General Physics PHY 007B
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This 76 page Class Notes was uploaded by Horace McClure on Tuesday September 8, 2015. The Class Notes belongs to PHY 007B at University of California - Davis taught by John Conway in Fall. Since its upload, it has received 880 views. For similar materials see /class/191835/phy-007b-university-of-california-davis in Physics 2 at University of California - Davis.
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Physics 7B DLM 1 Overview Unit 4 Momentum and Force ModelApproach Galilean SpaceTime Model CG 41 Using Vectors to represent Force Velocity and Other Vector Quantities Act 411 Working with Position and Displacement Vectors 60 min Learning Goals 39 Practice drawing vectors and adding and subtracting them using the tailtohead method 39 Understand the difference between distance and displacement 39 Understand the difference between displacement and position vectors and how one arises from the other 39 Understand that the displacement vector does not depend on the choice of coordinate system 39 Understand the relationship between the displacement vector and the average velocity vector Act 412 Using Vectors to Represent the Motion of a Weight Moving in a Circle 60 min Learning Goals 39 Practice carefully observing motion and representing this motion using vectors Practice using perpendicular components to represent a vector Understand that a vector does not depend on the perpendicular axes chosen to define the components Practice finding components using the graphical method Start discussing FNTs 20 min Announcements 39 Buy your Physics 7B Course Notes and your clicker student response system at the MU Bookstore Before the next DL read Unit 4 through page 25 and Summary and Review pages 1071 1 l 39 Quizzes will be in lecture Quizzes are closed book and closed notes You will occasionally need a calculator so always bring one with you 39 Be sure to check the Physics 7B webpage periodically httpphysics7ucdavisedu Check often for new information and material The FNTs should be available at this site as well as various announcements and all office hours thsics 7B DL Activitv 411 DLM 01 Working with Position and Displacement Vectors Make sure everyone in your group fully understands the ideas behind each question or part in these activities before going on to the next part Phenomenon You are in a strange city that has streets that are laid out in a perfectly square grid Your job will be to move around to several locations as instructed and record your progress All members of your group must now get up go to the board and work together on the responses to the following 1 Make a drawing on the board showing the streets in the central city This should be a square grid with at least 10 streets going in each direction Decide as a group how you are going to label the streets and use your labeling system Make sure your diagram is large enough for everyone in the room to clearly see it 2 Near one of the edges of the diagram label a street corner 0 for origin Your origin should be dilTerent than any other group Imagine you are standing at this point Now imagine walking toward the center of the diagram making several 90 turns as you walk Using the labeled origin draw a position vector on your diagram that shows where you ended up Label this vector as R i stands for initial a Determine the length of your position vector you might need the Pythagorean Theorem for this How you can describe the direction of the position vector What units does it have Whole Class Sharing b Continuing from your location above now imagine walking a distance equal to three blocks in a direction in space that is straight up on the board Show the position vector for this new location Remember position vectors start at the origin Label this vector Rf Look around the room and check to see if you are doing what other groups are doing c Off to the side of your diagram graphically add a new vector labeled with the symbol AR to the initial position vector so that the sum is the nal position vector using the tailtohead method as shown on page 7 of the Course Notes Make sure you draw these vectors with the same length and direction that they have on your drawing AR is called the displacement vector What does it physically represent Transfer your displacement vector to your street diagram d Write a vector addition equation from part c near your diagram It should have the form A B C Which vectors are A and B and which is C Does B A also equal C Show this What are the differences between position vectors and displacement vectors Is this the distance you walked along your path represented by any of these vectors Write this on the board and be prepared to share with the whole class Whole Class Sharing D V f Suppose you walked the three blocks up in 60 seconds Draw the average velocity vector for this situation What is the length of this vector Include the units Draw another velocity vector for a situation where you took 180 seconds to walk the three blocks Which arrow is longer Why g If you now walked back to your starting point 0 and assuming the total journey took eight hours what is your average velocity for the entire trip Whole Class Sharing 3 Look around the room at the different street diagrams What is the same in each one What is different Discuss in your group how you would summarize the meaning of everything you did in this activity Be prepared to share with the whole class How is this information useful for describing motion thsics 7B DL Activitv 412 DLM 01 Using Vectors to Represent the Motion of a Mass Moving in a Circle Make sure everyone in your group fully understands the ideas behind each question or part in these activities before going on to the neXt part Phenomenon You are going to swing a ball in a circle and represent its velocity several di erent ways using vectors 1 a Get the ball swinging in a large horizontal circle going clockwise when viewed from above Imagine looking down on the weight Pick a direction in space that you will identify as the 12 o clock position Using a coordinate system that has the XaXis running from the 9 to the 3 o clock position draw X and yaXes and draw the circular path of the weight on the board b Draw a position vector identifying the position of the weight when it is in the 4 o clock position Show the X and y components of this vector on your diagram Discuss in your group how to do this Be prepared to share with the whole class c Draw another position vector when the weight is in the 5 o clock position To the side of your diagram graphically subtract the position vectors as accurately as possible to nd the displacement vector label it Ar Write an English sentence explaining the meaning of the delta Quick Whole Class Sharing 2 a Describe in words the velocity of the ball as it moves in its circle Be prepared to share b What is an instantaneous velocity vector How is it different from a position vector Does it always point in the same direction If you are having trouble answering these questions apply them to a speci c situation eg driving northwest on 15 at 65 mph and then curving toward the north 3 a Imagine that you are sitting on the ball and driving it in a circle Which way are you moving when you are at the 4 o clock position Draw a velocity vector put the tail of the vector at the weight s location showing the velocity ofthe weight at the 4 o clock position Show X and y components of this vector on your diagram Discuss in your group how you do this b Looking at the vectors you have drawn is a position vector or a displacement vector more closely related to a velocity vector Make sure this agrees with your knowledge of the de nition of velocity c Which vectors on your diagram would be different if you changed the origin Whole Class Sharing 4 Draw the velocity vector for the weight at the 6 o clock position Off to the side graphically determine the change in velocity AV between the 4 o clock position and the 6 o clock position 5 a Find the X and y components of the 6 o clock velocity vector b Find the X and y components of the vector Av 6 How can you use the X and y components of the 4 o clock and the 6 o clock positions to get the change in velocity Av Summarize your method and be prepared to share it with the class Whole Class Discussion Physics 7B Exit Handout DLM 01 FNTs We won t be covering these in detail so come with questions 1 Three force vectors are added together One has a magnitude of 9 N the second one a magnitude of 18 N and the third a magnitude of 15 N What can we conclude about the magnitude of the net force vector Explain I It must equal 42 N 11 It cannot be 0 N 111 Anything from 742 N to 42 N IV It can be anything from 0 N to 42 N V None of the above can be concluded 2 Alice Bob and Chuck are three friends standing around talking We know that Alice is standing Ul V 9 m away from Bob and that Bob is standing 3 m away from Chuck Let AR AB be the vector that starts at Alice and ends at Bob and ARBC be the vector that starts at Bob and ends at Chuck a Write an equation to find the displacement between Alice and Chuck b What is the furthest Chuck can be from Alice given the information in the question What is the closest they can be c Draw a picture where Chuck is neither as close as he can be or as far as he can be from Alice Draw and label the vectors AR AB and ARBC Using a piece of graph paper carry out the operations listed below again on the vectors shown to the upper right Label all vectors a 2F A B b 2F B A c Ap A B d 2FABC e quotAP B 39 A unit length Vectors F1 an object F2 an object and F3 an object are all exerted on an object adding together to form a net force vector 2F as shown in the second graph to the right However only vectors F1 an object F2 an object and SF are known On a separate piece of graph paper use the properties of vector addition to graphically determine the vector F3 on object VeCtorS F1 on object F2 on object F3 on object and F4 on object are all exerted on an object adding together to form a net force vector 2F 0 as shown to the right However only vectors F1 an object F2 an object and SF which is zero are known It is known that F3 an object is completely vertical and F4 an object is completely horizontal On a separate piece of graph paper use the properties of vector addition to determine the magnitudes of the vertical vector F3 an object and the horizontal vector F4 an object Slcs Xi an ou Phy39 7B E tH d t DLM01 6 Two force vectors F1 and F2 as shown to the right act on a 2 kg object that has an initial velocity v I of 3 ms in the x direction a Find the x and y components of the net force use trigonometry b Find the magnitude and direction of the net force 7 Two rolling carts are moving toward each other at the same speed Cart 1 has a mass m1 200 g and cart 2 has a mass m 400 g a Draw a velocity vector V for each cart b Momentum p is a vector de ned as p mV Draw a momentum vector for each cart 0 Add the two momentum vectors together to find the total momentum ptotal P1 P2 Physics 7B DLM 5 Overview Unit 4 Angular Momentum and Torque ModelApproach Angular Momentum Conservation Model Act 426 Wrap up of linear momentum FNTs 1 6 from DLM 4 70 min Learning Goals 39 Get more practice with both qualitative and analysis of inelastic collisions in both 1D amp 2D 39 Get more practice working with vector components of momentum and related quantities 39 Develop a deeper understanding of the independence of motion in perpendicular directions 39 Get more practice applying the impulse Amomentum relation Act 431 Introduction to Angular Momentum and Torque 70 min Learning Goals 39 Become comfortable with the constructs of angular velocity and angular momentum including their vector nature as they apply to spinning a bike wheel 39 Become comfortable with the constructs of torque and angular impulse including their vector nature as they apply to spinning a bike wheel Develop understanding of the dependence of torque on how a force is applied Get practice using the righthand rule to determine directions of 0 1 angJ AL and L Announcements 39 Be sure to check the Physics 7B webpage periodically for new information and material various announcements and all office hours Physics 7B A Activity 426 DLM 05 Wrapup of Linear Momentum FNTs 1 3 from DLM 4 Phenomenon Inelastic collisions in both one and two dimensions 1 2 V 3 4 In your group compare your individually created momentum charts for FNT l and 2 Note that you do not need to complete the entire chart to simply nd the final velocity That is you don t need to nd the change in momentum of each asteroid separately However nding the individual changes is a good opportunity to hone your skills with momentum conservation Note You may nd it useful to choose an xy coordinate system and ll in the chart for FNT2 with x and y components On the board put up a complete momentum charts with equations worked through to numerical values for both the linear collision in FNT l and the rightangle collision in FNT 2 Compare your answers to FNT3 come to a consensus and put your responses on the board Many body problem Three asteroids collide and stick together diagram a possible momentum chart Whole Class Discussion B FNTs 4 6 from DLM 4 Phenomenon Horizontally thrown and dropped balls 1 2 VV 4 In your group compare your individually created momentum charts for FNT 4 and 5 It is simplest to put one header row with labels pi Ap and pf and then start a new row in your chart for each time step Put your charts for FNT 4 amp 5 on the board Discuss and be prepared to share with the class the question Descn be in words how the motion changes in the two directions Compare your diagrams from part 4c amp 5 Are the paths you constructed still the same and the question in FNT 68 Wrap up what you have done in questions 4 and 5 by explaining in as few words as possible why the ball moves in this path Come to a consensus on the questions in FNT 6b and 6c and be prepared to share them with the whole class Whole Class Discussion Prac comp C tice quiz spend no more than 5 minutes as a WC explaining to your DL instructor how to lete a momentum chart and the answers to this former FNT Two hockey goalies on ice skates block speeding pucks Their ice skates point in the direction they move so we can model them as on a frictionless surface Goalie A catches the puck in her glove while goalie B lets it bounce off of his stomach and return with a slower speed in the direction it came from Both goalies have the same mass and were initially stationary and both pucks have the same mass and were initially moving at the same speed a Explain which goalie exerts the greater impulse on the puck b Explain which goalie has the greater impulse exerted on himher by the puck c After blocking the puck which goalie will be moving the fastest Use impulse to explain why Physics 7B Activity 431 DLM 05 Angular Velocity Momentum and Torque With a Bike Wheel Phenomenon Change of Angular momentum of a stationary wheel clamped to a table All of these questions and activities center around the bike wheel with an axle clamped to the table Reference You will need to be constantly referring to your Course Notes on Angular Momentum Conservation in this activity Make sure you have these notes out in front of you A Rotation with constant speed Spin the wheels at your table so that they are rotating with a constant speed 1 Describe in everyday English words precisely what a is and what L is Which can you observe 2 Draw the wheel and axle and show the vectors a and L on your drawings 3 Does a change over time Why or why not US Speeding up the rotation of the wheel and slowing it down Carefully observe how a member of your group uses herhis hand to increase the angular velocity of the wheel and then to decrease the angular velocity of the wheel You should note precisely what the hand is doing to make the wheel i speed up and ii slow down 1 Describe in everyday English words precisely what the hand did to make the wheel speed up and to slow down N V Use the information on pages 29 to 32 of the Notes as needed to describe what the hand did to the wheel in each case using the technical language of torque angular impulse and change in angular momentum 3 For the case of the wheel speeding up draw a set of sketches showing the vectors a 1 angJ AL and Li amp Ly Draw an initial wheel showing Li and Di a change wheel showing 1 angJ and AL and a nal wheel showing L and m Repeat for the wheel slowing down 4 Put your two descriptions and vectors from 1 2 and 3 on the board Be prepared to explain to the whole class Make a simple angular momentum chart Whole Class Discussion Turn page over and complete activity Physics 7B Activity 431 page 2 DLM 05 C Creating the torque with your finger Each group member must do this Start the wheel turning by using one nger pushing hard against a spoke near the outer rim of the wheel and observe and feel how an initially stationary wheel responds Observe how this compares to pushing hard very close to the central hub to start the wheel turning Also try to stop the wheel by placing your nger at either of the previous locations Which location is it easier to stop the wheel If you are having trouble noticing a distinction use one finger to push open the DL door near the hinge versus near the door handle 1 Describe in everyday English words precisely what was different about how you pushed and how the wheel responded in each case pushing near the rim and pushing near the hub 2 Use the information on pages 29 to 32 of the Notes as needed to explain the differences using technical language that you previously described in 1 using everyday language Make a simple drawing to illustrate your technical description and to show explicitly the difference between the two cases of pushing near the rim and near the hub 3 Summarize on the board what torque is what it depends on and what it does Whole Class Discussion 4 Now you will determine the difference between net Torque net Force angular Impulse and Impulse When you are pushing on the spoke of the wheel as in B is there a net Force acting on the wheel Is there a net Torque To answer do the following a b d You should already know how to answer the net force question by examining the change in the linear momentum of the wheel The wheel s linear momentum is zero before you push on it Does its linear momentum change when you push on it In other words does the wheel y up into the air or drop to the oor or fly across the room in any way What does your answer imply about the net force SF on the wheel when you are pushing on it Draw a properly scaled force diagram for the wheel when you are pushing on it Now draw a simple picture of the wheel and transfer the forces from your force diagram onto this picture of the wheel Place the forces on your picture at the points where they are actually acting We call this an extended force diagram Determine the direction of 21 from the vectors drawn in 2 on the extended force diagram Are your values for SF and 21 consistent with what you observed for Ap and AL When you are not pushing on the wheel is there a net Torque or net Force Why does the wheel continuing rotating Whole Class Discussion Physics 7B DLM 2 Overview Unit 4 Momentum and Force ModelApproach Galilean SpaceTime Model CG 41 Using Vectors to represent Force Velocity and Other Vector Quantities Act 413 Net Force includes DLMOl FNT s 70 min Learning Goals 39 Get practice identifying the net Force in various situations 39 Get more practice working with vectors addingsubtracting using the tailtohead method 39 Get more practice working with components and using components to determine net force 39 Get practice using sine and cosine to determine components ModelApproach Momentum Conservation Model CG 42 Momentum and Impulse Act 421 Momentum and Change in Momentum in One Dimension 70 min Learning Goals 39 Get practice identifying momentum and change in momentum in simple lD collisions Get practice identifying impulse Get practice representing impulse momentum and change in momentum as vectors using the scaledarrow representation 39 Get practice defining and analyzing collisions interactions representing conservation of momentum using both the scaledarrow and vectorequation representations Announcements 39 Buy your Physics 7B Course Notes at the Memorial Union Bookstore At the bookstore you will also need to buy the clicker for the lecture problems activities keep your receipt and rebate coupon and then send for your rebate on the clicker The first quiz will be in lecture 2 The quiz will cover all material from the activities and FNTS of the first 2 DLs DiscussionLabs Be sure to check the Physics 7B webpage periodically for new information and material The FNTs will be available at this site as well as various announcements and all office hours thsics 7B DL Activitv 413 DLM 02 Net Force Force Diagrams A DLM 1 FNTs In Your Small Group 1 Compare your responses to FNTs l7 with other members of your small group For FNT 2 also determine the length ofEF Ap amp Ap 2 Come to a consensus on an appropriate response to each FNT and be able to make convincing reasons why you know for sure your response is appropriate Each person in your group should be prepared to do this for any of the FNTs Whole Class Sharing Fby scale 2 B Phenomenon You are going to pull on a metal ring with three ropes F 0 quotg attached A spring is attached to each rope to determine the tension in the rope byscalel 0quot quotg 1 Helpful hints Before you start make sure each scale is set to zero by twisting the knob Work together as a group This takes multiple hands Persons 1 and 2 will change their pull direction and magnitude Person 3 should always pull in the same direction and always read 30N Attach the ropes with the scales white side up to read values in Newtons Scenario 1 As shown in the upper picture to the right have person 1 and person Fby scale 3 on ring 2 pull directly opposite to person 3 with enough force to make the yellow spring scale for the third person read 30N Record the values on all three scales Fb l ysca e 2 Scenarlo 2 Persons 1 and 2 each change the d1rectron they are pulling unt11 the1r Fby seal on ring on ring ropes make an angle of about 130 with respect to person 3 as shown in the lower picture to the right Record the readings of all scales To get the angles to be 130 trace the ring on a sheet of paper and make a vertical line pointing down from it then use the plastic protractor to mark the 130 angles from the vertical line Now you can line up the ropes along the lines you drew and then note the force scale readings a Use the paper with the angles to graphically add the appropriately scaled F force vectors arrows as accurately as possible Also show the components by scale 3 0quot mg b Make two properly labeled and scaled force diagrams on the board one for each scenario above See Course Notes pages 89 for labeling conventions c On the board off to the side of each force diagram reproduce the vector addition from a What is the net force in this case Why does this make sense Quick Whole Class Sharing d Use trigonometry sine and cosine as appropriate to nd the magnitude of the components X and y of the two forces F1 ourng and F2 on ring i What is the relation between the components of these two forces and those of the third force ii Develop an explanation for this relationship in your small group and be ready to share iii Explain why the scale readings changed when the two people pulling parallel separated Whole Class Sharing 2 You are again going to pull on the metal ring with the three ropes Start at the same 130 angles as before With person 3 still reading 30N have persons 1 and 2 pull with a significantly greater force and your angle apart should widen Draw a force diagram and repeat the three parts of 1d above 3 Could two people pull their ropes all the way to 180 apart with the third person still pulling at 30N Include a force diagram in your response Whole Class Sharing thsics 7B DL Activitv 421 page 1 DLM 02 Momentum and Change in Momentum In Two and Three Dimensions Phenomenon Impulses and changes in momentum in more than onedimension A The essential elements of the Momentum Conservation Model See page 2325 of the Course Notes Meaning of Impulse Net Impulseext 2F At f2Fexta dt avg ext APsystem Pf Sys pi Sys 2 Net Impulseext Statement of Conservation of Linear Momentum APsystem Pf Syg pi Sys 0 01 Net Impulseext 0 or ZFeXt 0 1 In your small group describe an example of an impulse Identify two ways you can change this impulse For instance how could you make the impulse greater Describe both a a system that involves one object and b a system that involves two interacting objects Explain what Apsys is for each of your physical systems Put this on the board 2 In your small group develop a statement in your own words of what conservation of momentum means for your two systems in l B Representation of Conservation of Momentum with Vectors See page 25 of Course Notes Just as an energy system diagram is useful in helping us work through conservation of energy questionsproblems a momentum chart is useful for questionsproblems using conservation of momentum The momentum chart like an energysystem diagram helps us keep track of what we know about the interaction and also helps us see what we do not know All momentum charts are to be lled in with scaled arrows representing momentum vectors Typically used for collisionsinteractions T 39 ll al n th h involving two or more objects also useful yplca y use w en ep enomenon involves a net impulse acting on the system also useful for vector components Pi AP Pf Net Impulse pi AP pf System Object 1 Total System Object 2 For total system Ap 2 Net ImpulseexL piAppf Total System 0 Show the force diagram here that For total system Ap 0 determines EFAt 2 Net Impulseext For each object pi Ap pf Can also write expressions for each momentum vector such as p mv in additional equations related to the chart 1 On your boards complete momentum charts for both of your examples from part A 2 Each row and each column is a separate vector equation Check that every row equation and every column equation is added correctly and is consistent 3 Which column is signi cant for momentum conservation 4 Determine what parts of the momentum charts are analogous to energy system diagrams List the analogous parts on the board In what fundamental way do these diagrams differ thsics 7B DL Activitv 421 page 2 DLM 02 Momentum and Change in Momentum in One Dimension I C Phenomenon collisions of one cart with another obiect You are going to observe talk about in your group and analyze simple collisions in onedimension using aluminum carts that slide on a long track PLEASE be gentle with the carts and track Thank you Make sure everyone in your group fully understands the ideas behind each question or part in these activities before going on to the next part 1 Arrange a collision so a cart so it sticks to the bumper Observe the collision several times Analyze the collision using the Momentum Conservation Model a Use vectors to represent the various momenta Draw on the board a momentum chart Each column should have appropriately scaled and labeled vectors for either the momentum or the impulse that acted during the collision b Write a vector equation for the momentum off to the side of your chart with appropriate subscripts on the symbols to express what you observed during this collision c Describe in words what physically happened and how conservation of momentum applies in this situation Draw a force diagram for the cart for the time when its momentum was changing Put your diagrams equation and word statement on the board but leave enough space so you can write the responses to 2 for comparison 2 Arrange a collision so a cart bounces off the bumper Observe the collision several times Analyze the collision using the Momentum Conservation Model Do a b and c as in l 3 When total kinetic energy is conserved in a collision we call it an elastic collision If it is not conserved we call it an inelastic collision Classify the two collisions above as elastic or inelastic 4 Does an inelastic collision Violate conservation of energy If not which other energy systems could he kinetic energy have gone to 5 Is momentum conserved in the inelastic case and the elastic case Where in your momentum chart is this shown Physics 7B Exit Handout DLM 02 FNTs 1 2 3 4 5 Rework the parts of Activities 421 you are not sure of Bring your questions to the next DL meeting For each part of this problem suppose that an initial state consists of two of the carts each with the same mass m that you used in this DL In the initial state these two carts are moving toward each other with the same initial speed vi along the track These carts collide and the result is some final state The three parts of this question are concerned with three different final states a Assume that the carts hit each other and stop so that the final state of the system has both carts just sitting still not moving Draw a momentum chart for this situation Is the total kinetic energy before the collision equal to the total kinetic energy after the collision If not what energy system might have increased b Assume that the carts bounce off each other so that the final state of the system has each cart moving oppositely to its initial motion but with the same speed Draw a momentum chart for this situation Is the total kinetic energy before the collision equal to the total kinetic energy after the collision If not what energy system might have increased 0 As in b assume that the carts bounce off each other but now assume that the final speed is smaller than the initial speed but not zero Can both momentum and kinetic energy be conserved with these smaller speeds d In general is there any way to conserve both momentum and kinetic energy in this collision if the final speeds are not the same as the initial speeds Explain e Is the collision in b elastic or inelastic What about the collision in 0 Explain how you know A rocket works by expelling gas at a high velocity out of the back of the rocket We are going to treat the rocket as being far away from any gravitational objects a Draw a momentum chart for the rocket expelling gas in space b Use your chart to explain why the rocket accelerates 0 Does the rocket have to keep expelling gas to stay at a constant speed Explain Victoria is standing on a boat during a perfectly calm day Then Victoria walks from one end of the boat to the other Draw a momentum chart for this situation Does the boat move and if so which way Note This is a continuation of FNT 5 from DLM 1 Two force vectors F1 and F2 as shown at right act on a 2 kg object that has an initial velocity vi of 3 ms in the x direction 0 Use the xy components of the force you found in a to determine the xy components of impulse that would act on the 2 kg obj ect if the forces were applied for a time interval of 050 s Always include units with your answers Find separately for each component the change in velocity of the 2 kg object due to the impulse from c Find the magnitude of the velocity of the obj ect after the impulse as acted and the direction the velocity makes with the positive xaxis d e Welcome to Physics 78 Prof John Conway office 311 PhyGeo office hours Monday 9001030 phone 7547639 not recommended email conwayphysicsucdavisedu Course Organization Etc gt 10 lectures on Tuesdays gt quiz each week starting next week may be at the beginning middle or end of lecture will use PRS starting next week V two DL meetings per week final exam 19 March 1030 am VV main course website V httpllwwwph icsucdaviseduljconwaylteachingPhysics7BW07lcalendanhtml Vectors MotionForcesMomentum Prof John Conway Physics 78 Lecture 1 Vectors Defined A vector is a mathematical object which has gt magnitude length gt direction We often represent a vector by drawing an arrow Vectors Defined gt A vector does not have a position in space gt These two vectors are the same since they have the same direction and magnitude Vectors are Useful We can use vectors to represent gt position in space gt velocity gt forces gt fields electric magnetic gt what else For the mathematically inclined gt scalar single number 473 gt vector 1D array 341 152 221 390 gt tensor nD array 5 1 O 5 3 2 11 201 65 335 04 O5 Representing Vectors gt Let s start with a twodimensional vector which is five units in magnitude and oriented at an angle of 531 degrees from horizontal y A 35s4g E 3H4 34 I 3 units What are the funny rhings with hais Representing Vectors gt A vector is expressed in a particular coordinate system also called basis gt We often use Cartesian coordinates Xyz as the basis for vectors gt Coordinate system has origin scale orientation and handedness gt To understand how to write vectors though we need to first understand how to scale and add them Scaling Vectors gt Scaling a vector multiplying it by a scalar number makes it either longer or shorter 23 06 B Scaling a vector does not change its direction n Adding Vectors It isn t obvious at first how vectors add toghether Adding Vectors You simply put them head to tail to get the sum The sum of two vectors is another vector Unit Vectors v k The funny Things wi rh ha rs are uni r vec rors which lie along The coordina re axes and have leng rh 1 mm leng rh Building a Vector gt To build a vector we scale and add the unit vectors in our coordinate basis lt gtltgt Adding Vectors gt It is trivial to add vectors just add them component by component 32343 2iamp AB f517 Just remember to keep the x components distinct from the y components No mixing QI WI ibl Subtracting Vectors gt It is trivial to substract vectors just add the opposite component by component A 3934g B 2i 7 A B 5e3g Just remember to keep the x components distinct from the y components No mixing AddingSubtracting Graphically gt head to tail tail to head To add vec ror B To A 7 7 7 I quotmovequot rail of B To head of A 2 connect rail of A To head of B AddingSubtracting Graphically gt head to tail tail to head To sub rrac r vec ror B from A I quotmovequot rail of B To head of A B 2 connec r rail of A To head of B Vector Magnitude gt To find the magnitude length of a vector you simply take the sum of the squares of its components and then take the square root of that Pythagoras 3e4g V3242V255 Note that the magnitude of a vector is not a vector it s a scalar a single number gt 0 thin I A E Vector Components gt yes we have to use sines and cosines Y Av A sine 6 Ax A case Vector Projections b Often in physics we just want to know the projection of a vector parallel or perpendicular to some axis gt To do this you simply use the appropriate right triangle and noodle it out Position Vectors gt We use a vector with one end at the origin to indicate a position in space Y i In fact the position vector may be a function of time Momentum gt The product of the mass of an object and its velocity is its momentum Clearly since velocity is a vector momentum is also a vector Conservation of Momentum gt The total momentum of a closed system is always conserved constant d ZFemt dil gt This is true even if energy is not conserved example inelastic collision gt For various physical systems cars planets must consider conservation of both energy and momentum gt Closed system no external forces Vemo carts on frack Impulse gt For a nonclosed system subject to an external force the change in momentum is equal to the force times the time it acts Ap FAt impulse gt Review question what happens to a body subject to a force acting through a certain distance Physics 7B DLM 3 Overview Unit 4 Momentum and Force ModelApproach Momentum Conservation Model Act 422 Momentum and Change in Momentum in One Dimension 11 50 min Learning Goals 39 Get practice identifying momentum and change in momentum in simple lD collisions Get practice identifying impulse Get practice representing impulse momentum and change in momentum as vectors using the scaledarrow representation Get practice defining and analyzing collisions interactions representing conservation of momentum using both the scaledarrow and vectorequation representations Act 423 Momentum and Change in Momentum in Two and Three Dimensions 80 min Learning Goals Get practice identifying momentum and change in momentum in two and three dimensions Get practice identifying impulse from the net force in multiples spatial dimensions Get practice representing impulse momentum and change in momentum as vectors using the scaledarrow representation in more than one dimension Get practice defining and analyzing phenomena that occur in more than one dimension using the Momentum Conservation Model To understand that the net force causes changes in momentum To understand that 2F and Ap are parallel Announcements 39 Reading Assignment Read Summary and Review pages 96101 39 The rst quiz will be during the lecture on Tues Jan 16 thsics 7B DL Activitv 422 DLM 03 Momentum and Change in Momentum in One Dimension 11 FNTs 3 4 amp 2 from DLM2 In Your Small Group Compare your responses to FNT 3 and 4 from DLM2 with other members of your small group come to a consensus on the appropriate momentum charts and answers and put these on the board Added problem Draw an appropriately scaled force diagram for some time during the interaction for each problem Whole Class Discussion In Your Small Group Repeat the exercise as stated above for FNT 2 Added problem Draw an appropriately scaled force diagram for some time during the collision for each cart in part a and each cart in part b Whole Class Discussion Phenomenon Collisions of 2 carts in one dimension both elastic and inelastic collisions Finish Activity DLM2 Activity 421 as directed by your DL instructor Then use the Momentum Conservation Model discussed in 421 to analyze each of the collisions between two carts in the situations 1 5 below We treat the system made up of the two carts as a closed physical system because there is no net external impulse transferred to the system during the collision For each collision a Draw and fill in a momentum chart to help you describe momentum conservation in this closed physical system b Write an algebraic vector equation with appropriate subscripts on the symbols to express what happened during the collision c Draw a force diagram for each cart that shows the forces during the collision Describe in words what physically happened and then how conservation of momentum applies in to each cart and to the system as a whole Put your diagrams equation and word statement on the board d Classify each case as elastic or inelastic Use two carts of equal mass arranged so the carts will bounce off one another Start with one cart stationary 2 Repeat 1 but with the carts turned so the collision ends with the carts locked 3 Use two carts of equal mass carts initially moving toward each other with equal speed and still ending with the carts locked As you work through these scenarios observe what is the same in the scenarios and what is different What patterns can you observe Describe any rules you come up with for the patterns Whole Class Discussion p t V Optional if time permits else go to office hours if you have trouble working these on your own 4 Use two carts of unequal mass turned so the collision ends with the carts locked again one cart is stationary before the collision 5 Repeat 3 but with both carts initially moving toward each other with the same speed thsics 7B DL Activitv 423 DLM 03 Momentum and Change in Momentum in Two and Three Dimensions Phenomenon You are going to observe talk about in your group and analyze motions in three dimensions This will give you practice working with vectors and the concept of impulse in more than one dimension and practice determining the net force 1 Compare with your group your responses to FNT 5 from DLM 2 Come to a consensus and put your response on the board 2 Mass swinging in a horizontal circle Arrange the weight and give it a push so it swings in a horizontal circle as in DLM 1 Focus on a small section of the arc of the circle If we treat the moving mass as our physical system is there a net impulse on the system How do you know What must be the direction of the total or net Impulse acting on the mass How do you know this from the motion EXplicitly show the vectors Vi Vf and AV on a diagram drawn from above looking down on the motion What must the direction of the net force 2F be How do you know this from the motion E Fquot 90 Analyze the forces acting on the swinging object What objects exert forces on the swinging mass Remember these could be contact forces or long range forces Now think about a side view of the swinging ball Use what you know about the directions of these forces and the direction of EF to draw a force diagram of the swinging object in this side View Be sure to label all forces with two subscripts and make their lengths appropriately side View scaled with respect to the each other Show the net force separately with a double arrow Write a few sentences explaining why the mass swings in a circle in terms of momentum impulse and forces D quot1 Still treating our physical system as the moving mass is mechanical energy conserved that is does the mechanical energy of the system remain constant How do you know You should answer this in terms of energy systems changing or not as well as whether energy is transferred as work g Is the net force more closely related to Vi Vf or Av Put your responses on the board and be prepared to discuss them with the whole class Whole Class Sharing Physics 7B Exit Handout DLM 03 Refer to the Summary of Relationships in the Momentum Conservation Model on page 24 of the notes Use these relationships first to analyze each physical situation and second to make logical arguments that would convince another Physics 7 student of the correct response to the following multiple choice questions Remember that any conservation law requires you to compare a quantity at two times so you must always have in mind an initial time and a final time 1 A heavy ball is attached to a string and swung in a circular path in a horizontal plane as illustrated in the diagram to the right At the point indicated in the diagram the string suddenly I breaks at the ball If these events were observed from directly above indicate the path of the ball after the string breaks Use the statements and diagram below to answer the next four questions The diagram depicts a hockey puck sliding with a a b constant velocity from point quotaquot to point quotbquot along a frictionless horizontal surface The gt gt gt gt gt gt gt picture is a view from above looking down on the horizontal surface When the puck reaches point quotbquot it receives an instantaneous horizontal kick in the direction of the heavy print arrow 2 Along which of the paths below will the hockey puck move after receiving the quotkickquot A 3 Along the frictionless path you have chosen T how does the speed of the puck vary after receiving the quotkickquot A No change C Continuously increasing B Continuously decreasing D Increasing for a while and decreasing thereafter E Constant for a while and decreasing thereafter 4 The main forces acting on the puck after the kick as the puck moves along the path you have chosen are A the downward force due to gravity and the effect of air pressure B the downward force of gravity and the horizontal force of momentum in the direction of motion C the downward force of gravity the upward force exerted by the table and a horizontal force acting on the puck in the direction of motion D the downward force of gravity and an upward force exerted on the puck by the table E gravity does not exert a force on the puck it falls because of the intrinsic tendency of the object to fall to its natural place Announcemen rs we have organized reviews This Thursday and Friday all in 166 Roessler prac rice exam on The web I will be here all week my door is as always open To you nal exam 1216 1030 am 1230 pm in Chem 179 DL 13 and 194 DL 411 Oscillafions Physics 78 Lec rure 8 Prof Conway The Basic Idea IF you have a sys rem described by a parame rer posi rion For example where There is a Force coun rerac ring rhe parame rer rhe sys rem oscilla res Oscilla rions are of course The basis of waves Examples 1D Oscilla rions mass on a spring vibra ring s rring dia romic molecule pendulum RLC circui r All of These can be described as simple harmonic oscilla rors Mass and Spring 0 consider a mass connecfed 0 1 spring 0 fhe mass slides Fric onlessly on a horizom al surface 0 fhe mass oscillafes in 1 dimension Mass and Spring IF mass is displaced From compressed i rs ini rial uns rre rched F gt posi rion rhere is a Force WW fending to push i r back to unstretched the middle where rhere F0 is zero Force WW extended lt F 7W Hooke s Law wi rh rhe ex rension goes the Force k L 0 rhe Force is iWWV propor rional 0 and opposed ro rhe displacemen r of F the mass 4W L O k spring constant Nm O X Double Spring PRS Suppose we have a mass k k connec red ro rwo springs wi rh spring cons ran r k Then A F kgtlt B F kgtlt2 C F 0 D F 2kgtlt E F kgtlt2 Simple Harmonic Oscilla ror F cz d2 F ma md t d211 k a dt2 m We seek a function Xt whose second derivative is that function multiplied by a negative constant Simple Harmonic Oscilla ror O we can wri re a general solu rion of The Form ACOSltLUt gb amplitude Phase angular frequency Simple Harmonic Oscilla ror O we can wri re a general solu rion of The Form 3375 Acoswt gb 0 check if if solves rhe equa rion Simple Harmonic Oscilla ror 0 can also wri re if as tACOS w m A period AFAAA vavvvvt what about I Simple Harmonic Oscilla ror 0 value of I comes From ini rial condi rions O suppose rha r a r rO xAz 567520 ACOSO ACOS A gtgb 0 t A cod Frequency Period 0 Frequency number oF oscilla rions per second lperiod o F 1T 27Tt k A004 Acos at 21 E T m m 2 Frequency Period PRS Your car39s engine is ro ra ring wi rh 0 Frequency of 30 Hz The period of ro ra rion is A 30 sec B 130 sec C 2 sec D 12 sec Energy and Amplitude 0 Force is gradiemL deriva ve of pofem ial 0 spring pofem ial energy is guadrafic in fhe posi on Ux 1 UJc kx2 Energy and Ampli rude 0 kine ric energy depends on veloci ry 9075 vt 27Tt A COS T d 27Tt A dt COS T 27T Alt T mm 0 veloci ry is zero when mass is a r ex rreme x A or x A Energy and Amplifude 0 energy goes back and Forfh 00 be rween kine ric and po ren rial Kx 0 ro ral energy is cons ran r A Energy and Ampli rude O we Therefore know rha r rhe ro ral energy is 0 we can also ge r The maximum veloci ry XO EnergyAmpli rude PRS A cer rain massspring oscilla ror has an ro ral energy of 5 J IF we double the spring cons ran r and make The ampli rude halF as grea r rhe ro ral energy becomes A 25 J B 5 J no change C 10 J D 20 J E 25 J eph espringpenduumhtm httpwwwcooradoeduphysicsphetsimulations massspringlabMassSpringLab2swf Plane Pendulum 0 consider mass swinging From a s rring m 0 gravi ry is always pushing i r back To cen rer Plane Pendulum Pendulum can move only a E In dlrec rlon perpendicular to s rring Force along mo rion is mg Sine given by projec ring mg along rha r mg direc rion Plane Pendulum I we can now wri re down rhe equa rion governing rhe angle as a Func rion of lime 120 m6 7mg sin 9 small 9 sin0z0 Small Angles PRS Se r your calcula ror ro radians mode Now Take The sine of The angle 02 rad By wha r percen rage does This diFFer From 02 A 00035 B 00066 C 066 D 993 E 35 Plane Pendulum O we know how ro solve This one already QQ We wri re here The CW 6 2 solu rion in rhe case 615 Await where we pull The T pendulum bob ou r 27r g To some angle A rhe ampli rude and le r i r go General SH Oscilla ror 0 oscilla rions For m sys rem governed by an equa rion of This Form gt dZm w2r dt2 27T75 75 A m COS T 27 w T 27 gtT Cd Circular Mo rion Oscillo rions 0 simple harmonic oscilla rion in one dimension is jus r like considering only The x mo rion oF circular mo rion 1t yt 7quot Sin wt 7quot cos wt Thank You IT has been a privilege rhis pas r quar rer To be your ins rruc ror and I wish you all well For The res r OF The year in This course and in all oF your Fu rure endeavors Physics 7B DLM 16 Overview Model Linear Transport Model AC 633 Heat flow FNT from DLM 15 No activity sheet Model Exponential Change AC 712 Introduction to Capacitors and Capacitance Learning Goals Begin to think about nonsteadstate currents Get an introduction to a capacitor AC 713 Electrical Capacitor Discharging Learning Goals see AC 712 Get familiar with RC circuits Act 714 Physics of Exponential Change DLM 15 FNTs Learning Goals 15 min 25 min 40 min 60 min Understand how a physical system that behaves as dytdt 7 yt leads to exponential behavior Announcements 39 Reading Assignment Finish reading Unit 7 Practice seeing how the properties of a physical system determine the constant A know that U the time constant for a resistorcapacitor is the product RC 39 Begin studying for the nal now Do a little bit each day by going over the FNTs quizzes DLM activities and practice problems Physics 7B Activity 712 DLM 16 Introduction to Capacitors and Capacitance Consider the two pieces of metal very close to each other but not touching Each plate is suddenly connected to the two ends of a battery through a resister R use a light bulb as shown EK The two metal plates are called a capacitor The metal plates can accept electrons or lose electrons What happens when the battery is rst connected Usually we have thought about current in terms of positive charge but here it is useful to focus on the electrons themselves since they are the charge carriers in metal conductors a When the switch is closed connection is made can there be a steadystate current Explain Hint think about the conductivity of air as opposed to the conductivity of copper wire Does air normally conduct electricity b When the switch is closed connection is made can there be a current that lasts for a short time Explain c Focusing on the two metal plates what physically is happening to electrons immediately after the switch is closed Are they leaving a plate Going to a plate What is happening to the net charge on each plate d What would the time dependence of the current in the resistor be like immediately after closing the switch Make a sketch of the value of this current as a function of time What would its maximum value be Explain e Put your response to FNT 3 on the board after discussing each part in your SG Physics 7B Activity 713 DLM 16 Electrical Capacitor Discharging The phenomenon An electrical capacitor is charged and discharged a You have a capacitor a battery a resistor and a voltmeter along with some wires to hook these elements up You also have a holiday light bulb Make sure you have everything Set the voltmeter so it can read the voltage of the Resmor R battery and check to see that the battery is not dead Check to see that the bulb works as well Determine the value of your resistor and record it here b Charge up the capacitor by attaching it to the battery Then disconnect the l o battery and measure the voltage across the capacitor It should be similar to the voltage of the battery Why Battery 5 0 Hold the ends of the bulb across the capacitor and observe what happens Describe this in a sentence or two Based off this observation what can you infer from a capacitor does with electrical charge d The capacitance of a capacitor is defined to be C qAV The unit coulombvolt is given the name farad The capacitor you are using is extremely large It has a capacitance of nearly 1 farad Assuming your capacitor is 1 farad approximately how many electrons did you put on one plate of your capacitor when you charged it up with the battery Note one electron has a charge of 16 x10 19 Coulombs Collect the data Make sure to collect data so that you can do the next activity 714 and FNTs e Charge your capacitor up attach the voltmeter across the capacitor and then attach the resistor across the capacitor so you can measure the voltage of the capacitor as it discharges Measure the voltage of the capacitor as a function of time as it discharges through the resistor Take data until the voltage has reduced to below one tenth its initial value This is your Vt data f Plot your data on a sheet of graph paper Fit eyeball a smooth curve through your data points Remember you want a best fit curve the curve doesn t actually have to go through each point Then sketch your graph on the board g Determine the half life of your capacitorresistor combination from your graph The half life is the time taken for the initial voltage to decrease to one half that voltage h Determine the time for the voltage to drop to 1 e of its original value and record it here This is referred to as the time constant when the voltage is equal to 1 e 0368 of its original value i Now take and record measurements of charging your capacitor through the same resistor light bulb or other Do this by attaching the capacitor battery and resistor all in series see circuit in Act 712 Put the voltmeter across the capacitor to measure its voltage Start with the capacitor totally discharged use your bulb to discharge the capacitor and take time and voltage readings as the capacitor charges up Make a table of your readings and calculate a new variable Vfully charged Vt Plot this voltage as a function of time on the same graph What can you conclude about the curves as well as the half lives and time constants Whole Class Discussion Physics 7B Activity 714 DLM 16 Physics of Exponential Change DLM 15 FNTs A Experimentally seeing the fundamental condition that produces exponential behavior In your small group 1 FNT 2 using your Volume data a Use one of the graphs from your group members Determine from your graphs of Volt and dVoldt what number with units you would have to multiply the curve of Volt to make it equal the curve of dVoldt That is determine the constant A in the relation dVoldt A Volt Fill in the following chart for several time values I time I Volume I dVoldt I 9 I b What do you notice about the A Describe in your own words what you did in this FNT in the question above and the signi cance of what you found That is answer the question What does all this mean Whole class discussion 2 Using your Voltage data Using the data from part e of the last activity complete parts a and b that you just finished for the flow out of a standpipe Whole class discussion B Interpreting the time constant 1 The inverse of the constant A in e39M is called the timeconstant ie 1 timeconstant In electrical phenomena it is called the RC timeconstant a By how much has the voltage decayed from a series resistorcapacitor circuit when the time is equal to one time constant b How many RC time constants must pass for the voltage to have decayed to less than 5 of its original value use your calculator see how much it decreases for each successive time constant c Use the RC time constant you found in Activity 713 part h and the value of the resistor you found in part a to calculate the actual value of the capacitor you used 2 a Use a calculator to experimentally don t use a formula determine how many years it would take to reduce the radioactivity at a waste site to less than 001 of its initial value if the dominant radioactive isotope had a halflife of 50 years Put your work on the board b Draw a graph representing this decay and label your axes Plot a curve if the time constant were 5 years or 500 years Whole class discussion
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