Physics
Change of Momentum
Change of momentum refers to the difference between an object's initial momentum and its final momentum. It is calculated by subtracting the initial momentum from the final momentum. In physics, this change can occur due to the application of a force over a period of time, resulting in an acceleration or deceleration of the object.
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12 Key excerpts on "Change of Momentum"
eBook - PDF- Richard C. Hill, Kirstie Plantenberg(Authors)
- 2013(Publication Date)
- SDC Publications(Publisher)
These two examples, in a sense, capture the meaning of the precise mechanics/physics definition of momentum. These examples give you a sense that you and the team have some sort of forward motion that will carry you through your next obstacle. The more momentum you have, the more difficult it will be for you to stop. Momentum, as defined in mechanics, depends on mass and velocity. If you have a lot of mass and velocity, you have a lot of momentum that will carry you forward. Objects in motion have momentum. Linear momentum is defined as the product of the particle's mass and velocity as shown in Equation 9.1-1. Linear momentum is a vector quantity because it has both magnitude and direction. The direction derives from the particle's velocity. Because momentum is a vector, Equation 9.1-1 can be applied in each orthogonal coordinate direction. In many instances, linear momentum is referred to as just momentum. However, it is important to understand the difference between linear momentum, which is presented in this section, and angular momentum which is presented in a later section. Momentum does not indicate the velocity of an object, although velocity (v) is used in the calculation of momentum (G). Momentum gives you a sense of the force of an object and its ability to do work*. Even if a particle has a small velocity, it could still have significant momentum. For instance, consider a semi-truck. Its mass is very large, therefore, even if its velocity is small, it would still have a large amount of momentum. On the other hand, a bullet has a very small mass when compared to a semi-truck, but when fired from a gun, it also has a very large amount of momentum. What quantity or quantities are a measure of how hard a moving object is to stop? Momentum: The force of movement. Linear momentum of a particle: m G v (9.1-1) G = linear momentum of a particle m = particle's mass v = particle's velocity It should be noted that momentum is reference-frame dependent.
eBook - ePub- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
linear momentum, which is defined as follows:DEFINITION OF LINEAR MOMENTUM
The linear momentum(7.2)p →of an object is the product of the object's mass m and velocityv →:Linear momentum is a vector quantity that points in the same direction as the velocity.p →= mv →SI Unit of Linear Momentum: kilogram · meter/second (kg · m/s)Newton's second law of motion can now be used to reveal a relationship between impulse and momentum. Figure 7.3 shows a ball with an initial velocityapproaching a bat, being struck by the bat, and then departing with a final velocityv →0. When the velocity of an object changes fromv →ftov →0during a time interval Δt, the average accelerationv →fis given by Equation 2.4 asa →¯=a →¯−v →fv →0Δ tAccording to Newton's second law,(7.3)∑the average acceleration is produced by the net average force= mF →¯a →¯∑. HereF →¯∑represents the vector sum of all the average forces that act on the object. Thus,F →¯∑= mF →¯(=)−v →fv →0Δ tm− mv →fv →0Δ tIn this result, the numerator on the far right is the final momentum minus the initial momentum, which is the change in momentum. Thus, the net average force is given by the change in momentum per unit of time.* Multiplying both sides of Equation 7.3 by Δt yields Equation 7.4 , which is known as the impulse–momentum theorem.FIGURE 7.3When a bat hits a ball, an average forceis applied to the ball by the bat. As a result, the ball's velocity changes from an initial value ofF →¯(top drawing) to a final value ofv →0v →f
eBook - PDF- Daniel Kleppner, Robert Kolenkow(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
In this view, Newton’s third law is a simple consequence of the conservation of momentum for interacting particles. For our present purposes it is purely a matter of taste whether we wish to regard Newton’s third law or conservation of momentum as more fundamental. 4.6 Impulse and a Restatement of the Momentum Relation The relation between force and momentum is F = d P dt . (4.6) As a general rule, any law of physics that can be expressed in terms of derivatives can also be written in an integral form. The integral form of the force–momentum relationship is t 0 F dt = P ( t ) − P (0) . (4.7) The change in momentum of a system is given by the integral of force with respect to time. Equation ( 4.7 ) contains essentially the same phys-ical information as Eq. ( 4.6 ), but it gives a new way of looking at the e ff ect of a force: the change in momentum is the time integral of the force. To produce a given change in the momentum in time interval t 132 MOMENTUM requires only that t 0 F dt have the appropriate value; we can use a small force acting for much of the time or a large force acting for only part of the interval. The integral t 0 F dt is called the impulse . The word impulse calls to mind a short, sharp shock, as in Example 4.7 , where a blow to a mass at rest gave it a velocity v 0 . However, the physical definition of impulse can just as well apply to a weak force acting for a long time. Change of Momentum depends only on F dt , independent of the detailed time dependence of the force. Here are three examples involving impulse and momentum. Example 4.9 Measuring the Speed of a Bullet Faced with the problem of measuring the speed of a bullet, our first thought might be to turn to a raft of high-tech equipment—fast photodetectors, fancy electronics, whatever. In this example we show that a simple mechanical system can make the measurement, with the aid of conservation of momentum. We take a simplified model to emphasize the fundamental principles.
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
That change is related to the force by Newton’s second law written in the form F = d p∕dt. By rearranging this second‐law expression, we see that, in time interval dt, the change in the ball’s momentum is d p = F (t) dt. (9.20) We can find the net change in the ball’s momentum due to the collision if we integrate both sides of this equation: ∫ p 2 p 1 d p = ∫ t 2 t 1 F (t) dt. (9.21) The left side of this equation gives us the change in the momentum: p f − p i = Δ p. The right side, which is a measure of both the magnitude and the duration of the collision force, is called the impulse J of the collision: J = ∫ t 2 t 1 F (t) dt. (9.22) Pdf_Folio:150 150 Fundamentals of physics Thus, the change in an object’s momentum is equal to the impulse on the object: Δ p = J. (9.23) This expression, which is known as the impulse–linear momentum theorem, can also be written in the vector form p f − p i = J (9.24) and in component forms such as Δp x = J x (9.25) and p fx − p ix = ∫ t f t i F x dt. (9.26) Integrating the force If we have a function for F (t), we can evaluate J (and thus the change in momentum) by integrating the function. If we have a plot of F versus time t, we can evaluate J by finding the area between the curve and the t axis, such as in figure 9.9. In many situations we do not know how the force varies with time but we do know the average magnitude F avg of the force and the duration Δt (= t f − t i ) of the collision. Then we can write the magnitude of the impulse as J = F avg Δt. (9.27) The average force is plotted versus time as shown in figure 9.10. FIGURE 9.9 The curve shows the magnitude of the time-varying force F(t) that acts on the ball in the collision of fgure 9.8. The area under the curve is equal to the magnitude of the impulse J on the ball in the collision. t i F J F(t) t f Δ t t The impulse in the collision is equal to the area under the curve.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The collision is brief, and the ball experiences a force that is great enough to slow, stop, or even reverse its motion. Figure 9-8 depicts the collision at one instant. The ball experiences a force F → (t) that varies during the collision and changes the linear momentum p → of the ball. That change is related to the force by Newton’s second law written in the form F → = d p → /dt. By rearranging this second-law expression, we see that, in time interval dt, the change in the ball’s momentum is d p → = F → (t) dt. (9-28) The collision of a ball with a bat collapses part of the ball. Photo by Harold E. Edgerton. © The Harold and Esther Edgerton Family Trust, courtesy of Palm Press, Inc. Key Ideas impulse (and thus also the momentum change) by integrating the function. 9.24 Given a graph of force versus time, calculate the impulse (and thus also the momentum change) by graphical integration. 9.25 In a continuous series of collisions by projectiles, calculate the average force on the target by relating it to the rate at which mass collides and to the velocity change experienced by each projectile. ● Applying Newton’s second law in momentum form to a particle-like body involved in a collision leads to the impulse–linear momentum theorem: p → f − p → i = Δ p → = J → , where p → f − p → i = Δ p → is the change in the body’s linear momentum, and J → is the impulse due to the force F → (t) exerted on the body by the other body in the collision: J → = ∫ t f t i F → (t) dt. ● If F avg is the average magnitude of F → (t) during the col- lision and Δt is the duration of the collision, then for one- dimensional motion J = F avg Δt. ● When a steady stream of bodies, each with mass m and speed v, collides with a body whose position is fixed, the average force on the fixed body is F avg = − n Δt Δp = − n Δt m Δv, where n/Δt is the rate at which the bodies collide with the fixed body, and Δv is the change in velocity of each col- liding body.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Collisions occur commonly in our world, but before we get to them, we need to consider a simple collision in which a moving particle-like body (a projectile) collides with some other body (a target). Single Collision Let the projectile be a ball and the target be a bat. The collision is brief, and the ball experiences a force that is great enough to slow, stop, or even reverse its motion. Figure 9-8 depicts the collision at one instant. The ball experiences a force F → (t) that varies during the collision and changes the linear momentum p → of the ball. That change is related to the force by Newton’s second law written in the form F → = d p → /dt. By rearranging this second-law expression, we see that, in time interval dt, the change in the ball’s momentum is d p → = F → (t) dt. (9-28) The collision of a ball with a bat collapses part of the ball. Photo by Harold E. Edgerton. © The Harold and Esther Edgerton Family Trust, courtesy of Palm Press, Inc. Key Ideas impulse (and thus also the momentum change) by integrating the function. 9.24 Given a graph of force versus time, calculate the impulse (and thus also the momentum change) by graphical integration. 9.25 In a continuous series of collisions by projectiles, calculate the average force on the target by relating it to the rate at which mass collides and to the velocity change experienced by each projectile. ● Applying Newton’s second law in momentum form to a particle-like body involved in a collision leads to the impulse–linear momentum theorem: p → f − p → i = Δ p → = J → , where p → f − p → i = Δ p → is the change in the body’s linear momentum, and J → is the impulse due to the force F → (t) exerted on the body by the other body in the collision: J → = ∫ t f t i F → (t) dt. ● If F avg is the average magnitude of F → (t) during the col- lision and Δt is the duration of the collision, then for one- dimensional motion J = F avg Δt.
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
Chapter 9 Momentum and Impulse The language of science must be transparent like a window pane, that of poetry, like a stained glass window. A. Oz 9.1 Introduction In view of the knowledge we have accumulated so far in kinematics, dynamics and energy, we might be tempted to think that we can cope with almost any problem in mechanics. However, a cursory glance at collision problems will dispel this illusion. We consider two such problems, whose solution forces us to introduce new concepts. (a) Two bodies of the same mass approach each other with known velocities, collide and recoil. Assuming the total kinetic energy is conserved, find the velocities after the collision. (b) As in part a, but this time the bodies stick together and continue with a common velocity. Find this velocity. It is problems like these that make us realize that our 'toolbox' is not yet complete. This chapter will supply the missing item. Example 1 A body of mass m is traveling with constant initial velocity V}. Find the change in its velocity caused by a constant external force F 0 acting on it for a time At. Solution The solution is easily obtained by a combination of kinematics and dynamics. The acceleration is a = F(/m, so that the final velocity is V f = V t + aAt = V { + (F 0 /m)At Rearranging, we get F 0 At = mV f -mV t (9.1) This result suggests that we define the following vector quantities: Impulse J = FAt (a product of the force vector and the duration of its action). Momentump = mV (a product of the mass and the velocity vector of the body). We may therefore rewrite (9.1) as r = o r = Af rr y 0 1 m 1 v s x 1 1 m 280 Chapter 9 Momentum and Impulse 281 J=Pf-Pi = Ap (9.2) In words: The impulse acting on the body equals the change in its momentum. Note that the three vectors F, J and Ap are parallel to one another.
eBook - PDF- Donald F. Elger, Barbara A. LeBret, Clayton T. Crowe, John A. Roberson(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
156 The Momentum Equation CHAPTER ROAD MAP This chapter presents (a) the linear momentum equation and (b) the angular momentum equation. Both equations are derived from Newton’s second law of motion. CHAPTERSIX FIGURE 6.1 Engineers design systems by using a small set of fundamental equations, such as the momentum equation. (Photo courtesy of NASA.) LEARNING OUTCOMES NEWTON’S SECOND LAW (§6.1) ● Know the main ideas about Newton’s second law of motion. ● Solve problems that involve Newton’s second law by applying the visual solution method. THE LINEAR MOMENTUM EQUATION (§6.2 to §6.4) ● List the steps to derive the momentum equation and explain the physics. ● Draw a force diagram and a momentum diagram. ● Explain or calculate the momentum flow. ● Apply the linear momentum equation to solve problems. MOVING CONTROL VOLUMES (§6.5) ● Distinguish between an inertial and noninertial reference frame. ● Solve problems that involve moving control volumes. 6.1 Understanding Newton’s Second Law of Motion The momentum equation, which is the subject of this chapter, is useful for solving many types of engineering problems. For example, the momentum equation is applied to the design of rockets; see Fig. 6.1. The momentum equation is derived from Newton’s second law of motion, which is described next. Body and Surface Forces A force is an interaction between two bodies that can be idealized as a push or pull of one body on another body. A push/pull interaction is one that can cause acceleration. Newton’s third law tells us that forces must involve the interaction of two bodies and that forces occur in pairs. The two forces are equal in magnitude, opposite in direction, and colinear. EXAMPLE. To give examples of force, consider an airplane that is flying in a straight path at constant speed (Fig. 6.2). Select the airplane as the system for analysis. Idealize the air- plane as a particle. Newton’s first law (i.e., force equilibrium) tells us that the sum of forces must balance.
eBook - PDF- Donald F. Elger, Barbara A. LeBret, Clayton T. Crowe, John A. Robertson(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
188 The Momentum Equation CHAPTER ROAD MAP This chapter presents (a) the linear momentum equation and (b) the angular momentum equation. Both equations are derived from Newton’s second law of motion. CHAPTERSIX FIGURE 6.1 Engineers design systems by using a small set of fundamental equations, such as the momentum equation. (Photo courtesy of NASA.) LEARNING OUTCOMES NEWTON’S SECOND LAW (§6.1). ● Know the main ideas about Newton’s second law of motion. ● Solve problems that involve Newton’s second law by applying the visual solution method. THE LINEAR MOMENTUM EQUATION (§6.2 to §6.4). ● List the steps to derive the momentum equation and explain the physics. ● Draw a force diagram and a momentum diagram. ● Explain or calculate the momentum flow. ● Apply the linear momentum equation to solve problems. MOVING CONTROL VOLUMES (§6.5). ● Distinguish between an inertial and noninertial reference frame. ● Solve problems that involve moving control volumes. 6.1 Understanding Newton’s Second Law of Motion Because Newton’s second law is the theoretical foundation of the momentum equation, this section reviews relevant concepts. Body and Surface Forces A force is an interaction between two bodies that can be idealized as a push or pull of one body on another body. A push/pull interaction is one that can cause acceleration. Newton’s third law tells us that forces must involve the interaction of two bodies and that forces occur in pairs. The two forces are equal in magnitude, opposite in direction, and colinear. EXAMPLE. To give examples of force, consider an airplane that is flying in a straight path at constant speed (Fig. 6.2). Select the airplane as the system for analysis. Idealize the air- plane as a particle. Newton’s first law (i.e., force equilibrium) tells us that the sum of forces must balance. There are four forces on the airplane: • The lift force is the net upward push of the air (body 1) on the airplane (body 2).
eBook - PDF- Wm.H. Corcoran(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
CHAPTER I INTRODUCTION T O MOMENTUM TRANSFER The motion of any body of fluid involves momentum. The science of fluid mechanics is frequently concerned with the transfer of momentum from one part of the flowing system to another and to the surroundings. An understanding of the physical nature of momentum and the application of this concept to systems of variable weight are useful in the prediction of the physical behavior of fluids in motion. Throughout this discussion primary emphasis will be placed upon a consideration of the conservation of momentum rather than upon a balance of forces. The advantages of this approach become clear in the application of fluid mechanics to the evaluation of material and energy transfers. In describing the flow of fluids it is desirable to adopt certain conventions and symbols in order to simplify the application of Newtonian mechanics to such processes. In the present chapter an effort is made to familiarize the reader with these conventions and symbols. The general nature of flow processes and some of the simpler relationships also are described. Primary emphasis is given to a description of the general characteristics of isothermal fluid flow and to a presentation of the groundwork upon which a quantitative treatment may be based. It is assumed that the reader has some familiarity with mechanics and an understanding of the elements of thermodynamics. It is particularly necessary that the concept of dynamic physical equilibrium be developed along with an understanding of the physical nature of flow processes and the concepts of steady state. Throughout this treatment it will be assumed that local microscopic 1 equilibrium (/) obtains. This assumption does not imply in any way that the conditions at a point are invariant in respect to time but only that the properties of the fluid may be described in terms of the thermodynamic state of the system at any given time.
- T. Crouch(Author)
- 2016(Publication Date)
- Pergamon(Publisher)
Chapter 2 Dynamics 2.1.Newton's Laws of Motion Dynamics is concerned with the relationships between force, mass, energy and motion. For Engineering applications, except those dealing with nuclear and fast moving electron phenomena, the Newtonian model of mass, space, time and force is adequate. Newton (1642 - 1727) in his Philosophiae Naturalis Prinoipia Mathem-atical of 1687 enunciated three laws or axioms relating force and motion which can be stated as follows: 1 A particle will continue in a state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it. 2 A change of motion with respect to time is proportional to the motive force impressed. 3 For every force acting on a particle, there is a corresponding force exerted by the particle. These forces are equal in magnitude, but opposite in direction. The first law implies the existence of an inertial frame of reference. Consider the following hypothetical experiment. Erect a set of co-ordinate axes in deep space remote from any other matter and project a particle successively along each axis. If the axes are not accel-erating and not rotating, then the force free motion will persist along the axis which it was projected. Such a set of axes is said to be inertial. No set of axes is truly inertial, but a set of axis fixed in the f fixed' stars are very nearly inertial and must be used, for example,in space ballistics. For most Engineering applications forces can be predicted assuming that a reference frame fixed in the earth is inertial. In this text the inertial reference is always designated 1. The motion of the second law is measured by the momentum of the particle, which is the product of its mass and inertial velocity.Thus, by Newton's second law 38
eBook - PDF- Bogdan Skalmierski(Author)
- 2013(Publication Date)
- Elsevier(Publisher)
CHAPTER 2 The Dynamics of a Particle 2.1 Fundamental definitions and theorems In this chapter we shall discuss the dynamic aspects of a particle in motion. We begin with the basic laws of dynamics. Axioi 1 (Newton's second law). If a force P acts on a particle, the acceleration thus produced is proportional to that force, which can be written as follows: P= m a, (1) where m is the mass of the particle. We shall treat mass as a primary concept. The force P should be regarded as the resultant of the forces acting on the particle, that is, n P = R j . (la) f= 1 The cited law brings into association three basic concepts (force, mass and motion) of mechanics. It is valid in inertial systems (see Chapter 5). Under the SI system, the unit of force is the newton: 1 kgms -2 = 11. For a unit of force we can also take the force with which the earth attracts 1 kg of mass: 1 kgf = 1 kg • g, where g = 9.80665 m s — 2 and is the normal value of acceleration of gravity. The equality sign is valid between inert and heavy mass. Axioi 2 (Newton's third law). If a particle A acts on another particle B with force P AD , then simultaneously B acts on A with force PBA of equal absolute value but with an opposite sense, i.e. R A B + PBa = = 0 . (2) (4) a V P i = 70 THE DYNAMICS OF A PARTICLE Ch. 2 This is known as the law of action and reaction. We shall now introduce the definition of work. By work one should understand a process in which resistance is being overcome along a certain route. This definition is, however, imprecise, and for that reason it is better to define work in concise mathematical notation: Work analytically formulated is a curvilinear integral df B W = P • ds . (3) A If under the integral (3) a total differential occurs, then the forces doing the work are said to have potential V. A decrease in potential is tantamount to an increase in work: — ~ V = d W. Therefore Potential forces, as it will easily be seen, act in the direction of the maximum drop of the potential.
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