Physics
Momentum
Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is calculated as the product of an object's mass and its velocity. Momentum is a vector quantity, meaning it has both magnitude and direction, and is conserved in a closed system, making it a crucial factor in understanding the behavior of moving objects.
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12 Key excerpts on "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 - PDFThe Mechanical Universe
Mechanics and Heat, Advanced Edition
- Steven C. Frautschi, Richard P. Olenick, Tom M. Apostol, David L. Goodstein(Authors)
- 2008(Publication Date)
- Cambridge University Press(Publisher)
11.6 ENERGY AND Momentum CONSERVATION IN COLLISIONS In deriving the law of Momentum conservation, we did not need to specify the nature of the forces acting between the bodies. These forces between the bodies could have been electrical, gravitational, or other, and it would make no difference whatsoever; Momentum is conserved regardless of the nature of participating forces. The law of conservation of Momentum, like that of conservation of energy, is avast and powerful principle. It makes an overall statement about nature without fussing over the details of the force involved. We now have two conservation laws which may be applied together as a potent aid to understanding certain kinds of problems. These are problems in which two objects initially moving in some direction briefly interact, as in a collision, and move off in different directions. The word interact in physics means that some force is applied between the objects. We may not know the details of the force, and we may not know whether the objects have touched or interacted at a distance without direct contact, but we can conclude that for a brief time the particles have exerted forces on each other from the fact that the motion of each object is changed. By using the laws of conservation of energy and Momentum, we can predict much about the subsequent motion of particles in this type of encounter. Although it might not appear that the world is simply a collection of bodies colliding with each other, this type of problem is important in modern physics. Twentieth-century physics is an exploration of quantum mechanics - the physics of atoms and elementary particles. What physicists want to know in quantum mechanics is the hidden, invisible 276 THE CONSERVATION OF Momentum internal structure of atoms, nuclei, and even protons and neutrons themselves. But there are very few ways to find what goes on inside a nucleus.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Impulse is a vector that points in the same direction as the average force. The linear Momentum p B of an object is the product of the object’s mass m and velocity v B , accord- ing to Equation 7.2. Linear Momentum is a vector that points in the same direction as the velocity. The total linear Momentum of a system of objects is the vector sum of the momenta of the individual objects. The impulse–Momentum theorem states that when a net average force SF B acts on an object, the impulse of this force is equal to the change in Momentum of the object, as in Equation 7.4. 7.2 The Principle of Conservation of Linear Momentum External forces are those forces that agents external to the system exert on objects within the system. An isolated system is one for which the vector sum of the external forces acting on the system is zero. The principle of conservation of linear Momentum states that the total linear Momentum of an isolated system remains constant. For a two-body system, the conservation of linear Momentum can be written as in Equation 7.7b, where m 1 and m 2 are the masses, v B f1 and v B f2 are the final velocities, and v B 01 and v B 02 are the initial velocities of the objects. 7.3 Collisions in One Dimension An elastic collision is one in which the total kinetic energy of the system after the collision is equal to the total kinetic energy of the system before the collision. An inelastic collision is one in which the total kinetic energy of the system is not the same before and after the collision. If the objects stick together after the collision, the collision is said to be completely inelastic. 7.4 Collisions in Two Dimensions When the total linear Momentum is conserved in a two- dimensional collision, the x and y components of the total linear Momentum are conserved separately. For a collision between two objects, the conservation of total linear Momentum can be written as in Equations 7.9a and 7.9b.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Since each weight is balanced by a normal force, the sum of the external forces is zero, and the total Momentum of the system is conserved, as Equation 7.7b indicates. Momentum is a vector quantity, however, and in two dimensions the x and y components of the total Momentum are conserved separately. In other words, Equation 7.7b is equivalent to the following two equations: x Component m 1 υ f 1 x + m 2 υ f 2 x = m 1 υ 01x + m 2 υ 02x (7.9a) P fx P 0x y Component m 1 υ f 1y + m 2 υ f 2y = m 1 υ 01y + m 2 υ 02y (7.9b) P fy P 0y These equations are written for a system that contains two objects. If a system contains more than two objects, a mass-times-velocity term must be included for each additional object on either side of Equations 7.9a and 7.9b. 7.5 Center of Mass In previous sections, we have encountered situations in which objects interact with one another, such as the two skaters pushing off in Example 7. In these situations, the mass of the system is located in several places, and the various objects move relative to each other before, after, and even during the interaction. It is possible, however, to speak of a kind of average location for the total mass by introducing a concept known as the center of mass (abbreviated as “cm”). With the aid of this concept, we will be able to gain additional insight into the principle of conservation of linear Momentum. The center of mass is a point that represents the average location for the total mass of a system. Figure 7.15, for example, shows two particles of mass m 1 and m 2 that are located on the x axis at the positions x 1 and x 2 , respectively.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
180 Physics REASONING STRATEGY Applying the principle of conservation of linear Momentum 1. Decide which objects are included in the system. 2. Relative to the system that you have chosen, identify the internal forces and the external forces. 3. Verify that the system is isolated. In other words, verify that the sum of the external forces applied to the system is zero. Only if this sum is zero can the conservation principle be applied. If the sum of the average exter- nal forces is not zero, consider a different system for analysis. 4. Set the total final Momentum of the isolated system equal to the total initial Momentum. Remember that linear Momentum is a vector. If necessary, apply the conservation principle separately to the various vector components. 7.3 Collisions in one dimension LEARNING OBJECTIVE 7.3 Analyse one-dimensional collisions. As discussed in the previous section, the total linear Momentum is conserved when two objects collide, provided they constitute an isolated system. When the objects are atoms or subatomic particles, the total kinetic energy of the system is often conserved also. In other words, the total kinetic energy of the particles before the collision equals the total kinetic energy of the particles after the collision, so that kinetic energy gained by one particle is lost by another. In contrast, when two macroscopic objects collide, such as two cars, the total kinetic energy after the collision is generally less than that before the collision. Kinetic energy is lost mainly in two ways: (1) It can be converted into heat because of friction, and (2) it is spent in creating permanent distortion or damage, as in an automobile collision. With very hard objects, such as a solid steel ball and a marble floor, the permanent distortion suffered upon collision is much smaller than with softer objects and, consequently, less kinetic energy is lost.
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.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Because of Momentum conservation, the velocity of the center of mass of the balls is the same before and after the collision (see the vectors labeled v cm ). As a result, the center of mass moves along the same straight-line path before and after the collision. CONCEPT SUMMARY 7.1 The Impulse–Momentum Theorem The impulse J B of a force is the product of the average force F B and the time interval Dt during which the force acts, according to Equation 7.1. Impulse is a vector that points in the same direction as the average force. The linear Momentum p B of an object is the product of the object’s mass m and velocity v B , accord- ing to Equation 7.2. Linear Momentum is a vector that points in the same direction as the velocity. The total linear Momentum of a system of objects is the vector sum of the momenta of the individual objects. The impulse–Momentum theorem states that when a net average force SF B acts on an object, the impulse of this force is equal to the change in Momentum of the object, as in Equation 7.4. 7.2 The Principle of Conservation of Linear Momentum External forces are those forces that agents external to the system exert on objects within the system. An isolated system is one for which the vector sum of the external forces acting on the system is zero. The principle of conservation of linear Momentum states that the total linear Momentum of an isolated system remains constant. For a two-body system, the conservation of linear Momentum can be written as in Equation 7.7b, where m 1 and m 2 are the masses, v B f1 and v B f2 are the final velocities, and v B 01 and v B 02 are the initial velocities of the objects. J B 5 F B Dt (7.1) p B 5 m v B (7.2) ( SF B ) D t 5 mv B f 2 mv B 0 (7.4) m 1 v f1 B 1 m 2 v f2 B 5 m 1 v 01 B 1 m 2 v 02 B (7.7b) Initial total linear Momentum Final total linear Momentum μ μ
eBook - PDFEngineering Mechanics
Dynamics
- L. G. Kraige, J. N. Bolton(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
We found that the velocity changes could be expressed directly in terms of the work done or in terms of the overall changes in energy. In the next two articles, we will integrate the equation of motion with respect to time rather than displacement. This approach leads to the equations of impulse and Momentum. These equations greatly facilitate the solution of many problems in which the applied forces act during extremely short periods of time (as in impact problems) or over specified intervals of time. 3/9 Linear Impulse and Linear Momentum Consider again the general curvilinear motion in space of a particle of mass m, Fig. 3∕11, where the particle is located by its position vector r measured from a fixed origin O. The velocity of the particle is v = r ˙ and is tangent to its path (shown as a dashed line). The resultant ΣF of all forces on m is in the direction of its acceleration v ˙ . We may now write the basic equation of motion for the particle, Eq. 3∕3, as ΣF = mv ˙ = d dt ( mv) or ΣF = G ˙ (3∕ 25) where the product of the mass and velocity is defined as the linear Momentum G = mv of the particle. Equation 3 ∕25 states that the resultant of all forces acting on a particle equals its time rate of change of linear Momentum. In SI the units of linear Momentum mv are seen to be kg ∙ m∕ s, which also equals N ∙ s. In U.S. customary units, the units of linear Momentum mv are [lb∕(ft∕ sec 2 )][ft∕ sec] = lb-sec. Because Eq. 3∕ 25 is a vector equation, we recognize that, in addition to the equality of the magnitudes of ΣF and G ˙ , the direction of the resultant force coin- cides with the direction of the rate of change in linear Momentum, which is the direction of the rate of change in velocity. Equation 3∕ 25 is one of the most useful and important relationships in dynamics, and it is valid as long as the mass m of the particle is not changing with time. The case where m changes with time is dis- cussed in Art.
eBook - PDF- Wm.H. Corcoran(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
From this definition it follows that the dimensions of an element of volume may vary widely, depending on the conditions of the problem. 1-4. Conservatio n o f Momentu m The Momentum per unit volume of fluid is given by the following defining expression: M x , v : au x pu x . (1.01) Equation (1.01) is limited to movement in the ^-direction. Momentum is a vector quantity and must be evaluated in both magnitude and direction. Throughout the following discussion all flows will be limited to the ^-direction and the subscripts indicating this fact will be omitted. In dealing with a system of variable weight which is fixed in space, it is convenient to consider the Momentum flux to or from the element of volume. The Momentum flux at a section per unit area may be expressed as Μ = = g a w (1.02) F L O W I N X -D I R E C T I O N Δ Υ Δ Ζ -Α ο F I G . I-l. Element of volume stream. in a flowing The weight of the system of fixed total volume may vary with time as a result of change in specific weight of the fluid. Equation (1.02) serves as the defining relationship for the rate of transfer of Momentum across a section per unit area. A cross section of area A 0 constitutes one side of an element of volume with length ox shown in Fig. I-l. The Momentum of the fluid in the element of volume shown in this figure may be indicated in the following way where the velocity and specific weight are functions of position in the yz-plane: SM δχ. (1.03) 4 I. INTRODUCTION TO Momentum T R A N S F E R In the case for which the velocity and specific weight do not vary from one part to another of this section of area A 0> Eq. (1.03) simplifies to <5M = A 0 — δχ. (1.04) Newton postulated that the time rate of change of Momentum of a system is equal to the forces acting on it. This fact may be expressed for a system of volume A 0 δ χ in the following way: , „ (dM I -T * -I & = A o I. * * +1 ° 3 I **·( ·05 ) Equation (1.05) applies to the volume element of area A 0 and length δχ shown in Fig.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
LEARNING OBJECTIVES After reading this module, you should be able to... 7.1 Apply the impulse–Momentum theorem. 7.2 Apply the law of conservation of linear Momentum. 7.3 Analyze one-dimensional collisions. 7.4 Analyze two-dimensional collisions. 7.5 Determine the location and the velocity of the center of mass. Stephan Goerlich/Age Fotostock America CHAPTER 7 Impulse and Momentum In the sport of jousting in the Middle Ages, two knights in armor rode their horses toward each other and, using their lances, attempted to knock each other to the ground. Sometimes, however, the collision between a lance and an opponent’s shield caused the lance to shatter and no one was unseated. In physics such a collision is classified as being inelastic. Inelastic collisions are one of the two basic types that this chapter introduces. 7.1 The Impulse–Momentum Theorem There are many situations in which the force acting on an object is not constant, but varies with time. For instance, Figure 7.1a shows a baseball being hit, and part b of the figure illustrates approximately how the force applied to the ball by the bat changes during the time of contact. The magnitude of the force is zero at the instant t 0 just before the bat touches the ball. During contact, the force rises to a maximum and then returns to zero at the time t f when the ball leaves the bat. The time interval ∆t = t f ‒ t 0 during which the bat and ball are in contact is quite short, being only a few thousandths of a second, although the maximum force can be very large, often exceeding thousands of newtons. For comparison, the graph also shows the magnitude F of the average force exerted on the ball during the time of contact. Figure 7.2 depicts other situations in which a time-varying force is applied to a ball. To describe how a time-varying force affects the motion of an object, we will introduce two new ideas: the impulse of a force and the linear Momentum of an object.
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
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. When the force varies with time, we may divide At into many short intervals in each of which the force may be considered constant, and then sum over all contributions as an integral. We do this for each component At J k = JF k (t)dt (k = x 9 y,z) The areas under the two graphs are equal and represent the impulse in the jc-direction. The grey area in the adjoining graph represents the impulse 7 X generated by the force during At. The dotted rectangle, having the same area, represents the identical impulse that would be generated by an average (constant) force acting for the same time. The above relations follow from the second law of Newton in its original form /Ap dp F = lim-*->° At dt (9.3) 9.2 The law of conservation of Momentum It follows from (9.1)-(9.2) that in the absence of external forces the impulse vanishes, Ap = 0, so that Momentum is conserved Pi = Pi = const (9.4) This is a vector equation, which means that all three components of the Momentum are conserved. We may write (9.2) in component form Jx=*Px J y= A Py J z =A Pz (9.5) We conclude that if no force acts on the body in a certain direction, say x, then the Momentum component in that direction is conserved, p x = const. It is interesting to compare the pairs {work-energy} and {impulse-Momentum}, as shown in the table: work and energy impulse-Momentum work W = F-Ax impulse / = FAt W = E f -E x ,- AE work-energy impulse-Momentum J =p i -p i = Ap when F = 0 E = const when F = 0 p = const 282 Classical and Relativistic Mechanics 9.3 Momentum conservation in many-body systems So far we have dealt with Momentum conservation for a single particle.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Depend- ing on the forces acting on a system, linear Momentum might be conserved in one or two directions but not in all directions. However, If no net external force acts on a system of particles, the total linear Momentum P → of the system cannot change. 241 In a homework problem, how can you know if linear Momentum can be conserved along, say, an x axis? Check the force components along that axis. If the net of any such components is zero, then the conservation applies. As an example, suppose that you toss a grapefruit across a room. During its flight, the only external force acting on the grapefruit (which we take as the system) is the gravitational force F → g , which is directed vertically downward. Thus, the vertical component of the linear Momentum of the grapefruit changes, but since no hori- zontal external force acts on the grapefruit, the horizontal component of the lin- ear Momentum cannot change. Note that we focus on the external forces acting on a closed system. Although internal forces can change the linear Momentum of portions of the system, they cannot change the total linear Momentum of the entire system. For example, there are plenty of forces acting between the organs of your body, but they do not propel you across the room (thankfully). The sample problems in this module involve explosions that are either one- dimensional (meaning that the motions before and after the explosion are along a single axis) or two-dimensional (meaning that they are in a plane containing two axes). In the following modules we consider collisions. Checkpoint 9.5.1 An initially stationary device lying on a frictionless floor explodes into two pieces, which then slide across the floor, one of them in the positive x direction.
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