Physics
Conservation of Momentum
Conservation of Momentum states that the total momentum of a closed system remains constant if no external forces act on it. This principle is based on the law of inertia and is a fundamental concept in physics. It is often used to analyze collisions and interactions between objects, providing valuable insights into the behavior of physical systems.
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11 Key excerpts on "Conservation of Momentum"
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The principle of conservation of linear momentum states that the total lin- ear momentum of an isolated system remains constant. For a two-body sys- tem, the conservation of linear momentum can be written as in Equation 7.7b, where m 1 and m 2 are the masses, v → f1 and v → f2 are the final velocities, and v → 01 and v → 02 are the initial velocities of the objects. m 1 v f1 → + m 2 v f2 → = m 1 v 01 → + m 2 v 02 → (7.7b) 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 Initial total linear momentum ⏟⎵⎵⎵⎵ ⏟⎵⎵⎵⎵ ⏟ ⏟⎵⎵⎵⎵ ⏟⎵⎵⎵⎵ ⏟ Final total linear momentum 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. m 1 υ f1x + m 2 υ f 2x = m 1 υ 01x + m 2 υ 02x (7.9a) m 1 υ f1y + m 2 υ f 2y = m 1 υ 01y + m 2 υ 02y (7.9b) 7.5 Center of Mass The location of the center of mass of two particles lying on the x axis is given by Equation 7.10, where m 1 and m 2 are the masses of the particles and x 1 and x 2 are their positions relative to the coordinate origin. If the particles move with velocities υ 1 and υ 2 , the velo- city υ cm of the center of mass is given by Equation 7.11. If the total linear momentum of a system of particles remains constant during an interac- tion such as a collision, the velocity of the center of mass also remains constant.
eBook - PDFQuestioning the Universe
Concepts in Physics
- Ahren Sadoff(Author)
- 2008(Publication Date)
- Chapman and Hall/CRC(Publisher)
61 8 Conservation Laws Conservation laws are extremely important in physics. They are relevant to a wide variety of physical phenomena. They are very powerful and hence very useful. And interestingly, they follow directly from the symmetry principles we discussed in the last chapter. We can see this by just considering what it means if something is conserved. If something is conserved, it does not increase or decrease; it does not change; it does not vary. In other words, it remains invariant . So, we can give the following definition to conservation: Something is conserved if we can do something to it such that afterwards it remains invariant. Of course, this sounds almost identical to our definition of symmetry. It is no acci-dent. Emmy Noether, a woman mathematician, proved that symmetry principles lead directly to conservation laws. For instance, space translation symmetry leads to the Conservation of Momentum, and time translation symmetry leads to the conserva-tion of energy. We will discuss both of these conservation laws in this chapter. Using these as examples, we will be able to see why conservation laws are so powerful. 8.1 Conservation of Momentum The word momentum is often used in everyday speech to imply that something is in motion. But in physics, momentum has a very well-defined meaning. The momen-tum of a single object with a mass, m, is given by the equation p = m v (8.1) Momentum is a vector quantity since it is directly proportional to the velocity vec-tor. The units of momentum are [Kg][M]/[s]. Many quantities in physics have units named after some famous physicist. For instance, the unit of force is the Newton for Isaac Newton. But no one has been honored for momentum. If there is a group of objects and we are interested in the momentum of the entire system, then we just add up the momentum of the individual objects to get the momentum of the system.
- 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)
(9.33) If no net external force acts on an isolated and closed system of particles, the total linear momentum P of the system cannot change. 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 . (9.34) Pdf_Folio:154 154 Fundamentals of physics 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 problems of this module, momentum is conserved but mechanical energy is definitely not. Our last two equations are vector equations and thus each is equivalent to three equations corresponding to the conservation of linear momentum in the three mutually perpendicular directions in an xyz coordinate system. Depending on the forces acting on a system, linear momentum might be conserved in one or two directions but not in all directions. However, note the following. 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. 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 horizontal external force 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.
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.
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- William G. Gray, Genetha A. Gray(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
12 Statement of Conservation of Momentum 12.1 Introduction In the previous two chapters, we referred to the fact that a solution for the distribution of mass in a system of interest requires knowledge of the flow field that drives any redistribu-tion. Most are aware that the sum of forces acting on a system is equal to the mass times the acceleration. Furthermore, the acceleration is the rate of change of velocity with time. Thus, we want to formulate an equation that provides the rate of change of velocity in relation to the sum of forces acting on the system. The equation we seek provides a state-ment of the Conservation of Momentum of the material in the study region. This equation is sometimes referred to as a fundamental principle; but, in fact, it can be formulated as a statement based on observed physical behavior of a system. Then conversion of this state-ment to a mathematical form gives the desired result. Here, we develop the momentum conservation equation for a system and indicate how it can be used in conjunction with the mass conservation equation to model a system. 12.2 Elements of the momentum conservation equation We can construct the mathematical relation describing momentum conservation based on formulating terms for the word equation ⎡ ⎣ Rate of accumulation of momentum in a volume ⎤ ⎦ + ⎡ ⎣ Net rate of outflow of momentum by convection ⎤ ⎦ − ⎡ ⎣ Rate of supply of momentum by mechanisms other than convection ⎤ ⎦ = Rate of generation of momentum (12.1) This equation has dimensions of momentum of material per time ( ML / t 2 ). The amount of momentum that a system has is an extensive variable. Momentum is a vector quantity. A conservation equation may thus be applied for each vector component of momentum. However, it is convenient, since contributions to momentum behave the same regardless of direction, to develop the momentum equation in vector form. We will consider the momentum of the material in an arbitrary volume. 275
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
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). Conservation of Linear Momentum Suppose that the net external force F → net (and thus the net impulse J → ) acting on a system of particles is zero (the system is isolated) and that no particles leave or enter the system (the system is closed). Putting F → net in Eq. 9.3.6 then yields dP → /dt = 0, which means that P → = constant (closed, isolated system ). (9.5.1) In words, 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. 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). (9.5.2) 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 9.5.1 and 9.5.2 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, If no net external force acts on a system of particles, the total linear momentum P → of the system cannot change.
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)
- 2021(Publication Date)
- Wiley(Publisher)
→ J = ¯ → F Δ t (7.1) → p = m → v (7.2) (Σ ¯ → F ) Δ t = m → v f − m → v 0 (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 f 1 and → v f 2 are the final velocities, and → v 01 and → v 02 are the initial velocities of the objects. m 1 → v f 1 + m 2 → v f 2 = m 1 → v 01 + m 2 → v 02 (7.7b) 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 ine- lastic 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. m 1 υ f 1x + m 2 υ f 2x = m 1 υ 01x + m 2 υ 02x (7.9a) m 1 υ f 1y + m 2 υ f 2y = m 1 υ 01y + m 2 υ 02y (7.9b) 7.5 Center of Mass The location of the center of mass of two par- ticles lying on the x axis is given by Equation 7.10, where m 1 and m 2 are the masses of the particles and x 1 and x 2 are their positions relative to the coordinate origin.
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 - PDFWhy Toast Lands Jelly-Side Down
Zen and the Art of Physics Demonstrations
- Robert Ehrlich(Author)
- 2020(Publication Date)
- Princeton University Press(Publisher)
It is clear that the low damping of this cycle depends on the high co-efficient of restitution of the balls, meaning that the collisions are nearly elastic—an issue considered in the next demonstration. Here, we shall consider how the toy may be used to illustrate momentum conser-vation in completely inelastic collisions. In order to make quantitative observations, place a transparency made from a piece of graph paper on the OHP, and place the toy on top, with one axis of the graph paper along the direction the balls swing. Pull all but two of the balls aside, and rest them on an improvised shelf made from books, so that you can examine the collisions between a single pair of balls. Put a tiny amount of clay on one of the two balls, so that they will stick together on impact. Now, pull one ball aside and release it from a specific distance jtj 92 _ ,. . from the other stationary ball as seen on the trans-Conservation of / .... Momentum and parency graph paper. After the balls collide and stick -together, observe how far x 2 the two balls move after the collision before they swing back. If you cannot get a good reading of the maximum displacement on the first swing, see what you find on the second or third swing, as the two balls swing together. Be sure that you measure x l and x 2 as the distance a particular point on the balls moves. Since the collision between the two equal mass balls is completely inelastic, momentum conserva-tion requires that immediately after collision, their common velocity be half the initial velocity of the first ball just before collision. If we apply energy conservation during the two balls' swing following the collision, mgy = mv 2 , we find that the maxi-mum height they can rise to is proportional to the square of their velocity—so the two balls together should rise only a quarter as high as the first ball was initially.
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