Physics
Linear Momentum
Linear momentum is a fundamental concept in physics that describes the motion of an object. It is the product of an object's mass and its velocity, and it is a vector quantity, meaning it has both magnitude and direction. The principle of conservation of linear momentum states that the total linear momentum of a closed system remains constant if no external forces act on it.
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10 Key excerpts on "Linear Momentum"
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
The Linear Momentum p of an object is equal to the product of its mass m and its velocity v: = m p v (7.1.2) The SI unit of momentum is / kg m s ⋅ . Like impulse, Linear Momentum is a vector quantity, and an object’s momentum has the same direction as its velocity. 7.2 The Impulse–Momentum Theorem The change in an object’s momentum Dp is equal to the product of the average net force acting on the object F avg ∑ and the time interval Δt over which that force acts: ∑ ( ) D = D t F p avg (7.2.2) Equation 7.2.2 is called the impulse–momentum theorem, which can be stated in words as follows: The impulse imparted by the average net force acting on an object is equal to the change in the object’s momentum. The impulse–momentum theorem is simply a restate- ment of Newton’s second law. 7.3 Conservation of Momentum In physics, a system is collection of interacting entities (particles, objects, etc.) that is cho- sen for analysis. If the net external force on a system of objects is zero, then the total Linear Momentum of that system does not change: D = P 0 or = P P i f (7.3.3) Equation 7.3.3 is the mathematical statement of conservation of Linear Momentum, where capital “P” is used to denote the total momentum of the system. 7.4 One-Dimensional Collisions A collision is an interaction between two or more objects that results in a change in the momentum of the objects (but not necessarily the total momentum). Collisions can be classified as either elastic or inelastic. In physics, an elastic collision is one in which the total kinetic energy of the system is the same before and after the collision. Ine- lastic collisions, on the other hand, are those in which the kinetic energy of the system changes. In a completely inelastic collision, the objects do not “rebound” from one another—they stick together and essentially become a single physical object. If the 202 | Chapter 7 colliding objects constitute an isolated system, then Linear Momentum is conserved.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
The first definition concerns a familiar word—momentum—that has several meanings in everyday language but only a single precise meaning in physics and engineering. The Linear Momentum of a particle is a vector quantity p → that is defined as p → = m v → (Linear Momentum of a particle), (9.3.1) in which m is the mass of the particle and v → is its velocity. (The adjective linear is often dropped, but it serves to distinguish p → from angular momentum, which is introduced in Chapter 11 and which is associated with rotation.) Since m is always a positive scalar quantity, Eq. 9.3.1 tells us that p → and v → have the same direction. From Eq. 9.3.1, the SI unit for momentum is the kilogram-meter per second (kg · m/s). Force and Momentum. Newton expressed his second law of motion in terms of momentum: In equation form this becomes F → net = d p → ____ dt . (9.3.2) In words, Eq. 9.3.2 says that the net external force F → net on a particle changes the particle’s Linear Momentum p → . Conversely, the Linear Momentum can be changed only by a net external force. If there is no net external force, p → cannot change. As we shall see in Module 9.5, this last fact can be an extremely powerful tool in solving problems. Manipulating Eq. 9.3.2 by substituting for p → from Eq. 9.3.1 gives, for constant mass m, F → net = d p → ___ dt = d _ dt (m v → ) = m d v → ___ dt = m a → . Thus, the relations F → net = d p → / dt and F → net = m a → are equivalent expressions of Newton’s second law of motion for a particle. The time rate of change of the momentum of a particle is equal to the net force acting on the particle and is in the direction of that force. as the product of the particle’s mass and velocity. 9.3.3 Calculate the change in momentum (magnitude and direction) when a particle changes its speed and direction of travel. 9.3.4 Apply the relationship between a particle’s momentum and the (net) force acting on the particle.
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 - 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- 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- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
However, In a homework problem, how can you know if Linear Momentum can be con- served 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 exter- nal 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 horizontal external force If no net external force acts on a system of particles, the total Linear Momentum P → of the system cannot change. If the component of the net external force on a closed system is zero along an axis, then the component of the Linear Momentum of the system along that axis cannot change. Key Ideas ● If a system is closed and isolated so that no net external force acts on it, then the Linear Momentum P → must be constant even if there are internal changes: P → = constant (closed, isolated system). ● This conservation of Linear Momentum can also be written in terms of the system’s initial momentum and its momentum at some later instant: P → i = P → f (closed, isolated system), acts on the grapefruit, the horizontal component of the Linear 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).
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.
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 - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Putting F → net = 0 in Eq. 9-27 then yields d P → /dt = 0, which means that P → = constant (closed, isolated system). (9-42) In words, 9-5 CONSERVATION OF Linear Momentum Learning Objectives After reading this module, you should be able to . . . 9.26 For an isolated system of particles, apply the conservation of linear momenta to relate the initial momenta of the particles to their momenta at a later instant. 9.27 Identify that the conservation of Linear Momentum can be done along an individual axis by using com- ponents along that axis, provided that there is no net external force component along that axis. This result is called the law of conservation of Linear Momentum and is an extremely powerful tool in solving problems. In the homework we usually write the law as P → i = P → f (closed, isolated system). (9-43) In words, this equation says that, for a closed, isolated system, ( total Linear Momentum at some initial time t i ) = ( total Linear Momentum at some later time t f ) . Caution: Momentum should not be confused with energy. In the sample prob- lems of this module, momentum is conserved but energy is definitely not. Equations 9-42 and 9-43 are vector equations and, as such, each is equivalent to three equations corresponding to the conservation of Linear Momentum in three mutually perpendicular directions as in, say, an xyz coordinate system. 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, In a homework problem, how can you know if Linear Momentum can be con- served 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 exter- nal force acting on the grapefruit (which we take as the system) is the gravitational force F → g , which is directed vertically downward.
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