Projects. Assignments. Plan. Motivational Films. Motivational Film 1/6/09. Dynamics II. Forces, Collisions and Impulse
|
|
- Pierce Robertson
- 5 years ago
- Views:
Transcription
1 Projects Dynamics II Forces, Collisions and Impulse Presentations: Dates: Week 0: Mon, Feb 6 Week 0: Wed, Feb 8 Finals Week: Tues, Feb 4 (:30 :30) room TBA 5 minutes / presentation Schedule now on Web Please send me choice of time/day Assignments Assignment -- Framework Some have been graded If not submitted please do so. Assignment -- Keyframing Due Mon, Jan. Questions? Assignment 3 -- Billiards To be given Monday. Plan Physics 0 for rigid body animation Monday: translation and rotational dynamics Today: Forces, impacts, and collisions Next Monday: Numerical integration. Next Wednesday: Applications (including particle systems) But first Motivational Films Educational animations by Jim Blinn Motivational Film Mathematica: The Theorem of Pythagoras (988) JPL Cal Tech Now at Microsoft
2 Motivational Film The Mechanical Universe (984) Let s get started Physics for Rigid Body Dynamics Last class: Linear/ Rotational Motion Today: Collisions Wednesday: Numerical Integration Laws of Motion Linear Motion Law I Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. (Inertia) Law II: The acceleration of a body is proportional to the resulting force acting on the body, and this acceleration is in the same direction as the force. Law III: For every action there is an equal and opposite reaction. For Linear Physical Motion Mass Measure of the amount of matter in a body From Law II: Measure of the a body s resistance to motion Velocity Change of motion with respect to time Acceleration Change of velocity with respect to time Force In short, force is what makes objects accelerate Momentum mass x velocity Another way of stating Law I: Momentum is conserved Rotational Motion For Rotational Physical Motion Inertia Measure of the amount/distribution of matter in a body From Law II: Measure of the a body s resistance to motion Angular Velocity Change of rotation with respect to time Angular Acceleration Change of velocity with respect to time Torque In short, torque is what makes objects rotate Angular Momentum Inertia x velocity Another way of stating Law I: Momentum is conserved Where we are Object properties Position, orientation Linear and angular velocity Linear and angular momentum mass Update object properties Calculate forces Calculate accelerations Using mass, momenta
3 Where we are State of object at any given time s( t) R( t) S( t) = M ( t) L( t) position rotation (in world coords) momentum angular momentum Where we are Derivative of object state s& ( t) v( t) R& ( t) ( t) R( t) S& ω ( t) = = M& ( t) F( t) L& ( t) τ ( t) Putting it all together Step Calculate Forces, F(t), τ(t) Step Integrate position/rotation s(t +Δt) = s(t) + v(t)δt R(t +Δt) = R(t) + (ω(t)*r(t)) Δt /* CAREFUL HERE */ r(t +Δt) = s(t +Δt) + r body R(t +Δt) Update Momentum (integrate accelleration) M(t +Δt) = M(t) + F(t) Δt L(t +Δt) = L(t) + τ(t) Δt Putting it all together Step Calculate Forces, F(t), τ(t) Step Integrate position/rotation s(t +Δt) = s(t) + v(t)δt q(t +Δt) = q(t) (ω(t)q(t)) Δt /* normalize to avoid problems */ R(t +Δt) = quattorot (q(t +Δt) ) r(t +Δt) = s(t +Δt) + r body R(t +Δt) Update Momentum (integrate accelleration) M(t +Δt) = M(t) + F(t) Δt L(t +Δt) = L(t) + τ(t) Δt Putting It all together Step 3 Calculate velocities (for next step) v(t +Δt) = M(t +Δt)/m I - (t +Δt) = R(t +Δt)I - body R(t +Δt)T ω(t +Δt) = + I - (t +Δt)L(t +Δt) Go to step Questions? Useful forces Gravity Friction Impulse Spring Wind Add your own 3
4 Gravity Gravity is an attractive force between all pairs of massive objects in the universe. The gravitational force between two objects is given by a (fairly) simple mathematical equation. m m F = G r Where m and m are the masses of the two objects r is the distance between the two objects G is the universal gravity constant = 6.67 x 0 - Nm / kg Gravity For objects interacting on this earth, the acceleration due to gravity can be calculated using the radius of the earth. g = 9.8 m / sec g = 3 ft / sec This acceleration is always towards the earth s surface. Friction Arises from interaction of surfaces in contact. Always works against the direction of relative motion of two objects. Friction Static Friction For objects not in motion Fraction of the normal component of force Amount of force need to get object from rest moving Kinetic friction For objects in motion Fraction of the normal component of force Amount of resistance due to friction Static Friction Kinetic Friction Supporting object Resting contact Normal force F F s F N Supporting object Resting contact Normal force F F k F N v Static friction F s = u s * F N Kinetic friction F k = u k * F N 4
5 Friction Surfaces u s u k Dry glass on glass Dry iron on iron. 0.5 Dry rubber on pavement Dry steel on steel Dry Teflon on Teflon Dry wood on wood Ice on Ice Oiled steel on steel Springs Force applied by stretching a spring. Given by Hooke s Law restoring force due to a spring is proportional to the length that the spring is stretched acts in the opposite direction. F = -kx Springs Hooke s Law F = -kx k = spring constant Given in N/m Large k stronger springs Small k looser springs Springs Damping Decreases spring force proportional to velocity Springs Damping Damping specified by damping coefficient, k d Damping and Springs F = kx k d dx dt Springs Useful for: Simulate collections of connected particles or rigid bodies Example: cloth, paper, etc. Damping required to make solutions stable Can also be used in collision detection (impulse forces) 5
6 Wind Define a force field Force varies in space and time Add your own Most difficult part is mathematically describing the force Wind, turbulence See Wejchert paper in READING LIST Physics for game development book has a bunch Questions Impulse Law III: For every action there is an equal and opposite reaction. Impulse is the equal and opposite reaction after a collision Impulse Impulse A force that acts over a very short period of time. Gives change in momentum + t Linear Impulse = F ( t) dt = M after M before t + t Angular Impulse = τ (t)dt = L after L before t Impulse and Collision At the time of collision, each object applies an impulse force on the other. Magnitude is the same Opposite in Direction Following Law III Collisions Law says that momentum is conserved When objects collide with masses m and m Velocities v and v ( v + mv) before = ( mv + mv m ) after Link topic=5.0 6
7 Kinetic Energy Energy required to accelerate a body from rest KElinear = mv KEangular = Energy is conserved Iω Types of collisions Elastic Kinetic energy is conserved No kinetic energy is lost Example: flubber super ball Inelastic All kinetic energy is lost Colliding objects stick together Energy converted to heat, sound, damage Example: clay falling on a hard floor Coefficient of restitution Most collisions lie somewhere between perfectly elastic and perfectly inelastic Gives a measure of the elasticity during an impact ( v v) e = ( v v ) after before Coefficient of restitution Perfectly elastic = Perfectly inelastic = 0 Most collisions Somewhere between 0 and Link index.php?topic=4.0 Line of action Line of action Line perpendicular to the colliding surfaces Central The line of action passes through the center of mass Always true for spheres Direct Velocities is along the line of action Oblique The velocities are not along the line of action Line of action 7
8 Impulse Force is a vector Only velocities along line of action need be considered Link Calculating Impulse Force Linear Remember J = m v v ) J = m v v ) ( after before ( v v) e = ( v v ) after before ( after before Calculating Impulse Forces Calculating Impulse Force Linear After doing some algebra ( v v) before( e + ) J = + m m Note: J and v s are scalar giving the values along the line of action Calculating new velocities Once we know J Calculating Impulse Force Angular Jn v after = v before + m v after = v before Jn m Where n is a normalized vector along the line of action. Here, J is a scalar giving the impulse along the line of action Two more unknowns to deal with (ω after for each object) 8
9 Calculating Impulse Force Separate velocity into components r body (t) = v(t) + ω(t) r body Calculating Impulse Force Luckily we have 4 equations from linear impulse v after + (ω after r ) = (Jn) m + v before + (ω before r ) v after + (ω after r ) = (Jn) m + v before + (ω before r ) from angular impulse (r Jn) = I (ω after ω before ) (r Jn) = I (ω after ω before ) Calculating Impulse Forces Calculating Impulse Force After doing some more algebra ( v v) before( e + ) J = + + n m m [ I ( r n) ] r + n [ I ( r n) ] r Where n a unit vector normal to the surface of contact The v s are scalar velocities along the line of action. Calculating new velocities Once we know J Jn v after = v before + v m after = v before Jn m Questions Let s take a break ω after = ω before + I (r Jn) ω after = ω before + I (r Jn) 9
10 Collisions and Animation Answers the question, was there a collision Collision Determination If there is a collision, when and where did it occur? Collision Response How does the collision affect the motion of the objects involved? Strictly a kinematic issue Based on position and orientation of objects and how they change over time. Can be determined when after a given time step one object penetrates another Penetrating objects Be careful positions are only approximations False positive okay Missed positive bad Penetrating objects Complexity will depend upon objects being tested for penetration There are many good algorithms and free libraries for collision detection of planar convex polyhedra See Web Page for Survey paper. Major Challenge in collision detection Must check collision of each pair of objects General problem is O(N ) Multiple collisions in a single time step. Efficient collision detection for real time applications is still an active research area Tips Similar to tips for intersection testing for ray tracing Use hierarchical bounding volumes Place simple objects (i.e sphere or box) around complex objects Do initial intersection tests on bounding objects. If ray intersects bounding volume, then test complex bounded object 0
11 hierarchical bounding volumes hierarchical bounding volumes Considerations Ease of detection Bounding efficiency Other Tips You need only test moving objects for collisions (remember Law ) However, you must test against all other objects, moving or stationary - Spatial Subdivision Subdivide your scene volume into hierarchical regions Create a tree structure that indicates for each region: if the region is empty the object present at that particular region Test collision with objects in volume - Spatial Subdivision Motivation Without spatial subdivision, you will need to query all objects/polygons and test for intersection With spatial subdivision, you know which objects are in which volume so you only test objects that are in the volumes where an object traveling. - Spatial Subdivision Examples Octrees Recursively subdivide volume into equal regions. If subregion is empty, then stop Otherwise further subdivide subregion. Continue until each subregion is empty or contains a single object. BSPTrees Like Octrees but divides space into a pair of subregions Subregions need not be equally spaced Planes separating regions can be placed at object boundaries.
12 -- Octrees -- Octrees Images from flipcode.com -- Octrees -- BSPTree - Spatial Subdivision Issues Must rebuild tree at each time step Tradeoff Time to build tree vs. Time to check for collisions BSP/OctTrees for static objects Questions? Collision Determination If a collision has been detected, the time and point of impact must be determined By the time a collision is detected, one object has already penetrated another I.e. Collision occurred between time steps Need to backup simulation to the point when collision took place.
13 Collision Determination Binary search strategy No collision at time t Penetration at time t + Δt Test point at time t between t and t + Δt iterative. If No Collision, test at time between t and t + Δt Else test at time between t and t Iterate until a predefined tolerance is achieved. Point at t i- Collision Determination Linear interpolation strategy s L t = t i + s/l (t i t i- ) Point at t Collision Response Once we have the actual time and point of collision, we can use formula in first half to determine impulse force and new velocities. Collision Response Penalty Method Rather than applying kinematic equations, use a spring to approximate the impulse force. ( v v) before( e + ) J = + + n m m [ I ( r n) ] r + n [ I ( r n) ] r Collision Response Recall Spring force given by Hooke s Law F = -kx Where x is the displacement of the spring k is a spring constant Collision Response Penalty Method Pros Ease of implementation No need to step back to time of collision No need to mess with complex kinematic equations Cons How does one choose k? Must choose carefully else solution may become unstable or in accurate Simply a hueristic hack Your mileage may vary! 3
14 To sum up Answers the question, was there a collision Collision Determination If there is a collision, when and where did it occur? Collision Response How does the collision affect the motion of the objects involved? Questions? Putting it all together Object properties Position, orientation Linear and angular velocity Linear and angular momentum mass Update object properties Calculate forces Calculate accelerations Using mass, momenta Putting it all together Step Calculate Forces, F(t), τ(t) Step Integrate position/rotation s(t +Δt) = s(t) + v(t)δt q(t +Δt) = q(t) (ω(t)q(t)) Δt /* normalize to avoid problems */ R(t +Δt) = quattorot (q(t +Δt) ) r(t +Δt) = s(t +Δt) + r body R(t +Δt) Update Momentum (integrate accelleration) M(t +Δt) = M(t) + F(t) Δt L(t +Δt) = L(t) + τ(t) Δt Putting it all together Step 3 Calculate velocities (for next step) v(t +Δt) = M(t +Δt)/m + impulse velocity I - (t +Δt) = R(t +Δt)I - body R(t +Δt) T ω(t +Δt) = I - (t +Δt)L(t +Δt) + impulse velocity Go to step Questions? Video Let s see how this all looks when assembled properly Rigid Body Dynamics By James Hahn (also at Ohio State at the time) Accompanying video to his paper B-D Tree By Doug L. James (CMU) If falling chairs is your thing. Next Time Numerical Integration Assignment #3 Questions? 4
Collision Resolution
Collision Resolution Our Problem Collision detection (supposedly) reported a collision. We want to solve it, i.e. move back the colliding objects apart from each other. In which direction and with what
More informationPHYSICS I RESOURCE SHEET
PHYSICS I RESOURCE SHEET Cautions and Notes Kinematic Equations These are to be used in regions with constant acceleration only You must keep regions with different accelerations separate (for example,
More informationToday s lecture. WEST VIRGINIA UNIVERSITY Physics
Today s lecture Review of chapters 1-14 Note: I m taking for granted that you ll still know SI/cgs units, order-of-magnitude estimates, etc., so I m focusing on problems. Velocity and acceleration (1d)
More information2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity
2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity Energy 7 4 Kinematics Free fall Collisions 3 5 Dynamics
More informationIMPACT Today s Objectives: In-Class Activities:
Today s Objectives: Students will be able to: 1. Understand and analyze the mechanics of impact. 2. Analyze the motion of bodies undergoing a collision, in both central and oblique cases of impact. IMPACT
More informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Essential physics for game developers Introduction The primary issues Let s move virtual objects Kinematics: description
More informationIMPACT (Section 15.4)
IMPACT (Section 15.4) Today s Objectives: Students will be able to: a) Understand and analyze the mechanics of impact. b) Analyze the motion of bodies undergoing a collision, in both central and oblique
More informationPLANAR KINETIC EQUATIONS OF MOTION (Section 17.2)
PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2) We will limit our study of planar kinetics to rigid bodies that are symmetric with respect to a fixed reference plane. As discussed in Chapter 16, when
More informationThursday Simulation & Unity
Rigid Bodies Simulation Homework Build a particle system based either on F=ma or procedural simulation Examples: Smoke, Fire, Water, Wind, Leaves, Cloth, Magnets, Flocks, Fish, Insects, Crowds, etc. Simulate
More informationFinal Exam. June 10, 2008, 1:00pm
PHYSICS 101: Fundamentals of Physics Final Exam Final Exam Name TA/ Section # June 10, 2008, 1:00pm Recitation Time You have 2 hour to complete the exam. Please answer all questions clearly and completely,
More informationKinematics (special case) Dynamics gravity, tension, elastic, normal, friction. Energy: kinetic, potential gravity, spring + work (friction)
Kinematics (special case) a = constant 1D motion 2D projectile Uniform circular Dynamics gravity, tension, elastic, normal, friction Motion with a = constant Newton s Laws F = m a F 12 = F 21 Time & Position
More informationr CM = ir im i i m i m i v i (2) P = i
Physics 121 Test 3 study guide Thisisintendedtobeastudyguideforyourthirdtest, whichcoverschapters 9, 10, 12, and 13. Note that chapter 10 was also covered in test 2 without section 10.7 (elastic collisions),
More informationReview of Forces and Conservation of Momentum
Physics 7B-1 (A/B) Professor Cebra Winter 2010 Lecture 6 Review of Forces and Conservation of Momentum Slide 1 of 22 Vector Addition and Subtraction Vectors are added head to tail Note: a+b = b+a Vectors
More informationz F 3 = = = m 1 F 1 m 2 F 2 m 3 - Linear Momentum dp dt F net = d P net = d p 1 dt d p n dt - Conservation of Linear Momentum Δ P = 0
F 1 m 2 F 2 x m 1 O z F 3 m 3 y Ma com = F net F F F net, x net, y net, z = = = Ma Ma Ma com, x com, y com, z p = mv - Linear Momentum F net = dp dt F net = d P dt = d p 1 dt +...+ d p n dt Δ P = 0 - Conservation
More informationChapter 8: Momentum, Impulse, & Collisions. Newton s second law in terms of momentum:
linear momentum: Chapter 8: Momentum, Impulse, & Collisions Newton s second law in terms of momentum: impulse: Under what SPECIFIC condition is linear momentum conserved? (The answer does not involve collisions.)
More informationStudy Guide For Midterm - 25 weeks Physics Exam. d. the force exerted by a towing cable on the car. c. the upward force the road exerts on the car.
Name: Class: Date: ID: A Study Guide For Midterm - 25 weeks Physics Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which of the following is the
More informationPart Two: Earlier Material
Part Two: Earlier Material Problem 1: (Momentum and Impulse) A superball of m 1 = 0.08kg, starting at rest, is dropped from a height falls h 0 = 3.0m above the ground and bounces back up to a height of
More informationRigid Body Dynamics and Beyond
Rigid Body Dynamics and Beyond 1 Rigid Bodies 3 A rigid body Collection of particles Distance between any two particles is always constant What types of motions preserve these constraints? Translation,
More informationKinetics of Particles: Work and Energy
Kinetics of Particles: Work and Energy Total work done is given by: Modifying this eqn to account for the potential energy terms: U 1-2 + (-ΔV g ) + (-ΔV e ) = ΔT T U 1-2 is work of all external forces
More informationChapter 8- Rotational Motion
Chapter 8- Rotational Motion Assignment 8 Textbook (Giancoli, 6 th edition), Chapter 7-8: Due on Thursday, November 13, 2008 - Problem 28 - page 189 of the textbook - Problem 40 - page 190 of the textbook
More informationMechanics Topic B (Momentum) - 1 David Apsley
TOPIC B: MOMENTUM SPRING 2019 1. Newton s laws of motion 2. Equivalent forms of the equation of motion 2.1 orce, impulse and energy 2.2 Derivation of the equations of motion for particles 2.3 Examples
More informationDate Period Name. Write the term that correctly completes the statement. Use each term once. elastic collision
Date Period Name CHAPTER 11 Conservation of Energy Vocabulary Review Write the term that correctly completes the statement. Use each term once. elastic collision law of conservation of energy elastic potential
More informationPhysics 218 Lecture 23
Physics 218 Lecture 23 Dr. David Toback Physics 218, Lecture XXIII 1 Checklist for Today Things due Monday Chapter 14 in WebCT Things that were due yesterday Chapter 15 problems as Recitation Prep Things
More informationAPPLIED MATHEMATICS AM 02
AM SYLLABUS (2013) APPLIED MATHEMATICS AM 02 SYLLABUS Applied Mathematics AM 02 Syllabus (Available in September) Paper I (3 hrs)+paper II (3 hrs) Applied Mathematics (Mechanics) Aims A course based on
More informationSTEP Support Programme. Mechanics STEP Questions
STEP Support Programme Mechanics STEP Questions This is a selection of mainly STEP I questions with a couple of STEP II questions at the end. STEP I and STEP II papers follow the same specification, the
More informationExam II. Spring 2004 Serway & Jewett, Chapters Fill in the bubble for the correct answer on the answer sheet. next to the number.
Agin/Meyer PART I: QUALITATIVE Exam II Spring 2004 Serway & Jewett, Chapters 6-10 Assigned Seat Number Fill in the bubble for the correct answer on the answer sheet. next to the number. NO PARTIAL CREDIT:
More informationEN40: Dynamics and Vibrations. Final Examination Wed May : 2pm-5pm
EN40: Dynamics and Vibrations Final Examination Wed May 10 017: pm-5pm School of Engineering Brown University NAME: General Instructions No collaboration of any kind is permitted on this examination. You
More informationUNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics
UNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics Physics 111.6 MIDTERM TEST #2 November 15, 2001 Time: 90 minutes NAME: STUDENT NO.: (Last) Please Print (Given) LECTURE SECTION
More informationThe... of a particle is defined as its change in position in some time interval.
Distance is the. of a path followed by a particle. Distance is a quantity. The... of a particle is defined as its change in position in some time interval. Displacement is a.. quantity. The... of a particle
More informationConservation of Momentum
Conservation of Momentum Law of Conservation of Momentum The sum of the momenta before a collision equal the sum of the momenta after the collision in an isolated system (=no external forces acting).
More informationConservation of Momentum. Last modified: 08/05/2018
Conservation of Momentum Last modified: 08/05/2018 Links Momentum & Impulse Momentum Impulse Conservation of Momentum Example 1: 2 Blocks Initial Momentum is Not Enough Example 2: Blocks Sticking Together
More informationChapter 9 Linear Momentum and Collisions
Chapter 9 Linear Momentum and Collisions The Center of Mass The center of mass of a system of particles is the point that moves as though (1) all of the system s mass were concentrated there and (2) all
More informationChapter Work, Energy and Power. Q1. The co-efficient of restitution e for a perfectly elastic collision is [1988] (a) 1 (b) 0 (c) (d) 1 Ans: (a)
Chapter Work, Energy and Power Q1. The co-efficient of restitution e for a perfectly elastic collision is [1988] (a) 1 (b) 0 (c) (d) 1 Q2. A bullet of mass 10g leaves a rifle at an initial velocity of
More informationChapter 14 Periodic Motion
Chapter 14 Periodic Motion 1 Describing Oscillation First, we want to describe the kinematical and dynamical quantities associated with Simple Harmonic Motion (SHM), for example, x, v x, a x, and F x.
More informationWelcome back to Physics 211
Welcome back to Physics 211 Today s agenda: Impulse and momentum 09-2 1 Current assignments Reading: Chapter 10 in textbook Prelecture due next Tuesday HW#8 due this Friday at 5 pm. 09-2 2 9-2.1 A crash
More informationWORK, POWER AND ENERGY
WORK, POWER AND ENERGY Important Points:. Dot Product: a) Scalar product is defined as the product of the magnitudes of two vectors and the cosine of the angle between them. The dot product of two vectors
More informationPhysics 141. Lecture 18. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 18, Page 1
Physics 141. Lecture 18. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 18, Page 1 Physics 141. Lecture 18. Course Information. Topics to be discussed today: A
More informationMechanics. Time (s) Distance (m) Velocity (m/s) Acceleration (m/s 2 ) = + displacement/time.
Mechanics Symbols: Equations: Kinematics The Study of Motion s = distance or displacement v = final speed or velocity u = initial speed or velocity a = average acceleration s u+ v v v u v= also v= a =
More information3. Kinetics of Particles
3. Kinetics of Particles 3.1 Force, Mass and Acceleration 3.3 Impulse and Momentum 3.4 Impact 1 3.1 Force, Mass and Acceleration We draw two important conclusions from the results of the experiments. First,
More informationExam 3 Practice Solutions
Exam 3 Practice Solutions Multiple Choice 1. A thin hoop, a solid disk, and a solid sphere, each with the same mass and radius, are at rest at the top of an inclined plane. If all three are released at
More informationForces. Prof. Yury Kolomensky Feb 9/12, 2007
Forces Prof. Yury Kolomensky Feb 9/12, 2007 - Hooke s law - String tension - Gravity and Weight - Normal force - Friction - Drag -Review of Newton s laws Today s Plan Catalog common forces around us What
More informationparticle p = m v F ext = d P = M d v cm dt
Lecture 11: Momentum and Collisions; Introduction to Rotation 1 REVIEW: (Chapter 8) LINEAR MOMENTUM and COLLISIONS The first new physical quantity introduced in Chapter 8 is Linear Momentum Linear Momentum
More informationAPPLICATIONS. CEE 271: Applied Mechanics II, Dynamics Lecture 17: Ch.15, Sec.4 7. IMPACT (Section 15.4) APPLICATIONS (continued) IMPACT READING QUIZ
APPLICATIONS CEE 271: Applied Mechanics II, Dynamics Lecture 17: Ch.15, Sec.4 7 Prof. Albert S. Kim Civil and Environmental Engineering, University of Hawaii at Manoa Date: The quality of a tennis ball
More informationRigid body simulation. Once we consider an object with spatial extent, particle system simulation is no longer sufficient
Rigid body dynamics Rigid body simulation Once we consider an object with spatial extent, particle system simulation is no longer sufficient Rigid body simulation Unconstrained system no contact Constrained
More informationThe SI units of mass are kilograms (kg) and of velocity are meters / second (m/s). Therefore, the units of momentum are kg m/s.
Momentum Introduction As was pointed out in the previous chapter, some of the most powerful tools in physics are based on conservation principles. The idea behind a conservation principle is that there
More informationChapter 9- Static Equilibrium
Chapter 9- Static Equilibrium Changes in Office-hours The following changes will take place until the end of the semester Office-hours: - Monday, 12:00-13:00h - Wednesday, 14:00-15:00h - Friday, 13:00-14:00h
More informationAPPLIED MATHEMATICS IM 02
IM SYLLABUS (2013) APPLIED MATHEMATICS IM 02 SYLLABUS Applied Mathematics IM 02 Syllabus (Available in September) 1 Paper (3 hours) Applied Mathematics (Mechanics) Aims A course based on this syllabus
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationReview for 3 rd Midterm
Review for 3 rd Midterm Midterm is on 4/19 at 7:30pm in the same rooms as before You are allowed one double sided sheet of paper with any handwritten notes you like. The moment-of-inertia about the center-of-mass
More informationAnnouncements. 1. Do not bring the yellow equation sheets to the miderm. Idential sheets will be attached to the problems.
Announcements 1. Do not bring the yellow equation sheets to the miderm. Idential sheets will be attached to the problems. 2. Some PRS transmitters are missing. Please, bring them back! 1 Kinematics Displacement
More informationPHY131H1S - Class 20. Pre-class reading quiz on Chapter 12
PHY131H1S - Class 20 Today: Gravitational Torque Rotational Kinetic Energy Rolling without Slipping Equilibrium with Rotation Rotation Vectors Angular Momentum Pre-class reading quiz on Chapter 12 1 Last
More informationPH1104/PH114S MECHANICS
PH04/PH4S MECHANICS SEMESTER I EXAMINATION 06-07 SOLUTION MULTIPLE-CHOICE QUESTIONS. (B) For freely falling bodies, the equation v = gh holds. v is proportional to h, therefore v v = h h = h h =.. (B).5i
More informationPhysics 10 Lecture 6A. "And in knowing that you know nothing, that makes you the smartest of all. --Socrates
Physics 10 Lecture 6A "And in knowing that you know nothing, that makes you the smartest of all. --Socrates Momentum Which is harder to stop a small ball moving at 1 m/s or a car moving at 1 m/s? Obviously
More informationPhysics 8, Fall 2011, equation sheet work in progress
1 year 3.16 10 7 s Physics 8, Fall 2011, equation sheet work in progress circumference of earth 40 10 6 m speed of light c = 2.9979 10 8 m/s mass of proton or neutron 1 amu ( atomic mass unit ) = 1 1.66
More information1 Forces. 2 Energy & Work. GS 104, Exam II Review
1 Forces 1. What is a force? 2. Is weight a force? 3. Define weight and mass. 4. In European countries, they measure their weight in kg and in the United States we measure our weight in pounds (lbs). Who
More informationMomentum Conceptual Questions. 1. Which variable has more impact on an object s motion? Its mass or its velocity?
AP Physics I Momentum Conceptual Questions 1. Which variable has more impact on an object s motion? Its mass or its velocity? 2. Is momentum a vector or a scalar? Explain. 3. How does changing the duration
More informationAssignment 4: Rigid Body Dynamics
Assignment 4: Rigid Body Dynamics Due April 4 at :59pm Introduction Rigid bodies are a fundamental element of many physical simulations, and the physics underlying their motions are well-understood. In
More information= o + t = ot + ½ t 2 = o + 2
Chapters 8-9 Rotational Kinematics and Dynamics Rotational motion Rotational motion refers to the motion of an object or system that spins about an axis. The axis of rotation is the line about which the
More informationExtra credit assignment #4 It can be handed in up until one class before Test 4 (check your course outline). It will NOT be accepted after that.
Extra credit assignment #4 It can be handed in up until one class before Test 4 (check your course outline). It will NOT be accepted after that. NAME: 4. Units of power include which of the following?
More informationAP Physics C. Momentum. Free Response Problems
AP Physics C Momentum Free Response Problems 1. A bullet of mass m moves at a velocity v 0 and collides with a stationary block of mass M and length L. The bullet emerges from the block with a velocity
More informationhttps://njctl.org/courses/science/ap-physics-c-mechanics/attachments/summerassignment-3/
AP Physics C Summer Assignment 2017 1. Complete the problem set that is online, entitled, AP C Physics C Summer Assignment 2017. I also gave you a copy of the problem set. You may work in groups as a matter
More informationAP PHYSICS C Momentum Name: AP Review
AP PHYSICS C Momentum Name: AP Review Momentum How hard it is to stop a moving object. Related to both mass and velocity. For one particle p = mv For a system of multiple particles P = p i = m ivi Units:
More informationLecture 13 REVIEW. Physics 106 Spring What should we know? What should we know? Newton s Laws
Lecture 13 REVIEW Physics 106 Spring 2006 http://web.njit.edu/~sirenko/ What should we know? Vectors addition, subtraction, scalar and vector multiplication Trigonometric functions sinθ, cos θ, tan θ,
More informationwhere G is called the universal gravitational constant.
UNIT-I BASICS & STATICS OF PARTICLES 1. What are the different laws of mechanics? First law: A body does not change its state of motion unless acted upon by a force or Every object in a state of uniform
More informationAnnouncements. There will still be a WebAssign due this Friday, the last before the midterm.
Announcements THERE WILL BE NO CLASS THIS FRIDAY, MARCH 5 (We are 1 full lecture ahead of the syllabus, so we will still have review/problem solving on March 7 and 9). There will still be a WebAssign due
More informationEssential Physics I. Lecture 9:
Essential Physics I E I Lecture 9: 15-06-15 Last lecture: review Conservation of momentum: p = m v p before = p after m 1 v 1,i + m 2 v 2,i = m 1 v 1,f + m 2 v 2,f m 1 m 1 m 2 m 2 Elastic collision: +
More informationPhysics 8 Monday, October 28, 2013
Physics 8 Monday, October 28, 2013 Turn in HW8 today. I ll make them less difficult in the future! Rotation is a hard topic. And these were hard problems. HW9 (due Friday) is 7 conceptual + 8 calculation
More informationPhysics 12 Final Exam Review Booklet # 1
Physics 12 Final Exam Review Booklet # 1 1. Which is true of two vectors whose sum is zero? (C) 2. Which graph represents an object moving to the left at a constant speed? (C) 3. Which graph represents
More informationPhysics 5A Final Review Solutions
Physics A Final Review Solutions Eric Reichwein Department of Physics University of California, Santa Cruz November 6, 0. A stone is dropped into the water from a tower 44.m above the ground. Another stone
More informationWhen particle with mass m moves with velocity v, we define its Linear Momentum p as product of its mass m and its velocity v:
8. LINEAR MOMENTUM. Key words: Linear Momentum, Law of Conservation of Momentum, Collisions, Elastic Collisions, Inelastic Collisions, Completely Inelastic Collision, Impulse, Impulse Momentum Theorem.
More informationLab 8: Ballistic Pendulum
Lab 8: Ballistic Pendulum Caution In this experiment a steel ball is projected horizontally across the room with sufficient speed to injure a person. Be sure the line of fire is clear before firing the
More informationWhich iceboat crosses the finish line with more kinetic energy (KE)?
Two iceboats (one of mass m, one of mass 2m) hold a race on a frictionless, horizontal, frozen lake. Both iceboats start at rest, and the wind exerts the same constant force on both iceboats. Which iceboat
More informationPHY 101. Work and Kinetic Energy 7.1 Work Done by a Constant Force
PHY 101 DR M. A. ELERUJA KINETIC ENERGY AND WORK POTENTIAL ENERGY AND CONSERVATION OF ENERGY CENTRE OF MASS AND LINEAR MOMENTUM Work is done by a force acting on an object when the point of application
More informationCEE 271: Applied Mechanics II, Dynamics Lecture 17: Ch.15, Sec.2 4
1 / 38 CEE 271: Applied Mechanics II, Dynamics Lecture 17: Ch.15, Sec.2 4 Prof. Albert S. Kim Civil and Environmental Engineering, University of Hawaii at Manoa Tuesday, October 16, 2012 2 / 38 PRINCIPLE
More informationPassive Dynamics and Particle Systems COS 426
Passive Dynamics and Particle Systems COS 426 Computer Animation Animation Make objects change over time according to scripted actions Simulation / dynamics Predict how objects change over time according
More informationNorthwestern CT Community College Course Syllabus. Course Title: CALCULUS-BASED PHYSICS I with Lab Course #: PHY 221
Northwestern CT Community College Course Syllabus Course Title: CALCULUS-BASED PHYSICS I with Lab Course #: PHY 221 Course Description: 4 credits (3 class hours and 3 laboratory hours per week) Physics
More informationAP PHYSICS 1 Learning Objectives Arranged Topically
AP PHYSICS 1 Learning Objectives Arranged Topically with o Big Ideas o Enduring Understandings o Essential Knowledges o Learning Objectives o Science Practices o Correlation to Knight Textbook Chapters
More informationNorthwestern Connecticut Community College Course Syllabus
Northwestern Connecticut Community College Course Syllabus Course Title: Introductory Physics Course #: PHY 110 Course Description: 4 credits (3 class hours and 3 laboratory hours per week) Physics 110
More informationCh 7 Impulse-Momentum Theorem, Conservation of Momentum, and Collisions
Ch 7 Impulse-Momentum Theorem, Conservation of Momentum, and Collisions Momentum and its relation to force Momentum describes an object s motion. Linear momentum is the product of an object s mass and
More informationQuiz Number 4 PHYSICS April 17, 2009
Instructions Write your name, student ID and name of your TA instructor clearly on all sheets and fill your name and student ID on the bubble sheet. Solve all multiple choice questions. No penalty is given
More informationPSE Game Physics. Session (6) Angular momentum, microcollisions, damping. Oliver Meister, Roland Wittmann
PSE Game Physics Session (6) Angular momentum, microcollisions, damping Oliver Meister, Roland Wittmann 23.05.2014 Session (6)Angular momentum, microcollisions, damping, 23.05.2014 1 Outline Angular momentum
More information14300 Dynamics Carts w/o Hoops Teachers Instructions
14300 Dynamics Carts w/o Hoops Teachers Instructions Required Accessories o (2) Table stops (wooden bars) o (4) C-Clamps o (2) Recording Timers (#15210 or #15215) o (5) Bricks or Books (or other identical
More informationParticle Systems. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2017
Particle Systems CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2017 Particle Systems Particle systems have been used extensively in computer animation and special effects since their
More informationIII. Work and Energy
Rotation I. Kinematics - Angular analogs II. III. IV. Dynamics - Torque and Rotational Inertia Work and Energy Angular Momentum - Bodies and particles V. Elliptical Orbits The student will be able to:
More informationExam 2--PHYS 101--F11--Chapters 4, 5, & 6
ame: Exam 2--PHYS 101--F11--Chapters 4, 5, & 6 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Consider this figure. What is the normal force acting on
More informationDistance travelled time taken and if the particle is a distance s(t) along the x-axis, then its instantaneous speed is:
Chapter 1 Kinematics 1.1 Basic ideas r(t) is the position of a particle; r = r is the distance to the origin. If r = x i + y j + z k = (x, y, z), then r = r = x 2 + y 2 + z 2. v(t) is the velocity; v =
More information23. A force in the negative direction of an x-axis is applied for 27ms to a 0.40kg ball initially moving at 14m/s in the positive direction of the
23. A force in the negative direction of an x-axis is applied for 27ms to a 0.40kg ball initially moving at 14m/s in the positive direction of the axis. The force varies in magnitude, and the impulse has
More informationMotion. Argument: (i) Forces are needed to keep things moving, because they stop when the forces are taken away (evidence horse pulling a carriage).
1 Motion Aristotle s Study Aristotle s Law of Motion This law of motion was based on false assumptions. He believed that an object moved only if something was pushing it. His arguments were based on everyday
More informationLINEAR MOMENTUM AND COLLISIONS
LINEAR MOMENTUM AND COLLISIONS Chapter 9 Units of Chapter 9 Linear Momentum Momentum and Newton s Second Law Impulse Conservation of Linear Momentum Inelastic Collisions Elastic Collisions Center of Mass
More informationOCR Physics Specification A - H156/H556
OCR Physics Specification A - H156/H556 Module 3: Forces and Motion You should be able to demonstrate and show your understanding of: 3.1 Motion Displacement, instantaneous speed, average speed, velocity
More informationLecture 18. Newton s Laws
Agenda: l Review for exam Lecture 18 l Assignment: For Monday, Read chapter 14 Physics 207: Lecture 18, Pg 1 Newton s Laws Three blocks are connected on the table as shown. The table has a coefficient
More informationPhysics 8, Fall 2013, equation sheet work in progress
(Chapter 1: foundations) 1 year 3.16 10 7 s Physics 8, Fall 2013, equation sheet work in progress circumference of earth 40 10 6 m speed of light c = 2.9979 10 8 m/s mass of proton or neutron 1 amu ( atomic
More informationPHYSICS 221, FALL 2011 EXAM #2 SOLUTIONS WEDNESDAY, NOVEMBER 2, 2011
PHYSICS 1, FALL 011 EXAM SOLUTIONS WEDNESDAY, NOVEMBER, 011 Note: The unit vectors in the +x, +y, and +z directions of a right-handed Cartesian coordinate system are î, ĵ, and ˆk, respectively. In this
More informationPhysics 1C. Lecture 12B
Physics 1C Lecture 12B SHM: Mathematical Model! Equations of motion for SHM:! Remember, simple harmonic motion is not uniformly accelerated motion SHM: Mathematical Model! The maximum values of velocity
More informationPY205N Spring The vectors a, b, and c. are related by c = a b. The diagram below that best illustrates this relationship is (a) I
PY205N Spring 2013 Final exam, practice version MODIFIED This practice exam is to help students prepare for the final exam to be given at the end of the semester. Please note that while problems on this
More informationLecture 13. Collisions. and Review of material. Pre-reading: KJF 9.5. Please take an evaluation form
Lecture 13 Collisions and Review of material Pre-reading: KJF 9.5 Please take an evaluation form COLLISIONS KJF 9.5, 10.7 Conservation of momentum Recall from our discussion of momentum (Lecture 9), that
More informationRotational & Rigid-Body Mechanics. Lectures 3+4
Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions
More informationPleeeeeeeeeeeeeease mark your UFID, exam number, and name correctly. 20 problems 3 problems from exam 2
Pleeeeeeeeeeeeeease mark your UFID, exam number, and name correctly. 20 problems 3 problems from exam 1 3 problems from exam 2 6 problems 13.1 14.6 (including 14.5) 8 problems 1.1---9.6 Go through the
More informationFENG CHIA UNIVERSITY
FENG CHIA UNIVERSITY Fundamentals of Physics I (With Lab) PHYS114, Summer 2018 (May14-Jun15) Lecturer: TBA E-mail: TBA Time: Monday through Friday Contact hours: 60 (50 minutes each) Credits: 4 Office
More informationCS229 Programming Assignment 3: Dynamics
: Assignment Out: Wednesday, Oct. 12, 1999 Assignment Due: Fri Oct 22, 1999 (11:59 pm) 1 Introduction Physically based modeling has been an extremely active area of research in computer graphics for over
More information