DOCSLIB.ORG

One Dimensional Motion and Gravity When We Drop an Object It’S Velocity Continues to Increase

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
One Dimensional Motion and Gravity When We Drop an Object It’S Velocity Continues to Increase One dimensional motion and gravity When we drop an object it’s velocity continues to increase. That means there is an acceleration. Near the earths surface the value of this acceleration is g = 9.8m/s2. This is due to the attractive force of gravity. g varies slightly over the earth because the earth is not a perfect sphere and because of the rotation of the earth. On the moon g is much smaller (~1/6) For CONSTANT acceleration, The equations of motion are: 2 v = v0 + at , d = v0t + ½(at ) 2 2 + v = v0 + 2ad , d = ½(v +v0)t (starting at d =0) For this case all quantities are + at is the change in v for time interval t And g is rounded up to 10.0 m/s which is 2% off and might fail CHIP Week 2 Physics 214 1 Spring 2009 One dimensional motion and gravity d, vo, v, a, t For CONSTANT acceleration, For constant acceleration, if you The equations of motion are: are told the values of three of these quantities, you can solve for a v = v + at does not involve d 0 fourth quantity --- and not have to 2 solve for the fifth quantity if you d = v0t + ½(at ) does not involve v don’t need to, by picking the proper 2 2 v = v0 + 2ad does not involve t equation. d = ½(v0 + v)t does not involve a Displacement = avg. velocity * time (starting at d =0) Combine 1st and 4th eqns. Æ 2d eq. Combine eqs 1 & 4 differently, get 3d eqn., ‘cuz (v-v0)/a = t, plug into eq.4 2 2 d = ½(v0+v)(v-v0)/a = (v -v0 )/2a is same as eqn. 3 Week 2 Physics 214 2 Spring 2009 Throwing vertically In the example shown a ball is thrown vertically.The acceleration is minus + 9.8m/s2 and the motion is symmetric. v = v0 + at at t=0 v = v0 At the top v = 0 and t = v0/9.8 At the bottom t = 2v /9.8 and v = - v 0 0 g = -9.8m/s2 time up and down = 2 x time to top 2 d = v0t + 1/2at so d=0 when t = 0 or t = -2v0/a or 2v0/9.8 since a = -9.8 2 WHY? d = v0t+1/2at = t(v0+at/2) So d = 0 if t=0 or if ( ) = 0 Week 2 Physics 214 3 Spring 2009 Throwing vertically In the example shown a ball is thrown vertically.The acceleration is minus + 9.8m/s2 and the motion is symmetric. v = v0 + at at t=0 v = v0 At the top v = 0 and t = v0/9.8 At the bottom t = 2v /9.8 and v = - v 0 0 g = -9.8m/s2 time up and down = 2 x time to top 2 d = v0t + 1/2at so d=0 when t = 0 or t = -2v0/a or 2v0/9.8 since a = -9.8 2 2 v = v0 + 2ad At d = 0 v = +v0 (t = 0) Or v = -v0 ( t = 2v0/9.8) Week 2 Physics 214 4 Spring 2009 QUIZ Ch 3 E6 Ball is dropped What is the change in velocity between 1 and 4 seconds? A. 9.8 m/s B. 19.6 m/s v g C. 29.4 m/s D. 39.2 m/s + After 1 sec v = 9.8 t = 9.8 m/s After 4 sec v = 9.8 x 4 = 39.2 m/s Change in velocity is 29.4 m/s . Week 2 Physics 214 5 Spring 2009 Projectile Motion If we throw an object so that it’s initial velocity is horizontal then ignoring friction it will continue to move with this velocity in the horizontal direction but it will also start to fall so that it’s trajectory will be curved. Week 2 Physics 214 6 Spring 2009 Motion in two dimensions If we take two axes at right angles we can analyze the motion along each axis separately and determine properties of the whole motion. We will only deal with cases where there is a constant velocity along one axis and a constant acceleration along the other axis. This means that for up Axis 2 usually y Axis 1 usually x down axis 1 v1 = constant and d1 = v1t note: a1=0 2 2 2 axis 2 v2 = v02 + at d = v02t + ½ at v2 = v02 + 2ad Week 2 Physics 214 7 Spring 2009 Projectile Motion We will focus on projectiles with a = 9.8m/s2 9.8 v0v θ v1 R = range At the highest point the vertical velocity is zero vv = v0v + at so t = v0v/9.8 since a = -g 2 h = v0vt + ½ at = vertical height above launch At impact t = 2v0v/9.8 and R = v1 x 2v0v/9.8 And the vertical velocity is minus v0v o Maximum range R for 45 degree upward launch, easiest to see with trig. Week 2 Physics 214 8 Spring 2009 Projectile Motion We will focus on projectiles with a = 9.8m/s2 9.8 v2=vv v0v θ v1 R = range At impact t = 2v0v/9.8 and R = v1 x 2v0v/9.8 2 2 v1=v0cos(θ) v0v=v0sin(θ) R=2c.s.v0 /9.8m/s 2sin.cos = sin(2θ) , max when 2θ = 90o o Maximum range R for 45 degree upward launch, easiest to see with trig. Week 2 Physics 214 9 Spring 2009 Trajectories With no friction there are always two angles which give the same range for the same starting velocity 450 + X and 450 –X R varies as sin(2θ) http://www.physics.purdue.edu/academic_programs/courses/phys214/movies.php (anim0002.mov) http://faculty.tcc.fl.edu/scma/carrj/Java/baseball4.html http://www.mhhe.com/physsci/physical/giambattista/proj/projectile.html http://www.physics.purdue.edu/class/applets/phe/projectile.htm Week 2 Physics 214 10 Spring 2009 Throwing a ball horizontally from a building Use + down so v1 g is positive and h is positive 2 g h = v02t + ½ at h v = v02 + at 2 2 v = v02 + 2ah v2 2 R v02 = 0, t = 2h/a R = v1t = v1sqrt(2h/a) NOTE h is measured down Week 2 Physics 214 11 Spring 2009 Summary Chapter 3 Any motion in a plane can analyzed using two axes at right angles. The motion along each axis is independent of the motion along the other axis so the two dimensional motion can be analyzed as two one dimensional motions linked by time. Special case axis 1 v1 = constant and d1 = v1t 2 2 2 axis 2 v2 = v02 + at , h = v02t + 1/2at ,v = v02 + 2ah 2 v0v 9.8m/s v1 v1 v1 v For projectile motion a = 9.8m/s2 0v Week 2 Physics 214 12 Spring 2009 1D-20 Independence of Vertical & Horizontal Motions (Drop-Kick) One ball drops from rest. The other ball is simultaneously projected horizontally Which ball will hit the ground first ? Listen to the SOUND when they hit the ground and when they bounce. THE VERTICAL & HORIZONTAL MOTIONS ARE INDEPENDENT. THE HORIZONTAL VELOCITY DOES NOT AFFECT THE VERTICAL MOTION. THE VERTICAL FALL TIME IS THE SAME AS LONG AS THE BALLS DROP SIMULTANEOUSLY FROM THE SAME HEIGHT. Week 2 Physics 214 13 Spring 2009 1D-21 Independence of Vertical and Horizontal Motions A ball is projected vertically from a cart traveling horizontally The trajectory in the cart frame The trajectory in the room frame THE HORIZONTAL MOTION OF THE BALL IS UNAFFECTED BY ITS VERTICAL MOTION. Week 2 Physics 214 14 Spring 2009 1G-03 Measurement of g Measuring g by dropping an object d = ½ gt2 t = sqrt ( 2d/g ) g = 2d/t2 Let d = 1 m, What difficulties might be t = sqrt(2/9.8) = 0.45 s encountered in measuring h & t of fall ? Hand timing would not be accurate because of the short fall time Week 2 Physics 214 15 Spring 2009 1K-11 Coin and Feather DROPPING A COIN AND A FEATHER ? DO ALL OBJECTS HAVE THE SAME ACCELERATION WHEN DROPPED ? IN AIR, WEIGHT AND SURFACE AREA MAKE OBJECTS FALL AT DIFFERENT SPEEDS BECAUSE OF AIR FRICTION. Week 2 Physics 214 16 Spring 2009 1D-22 Water Jets & Projectile Motion PROJECTILE MOTION OF A WATER JET What angle gives the maximum range? g NEGLECTING FRICTION THE RANGE IS A MAXIMUM AT 450. TWO DIFFERENT ANGLES CAN GIVE THE SAME RANGE (ANGLES SYMMETRIC ABOUT 45°). A LARGER ANGLE MEANS A LONGER TIME OF FLIGHT, BUT LESS HORIZONTAL VELOCITY. A SMALLER ANGLE MEANS A LARGER HORIZONTAL VELOCITY, AND LESS FLIGHT TIME. THE TRAJECTORY IS SYMMETRIC. Week 2 Physics 214 17 Spring 2009 1D-23 Shoot the Monkey The monkey falls out of the tree at the instant the gun is fired v0y/v0x = h/d WHERE SHOULD ONE AIM, ABOVE, BELOW OR AT? y=0 . |--------------d-----------------| Turn off g, bullet hits monkey at height h. Turn on g, both bullet and monkey fall same amount 2 Ignoring friction y = v0yt – 1/2gt , t = d/v0x , v0yt = v0yd/v0x = h So at x = d y = h – ½ gt2 In the same time the monkey falls ½ gt2 So the bullet always hits the monkey no matter what the value of v0, Unless range R is less than d! THE VERTICAL MOTION IS INDEPENDENT OF THE HORIZONTAL MOTION THE EFFECT OF FRICTION IS MINIMIZED BY USING A HEAVY TARGET Week 2 Physics 214 18 Spring 2009 Questions Chapter 3 Q1 A small piece of paper is dropped and flutters to the floor.
Load more

Recommended publications
  • Newtonian Mechanics Is Most Straightforward in Its Formulation and Is Based on Newton’S Second Law
    CLASSICAL MECHANICS D. A. Garanin September 30, 2015 1 Introduction Mechanics is part of physics studying motion of material bodies or conditions of their equilibrium. The latter is the subject of statics that is important in engineering. General properties of motion of bodies regardless of the source of motion (in particular, the role of constraints) belong to kinematics. Finally, motion caused by forces or interactions is the subject of dynamics, the biggest and most important part of mechanics. Concerning systems studied, mechanics can be divided into mechanics of material points, mechanics of rigid bodies, mechanics of elastic bodies, and mechanics of fluids: hydro- and aerodynamics. At the core of each of these areas of mechanics is the equation of motion, Newton's second law. Mechanics of material points is described by ordinary differential equations (ODE). One can distinguish between mechanics of one or few bodies and mechanics of many-body systems. Mechanics of rigid bodies is also described by ordinary differential equations, including positions and velocities of their centers and the angles defining their orientation. Mechanics of elastic bodies and fluids (that is, mechanics of continuum) is more compli- cated and described by partial differential equation. In many cases mechanics of continuum is coupled to thermodynamics, especially in aerodynamics. The subject of this course are systems described by ODE, including particles and rigid bodies. There are two limitations on classical mechanics. First, speeds of the objects should be much smaller than the speed of light, v c, otherwise it becomes relativistic mechanics. Second, the bodies should have a sufficiently large mass and/or kinetic energy.
    [Show full text]
  • Chapter 04 Rotational Motion
    Chapter 04 Rotational Motion P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 04: Rotational Motion Angular Momentum Angular momentum of particle with respect to origin, O, is given by l⃗ = ⃗r × p⃗ Rate of change of angular momentum is given z by cross product of ⃗r with applied force. p m dl⃗ dp⃗ = ⃗r × = ⃗r × F⃗ = ⃗휏 r dt dt O y Cross product is defined as applied torque, ⃗휏. x Unlike linear momentum, angular momentum depends on origin choice. P. J. Grandinetti Chapter 04: Rotational Motion Conservation of Angular Momentum Consider system of N Particles z m5 m 2 Rate of change of angular momentum is m3 ⃗ ∑N l⃗ ∑N ⃗ m1 dL d 훼 dp훼 = = ⃗r훼 × dt dt dt 훼=1 훼=1 y which becomes m4 x ⃗ ∑N dL ⃗ net = ⃗r훼 × F dt 훼 Total angular momentum is 훼=1 ∑N ∑N ⃗ ⃗ L = l훼 = ⃗r훼 × p⃗훼 훼=1 훼=1 P. J. Grandinetti Chapter 04: Rotational Motion Conservation of Angular Momentum ⃗ ∑N dL ⃗ net = ⃗r훼 × F dt 훼 훼=1 Taking an earlier expression for a system of particles from chapter 1 ∑N ⃗ net ⃗ ext ⃗ F훼 = F훼 + f훼훽 훽=1 훽≠훼 we obtain ⃗ ∑N ∑N ∑N dL ⃗ ext ⃗ = ⃗r훼 × F + ⃗r훼 × f훼훽 dt 훼 훼=1 훼=1 훽=1 훽≠훼 and then obtain 0 > ⃗ ∑N ∑N ∑N dL ⃗ ext ⃗ rd ⃗ ⃗ = ⃗r훼 × F + ⃗r훼 × f훼훽 double sum disappears from Newton’s 3 law (f = *f ) dt 훼 12 21 훼=1 훼=1 훽=1 훽≠훼 P.
    [Show full text]
  • Analytical Mechanics
    A Guided Tour of Analytical Mechanics with animations in MAPLE Rouben Rostamian Department of Mathematics and Statistics UMBC [email protected] December 2, 2018 ii Contents Preface vii 1 An introduction through examples 1 1.1 ThesimplependulumàlaNewton ...................... 1 1.2 ThesimplependulumàlaEuler ....................... 3 1.3 ThesimplependulumàlaLagrange.. .. .. ... .. .. ... .. .. .. 3 1.4 Thedoublependulum .............................. 4 Exercises .......................................... .. 6 2 Work and potential energy 9 Exercises .......................................... .. 12 3 A single particle in a conservative force field 13 3.1 The principle of conservation of energy . ..... 13 3.2 Thescalarcase ................................... 14 3.3 Stability....................................... 16 3.4 Thephaseportraitofasimplependulum . ... 16 Exercises .......................................... .. 17 4 TheKapitsa pendulum 19 4.1 Theinvertedpendulum ............................. 19 4.2 Averaging out the fast oscillations . ...... 19 4.3 Stabilityanalysis ............................... ... 22 Exercises .......................................... .. 23 5 Lagrangian mechanics 25 5.1 Newtonianmechanics .............................. 25 5.2 Holonomicconstraints............................ .. 26 5.3 Generalizedcoordinates .......................... ... 29 5.4 Virtual displacements, virtual work, and generalized force....... 30 5.5 External versus reaction forces . ..... 32 5.6 The equations of motion for a holonomic system . ...
    [Show full text]
  • Fundamental Governing Equations of Motion in Consistent Continuum Mechanics
    Fundamental governing equations of motion in consistent continuum mechanics Ali R. Hadjesfandiari, Gary F. Dargush Department of Mechanical and Aerospace Engineering University at Buffalo, The State University of New York, Buffalo, NY 14260 USA [email protected], [email protected] October 1, 2018 Abstract We investigate the consistency of the fundamental governing equations of motion in continuum mechanics. In the first step, we examine the governing equations for a system of particles, which can be considered as the discrete analog of the continuum. Based on Newton’s third law of action and reaction, there are two vectorial governing equations of motion for a system of particles, the force and moment equations. As is well known, these equations provide the governing equations of motion for infinitesimal elements of matter at each point, consisting of three force equations for translation, and three moment equations for rotation. We also examine the character of other first and second moment equations, which result in non-physical governing equations violating Newton’s third law of action and reaction. Finally, we derive the consistent governing equations of motion in continuum mechanics within the framework of couple stress theory. For completeness, the original couple stress theory and its evolution toward consistent couple stress theory are presented in true tensorial forms. Keywords: Governing equations of motion, Higher moment equations, Couple stress theory, Third order tensors, Newton’s third law of action and reaction 1 1. Introduction The governing equations of motion in continuum mechanics are based on the governing equations for systems of particles, in which the effect of internal forces are cancelled based on Newton’s third law of action and reaction.
    [Show full text]
  • Energy and Equations of Motion in a Tentative Theory of Gravity with a Privileged Reference Frame
    1 Energy and equations of motion in a tentative theory of gravity with a privileged reference frame Archives of Mechanics (Warszawa) 48, N°1, 25-52 (1996) M. ARMINJON (GRENOBLE) Abstract- Based on a tentative interpretation of gravity as a pressure force, a scalar theory of gravity was previously investigated. It assumes gravitational contraction (dilation) of space (time) standards. In the static case, the same Newton law as in special relativity was expressed in terms of these distorted local standards, and was found to imply geodesic motion. Here, the formulation of motion is reexamined in the most general situation. A consistent Newton law can still be defined, which accounts for the time variation of the space metric, but it is not compatible with geodesic motion for a time-dependent field. The energy of a test particle is defined: it is constant in the static case. Starting from ”dust‘, a balance equation is then derived for the energy of matter. If the Newton law is assumed, the field equation of the theory allows to rewrite this as a true conservation equation, including the gravitational energy. The latter contains a Newtonian term, plus the square of the relative rate of the local velocity of gravitation waves (or that of light), the velocity being expressed in terms of absolute standards. 1. Introduction AN ATTEMPT to deduce a consistent theory of gravity from the idea of a physically privileged reference frame or ”ether‘ was previously proposed [1-3]. This work is a further development which is likely to close the theory. It is well-known that the concept of ether has been abandoned at the beginning of this century.
    [Show full text]
  • Chapter 6 the Equations of Fluid Motion
    Chapter 6 The equations of fluid motion In order to proceed further with our discussion of the circulation of the at- mosphere, and later the ocean, we must develop some of the underlying theory governing the motion of a fluid on the spinning Earth. A differen- tially heated, stratified fluid on a rotating planet cannot move in arbitrary paths. Indeed, there are strong constraints on its motion imparted by the angular momentum of the spinning Earth. These constraints are profoundly important in shaping the pattern of atmosphere and ocean circulation and their ability to transport properties around the globe. The laws governing the evolution of both fluids are the same and so our theoretical discussion willnotbespecifictoeitheratmosphereorocean,butcanandwillbeapplied to both. Because the properties of rotating fluids are often counter-intuitive and sometimes difficult to grasp, alongside our theoretical development we will describe and carry out laboratory experiments with a tank of water on a rotating table (Fig.6.1). Many of the laboratory experiments we make use of are simplified versions of ‘classics’ that have become cornerstones of geo- physical fluid dynamics. They are listed in Appendix 13.4. Furthermore we have chosen relatively simple experiments that, in the main, do nor require sophisticated apparatus. We encourage you to ‘have a go’ or view the atten- dant movie loops that record the experiments carried out in preparation of our text. We now begin a more formal development of the equations that govern the evolution of a fluid. A brief summary of the associated mathematical concepts, definitions and notation we employ can be found in an Appendix 13.2.
    [Show full text]
  • Lagrangian Mechanics - Wikipedia, the Free Encyclopedia Page 1 of 11
    Lagrangian mechanics - Wikipedia, the free encyclopedia Page 1 of 11 Lagrangian mechanics From Wikipedia, the free encyclopedia Lagrangian mechanics is a re-formulation of classical mechanics that combines Classical mechanics conservation of momentum with conservation of energy. It was introduced by the French mathematician Joseph-Louis Lagrange in 1788. Newton's Second Law In Lagrangian mechanics, the trajectory of a system of particles is derived by solving History of classical mechanics · the Lagrange equations in one of two forms, either the Lagrange equations of the Timeline of classical mechanics [1] first kind , which treat constraints explicitly as extra equations, often using Branches [2][3] Lagrange multipliers; or the Lagrange equations of the second kind , which Statics · Dynamics / Kinetics · Kinematics · [1] incorporate the constraints directly by judicious choice of generalized coordinates. Applied mechanics · Celestial mechanics · [4] The fundamental lemma of the calculus of variations shows that solving the Continuum mechanics · Lagrange equations is equivalent to finding the path for which the action functional is Statistical mechanics stationary, a quantity that is the integral of the Lagrangian over time. Formulations The use of generalized coordinates may considerably simplify a system's analysis. Newtonian mechanics (Vectorial For example, consider a small frictionless bead traveling in a groove. If one is tracking the bead as a particle, calculation of the motion of the bead using Newtonian mechanics) mechanics would require solving for the time-varying constraint force required to Analytical mechanics: keep the bead in the groove. For the same problem using Lagrangian mechanics, one Lagrangian mechanics looks at the path of the groove and chooses a set of independent generalized Hamiltonian mechanics coordinates that completely characterize the possible motion of the bead.
    [Show full text]
  • The Equations of Motion in a Rotating Coordinate System
    The Equations of Motion in a Rotating Coordinate System Chapter 3 Since the earth is rotating about its axis and since it is convenient to adopt a frame of reference fixed in the earth, we need to study the equations of motion in a rotating coordinate system. Before proceeding to the formal derivation, we consider briefly two concepts which arise therein: Effective gravity and Coriolis force Effective Gravity g is everywhere normal to the earth’s surface Ω 2 R 2 R R Ω R g* g g* g effective gravity g effective gravity on an earth on a spherical earth with a slight equatorial bulge Effective Gravity If the earth were a perfect sphere and not rotating, the only gravitational component g* would be radial. Because the earth has a bulge and is rotating, the effective gravitational force g is the vector sum of the normal gravity to the mass distribution g*, together with a centrifugal force Ω2R, and this has no tangential component at the earth’s surface. gg*R= + Ω 2 The Coriolis force A line that rotates with the roundabout Ω A line at rest in an inertial system Apparent trajectory of the ball in a rotating coordinate system Mathematical derivation of the Coriolis force Representation of an arbitrary vector A(t) Ω A(t) = A (t)i + A (t)j + A (t)k k 1 2 3 k ´ A(t) = A1´(t)i ´ + A2´ (t)j ´ + A3´ (t)k´ inertial j reference system rotating j ´ reference system i ´ i The time derivative of an arbitrary vector A(t) The derivative of A(t) with respect to time d A dA dA dA a =+ijk12 + 3 dt dt dt dt the subscript “a” denotes the derivative in an inertial reference frame ddAdAi′ ′ In the rotating frame a1=++i′′A ..
    [Show full text]
  • Chapter 8. Motion in a Noninertial Reference Frame
    Chapter 8. Motion in a Noninertial Reference Frame (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 10.) We have so far dealt only with problems situated in inertial reference frame, or if not, problems that could be solved with enough accuracy by ignoring the noninertial nature of the coordinate systems. There are, however, many problems for which it is necessary, or beneficial, to treat the motion of the system at hand in a noninertial reference frame. In this chapter, we will develop the mathematical apparatus that will allow us to deal with such problems, and prepare the way for the study of the motion of rigid bodies that we will tackle in the next chapter. 8.1 Rotating Coordinate Systems Let’s consider two coordinate systems: one that is inertial and for which the axes are fixed, and another whose axes are rotating with respect to the inertial system. We represent the coordinates of the fixed system by xi! and the coordinates of the rotating system by xi . If we choose some point in space P (see Figure 8-1) we have r! = R + r, (8.1) where R locates the origin of the rotating system in the fixed system. We assume that P is at rest in the inertial so that r! is constant. If during an infinitesimal amount of time dt the rotating system undergoes an infinitesimal rotation d! about some axis, then the vector r will vary not only as measured by an observer co-moving with the rotating system, but also when measured in the inertial frame.
    [Show full text]
  • Equations of Motion Lecture 19
    Equations of Motion Lecture 19 ME 231: Dynamics Question of the Day Determine the vertical acceleration of the 60- lb cylinder for each of the two cases. ME 231: Dynamics 2 Outline for Today # Question ofof the dayday # Free body diagram # Equations of motion # Two types of problems – Inverse dynamics – Forward dynamics # Constrained vs. unconstrained motion # Rectilinear motion # Answer your questions! ME 231: Dynamics 3 Free Body Diagram Perhaps the most important concept in mechanics! o o F2 = 50(cos30 j + sin30 k) F1 = 50 j 30o mz my y j F3 = -100 j y k z z x mg ME 231: Dynamics 4 Equations of Motion Perhaps the most important concept in dynamics! # Vector form of force-mass- acceleration equation B F ma # Component form of force- my mass-acceleration equation B mz mx Fx max mx y B Fy may my B F ma mz z x z z mg ME 231: Dynamics 5 Inverse Dynamics Problem inverse Velocities d d Forces F a Positions m dt dt # Accelerations are specified or determined directly from kinematic conditions # Determine the corresponding forces ME 231: Dynamics 6 Forward Dynamics Problem inverseinverse Velocities Forces F ma O O Positions forward # Forces are specified or determined determ from a FBD # Determine the corresponding acceleration If forces are functions of time, position, or velocity, then integrate differential equation to determine velocity and position ME 231: Dynamics 7 Constrained vs. Unconstrained Motion # Unconstrained – Free of mechanical guideses – Follows a path determineded by its initial motion andd applied external forces # Constrained – Path is partially or totallyy determined by restraining guides – All forces (applied and reactive) must be accounted for in F=ma ME 231: Dynamics 8 Outline for Today # Question ofof the dayday # Free bbodyody ddiagramiagram # E Equationsquations of motionmotion # Two ttypesypes ofof problemsproblems – Inverse dynamicsdynamics – ForwardForward dynamicsdynamics # ConstrainedConstrained vs.
    [Show full text]
  • Direct Application of the Least Action Principle to Solve (Human) Movement Dynamics Without Using Differential Equations of Motion
    Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be liable for copyright infringement, This institution reserves the right to refuse to accept a copying order if, in its judgment, fulfillment of the order would involve violation of copyright law. Please Note: The author retains the copyright while the New Jersey Institute of Technology reserves the right to distribute this thesis or dissertation Printing note: If you do not wish to print this page, then select “Pages from: first page # to: last page #” on the print dialog screen The Van Houten library has removed some of the personal information and all signatures from the approval page and biographical sketches of theses and dissertations in order to protect the identity of NJIT graduates and faculty. ABSTRACT DIRECT APPLICATION OF THE LEAST ACTION PRINCIPLE TO SOLVE (HUMAN) MOVEMENT DYNAMICS WITHOUT USING DIFFERENTIAL EQUATIONS OF MOTION by Amitha Kumar This thesis explores the numerical feasibility of solving motion 2-point bndr vl prbl (BVPs) by direct numerical minimization of the action without using the tn f tn (EOM) as an intermediate step.
    [Show full text]
  • Physics Formulas List
    Physics Formulas List Home Physics 95 Physics Formulas 23 Listen Physics Formulas List Learning physics is all about applying concepts to solve problems. This article provides a comprehensive physics formulas list, that will act as a ready reference, when you are solving physics problems. You can even use this list, for a quick revision before an exam. Physics is the most fundamental of all sciences. It is also one of the toughest sciences to master. Learning physics is basically studying the fundamental laws that govern our universe. I would say that there is a lot more to ascertain than just remember and mug up the physics formulas. Try to understand what a formula says and means, and what physical relation it expounds. If you understand the physical concepts underlying those formulas, deriving them or remembering them is easy. This Buzzle article lists some physics formulas that you would need in solving basic physics problems. Physics Formulas Mechanics Friction Moment of Inertia Newtonian Gravity Projectile Motion Simple Pendulum Electricity Thermodynamics Electromagnetism Optics Quantum Physics Derive all these formulas once, before you start using them. Study physics and look at it as an opportunity to appreciate the underlying beauty of nature, expressed through natural laws. Physics help is provided here in the form of ready to use formulas. Physics has a reputation for being difficult and to some extent that's http://www.buzzle.com/articles/physics-formulas-list.html[1/10/2014 10:58:12 AM] Physics Formulas List true, due to the mathematics involved. If you don't wish to think on your own and apply basic physics principles, solving physics problems is always going to be tough.
    [Show full text]