(t) = x(t + T)orAcos(ωt + φ) = Acos(ωt + ωT + φ)• In order for this to be true we must require ωΤ = 2π. Theperiod T is thus equal to 2π/ω.• The frequency ν is the number of oscillations carried out persecond (ν = 1/T). The unit of frequency is the Hertz (Hz).Per definition, 1 Hz = 1 s-1.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple harmonic motion.• The frequency of the oscillationis the number of oscillationscarried out per second:ν = 1/T• The unit of frequency is the Hertz(Hz). Per definition, 1 Hz = 1 s-1.Frank L. H. Wolfs Department of


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