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# Ch 4: Contact Interactions Study Guide PHYS 172

Purdue

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##### ENGR 0020: Probability and statistics for Engineers I

###### Emily Binakonsky

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This 6 page Study Guide was uploaded by Sara on Tuesday March 10, 2015. The Study Guide belongs to PHYS 172 at Purdue University taught by in Winter2015. Since its upload, it has received 48 views.

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Date Created: 03/10/15

Chapter 4 Contact Interactions 0 contact forces between solid objects are due to the compression or stretching of springlike interatomic bonds 0 contact forces 0 stretch forces tension 0 compression forces normal 0 friction forces 0 buoyancy forces 0 a spring mass system oscillates sinusoidally Q the speed of sound in a solid can be understood in terms of atomic oscillations 41 Tarzan and the Vine 42 A Model of a Solid Balls Connected by Springs 0 Properties of Atoms 0 all matter consists of atoms whose typical radius is about 1 X 10quot 10 meter 0 atoms attract each other when they are close to each other but not too close 0 atoms repel each other when they get too close to each other 0 atoms in solids liquids and gases keep moving even at very low temperatures 0 the perfect distance to keep something together is called the equilibrium distance between atoms 0 a chemical bond is like a spring 0 a ball spring model for a solid object O crystalline structure 0 cubic lattice atoms located at corners of adjacent cubes 0 body centered cubic additional atom at the center of each cube 43 Tension Forces 0 the magnitude of the tension force exerted by the wire depends on the mass of the ball 0 the bonds between the atoms in a wire or string stretch when weight is appliedwhen not sitting at equilibrium 0 the tension is the same throughout the wire equally dispersed throughout the atoms 44 Length of an lnteratomic Bond 0 we are gonna need the length of an interatomic bond to find its stiffness 0 density does not depend on size shape or mass of an object 45 The Stiffness of an lnteratomic Bond 0 now that we have the length of an interatomic bond the basic idea is to hang heavy masses on a long wire and measure the stretch of the wire then by figuring out how many interatomic springs there are in the wire we can determine the spring constant of a single interatomic bond 0 to do this we need to be able to related the stiffness of an object composed of many springs to the stiffness of each individual spring 46 Stress Strain and Young s Modulus Q Young s Modulus lnitial Definition 0 a property of a particular material that is independent of the shape or size of a particular object made of that material 0 microscopic measure of the stretchability of a solid material 0 relates the fractional change in length of an object to the force per square meter of crosssectional area applied to the object 0 Strain strain ALL AL amount of stretch of the wire L original length of wire 0 Stress stress FTA FT tension force A cross sectional area of wire 0 Young s Modulus O Y stressstrain FTA ALL O the stiffer the material the larger Y will be 0 Another form that relates stress and strain l FTA YALL 0 Limit of Applicability of Young s Modulus 0 can t have too much stress stretches too much to breaking point 0 stress and strain need to be proportional for it to work see graph figure 419 0 stress is the cause independent variable 0 strain is the effect dependent variable 0 Relating Young s Modulus to lnteratomic Spring Stiffness relating macroscopic properties to microscopic properties 0 Y s M in terms of Atomic Quantities I Y k3 d l ksi stiffness of an interatomic bond in a solid GOO GOO l d length of interatomic bond and diameter of an atom 47 Compression Normal Forces 48 Friction Q FNET O for a constant velocitynot just simply FAPPLIED 0 Friction Doesn t Always slow things Down 0 Sliding Friction O Ffriction UKFN 0 uk coefficient fo kinetic friction typically between 0 and 1 0 only an approximation 0 Static Friction O basically same as sliding friction but object is not moving 0 Ffriction USFN 0 when us gt uk there is a stick and slip motionlarge us force starts the motion and when the friction force slowly decreases to uk level the net force becomesi m a little confused here ask in SI 0 if the net force is constant vavg vi vf2 49 Speed of Sound in a Solid and lnteratomic Bond Stiffness Q The Propagation of Stretch or Compression O the equal stretch or compression of bonds for ex in a Tarzan vine does not happen instantaneously it starts at the closests bonds and quickly spreads 0 speed of sound the rate at which the bonds stretch or the rate of propagation of the boundary between the stretched and unstretched regions 0 sound disturbance in a material 0 wave propagation of a disturbance of sound 0 An Iterative Model of Sound Propagation in a Metal 0 an iterative model is good for understanding how the speed of sound works but its not quite as practical as sayan algebraic expression solets look into that 410 410 Derivative Form of the Momentum Principle 0 So farAp FnetAt and pf p FnetAt Q The Momentum Principle Derivative Form dpdt Fne O the instantaneous time rate of change of the momentum of an object is equal to the net force acting on the object 0 Acceleration 0 F ma can be used only when l the mass is constant I the speed is much less than the speed of light O a dvdt O the rate in change of velocity with respect to time Q knowing this we can say the approx rate of change of momentum is l dpdt ma I so ma Fnet nonrelativistic form constant mass 0 full form of momentum principle dpdt Fnet is more accurate tho even when mass is changing and correct at any speed 411 Analytical Solution SpringMass System 0 Applying the Momentum Principle to a SpringMass System 0 x Acoswt 0 w sqrtkSm angular frequency radianss O A amplitude maximum stretch of the spring during an oscillation Q Angular Frequency w Period and Frequency 0 Angular Frequency w 2piT sqrtkSm 0 Period T 2piw seconds 0 Frequency f 1T w2pi A Vertical MassSpring System Harmonic and Anharmonic Oscillators O harmonic oscillator l idealized massspring system I the position of the mass as a function of time is given by a cos function I the period round trip time is independent of the amplitude of oscillations O anharmonic example a freely bouncing ball Real Springs Have Mass Existence of Analytical Solutions lnitial Conditions for a MassSpring System 412 Analytical Expression for Speed of Sound 0 the lower mass of the atoms the faster a sound pulse disturbance might propagate through a rod 0 the stiffer the bond the greater the force the spring will exert on a neighboring atoms in response to the same displacementso stiffer interatomic bonds should make a sound pulse propagate faster through a rod 0 the larger the angular frequency is the faster an atom oscillates 0 v wd sqrtksima d O ksi stiffness of the interatomic bond 0 ma mass of one atom Q d length of interatomic bond 413 Contact Forces Due to Gases Q Buoyancy 0 density of the Earth s atmosphere decreases as you move towards space 0 there is an actual buoyant force upwards on a ball or balloon for example because the number of air molecules hitting the bottom of the ball per second is slightly greater than the number of molecules hitting the top of the ball persecond O buoyant force interatomic contact force due to fluid molecules striking atoms in the surface of the ball or air I can be usually neglected because buoyant force mairg compared to the gravitational force mg of ball is miniscule l Brownian motion observing small particles being jostled by the random collisions with waterair molecules 0 Pressure 0 P FA force per unit area 0 atmospheric pressure at sea level is about 1X1Oquot5 Nmquot2 Q A ConstantDensity Model of the Atmosphere 0 Suction 414 A Vertical SpringMass System 0 just shows that period is the same vertically as it is horizontally 415 General Solution for the MassSpring System 0 I really don t think we will need to know this stuff SUMMARY 0 Contact forces 0 A solid object may be modeled as a 3D lattic of balls atoms connected by springs interatomic bonds 0 Based on this model and given the density and atomic weight of an element one can find the length of an interatomic bond in a solid metal 0 From macroscopic measurements of stress and strain one can determine the stiffness of an interatomic bond in a solid material 0 Solid objects exert contact forces because the interatomic bonds in the solid are stretched a tension force or compressed a normal force 0 Friction forces arise when one object slides on another O The speed of sound in a solid is the speed at which a disturbance propagates through the material 0 DefinitionsEquationsMacroMicro Connections 0 strain ALL 0 stress FTA O Young s Modulus Y FTA ALL ksid 0 speed of sound in a solid v sqrtk3imatom d O Momentum Principle derivative form gt dpdt Fnet l Fnet ma for when v is much less than the speed of light and has a constant mass 0 Massspring oscillations l x Acoswt l Angular Frequency w sqrtkSm l Period T 2piw l Frequency f 1T w2pi

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