Physics
Conservation of Angular Momentum
Conservation of Angular Momentum states that the total angular momentum of a system remains constant if no external torque acts on it. This means that as long as no external forces are present, the total angular momentum of a system will remain unchanged. This principle is a fundamental concept in physics and is used to analyze the motion of rotating objects.
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10 Key excerpts on "Conservation of Angular Momentum"
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
This is the second of the major conservation laws we have discussed. Along with conservation of linear momentum, Conservation of Angular Momentum is a general result that is valid for a wide range of systems. It holds true in both the relativistic limit and in the quantum limit. No excep- tions have ever been found. Like conservation of linear momentum in a system on which the net external force is zero, Conservation of Angular Momentum applies to the total angular momentum of a sys- tem of particles on which the net external torque is zero. The angular momentum of individual particles in a system may change due to internal torques ( just as the linear mo- mentum of each particle in a collision may change due to internal forces), but the total remains constant. Angular momentum is (like linear momentum) a vector quantity so that Eqs. 10-15 is equivalent to three one- dimensional equations, one for each coordinate direction through the reference point. Conservation of angular mo- mentum therefore supplies us with three conditions on the motion of a system to which it applies. Any component of the angular momentum will be constant if the correspond- ing component of the torque is zero; it might be the case that only one of the three components of torque is zero, which means that only one component of the angular mo- mentum will be constant, the other components changing as determined by the corresponding torque components. For a system consisting of a rigid body rotating with an- gular speed about an axis (the z axis, say) that is fixed in an inertial reference frame, we have (10-16) where L z is the component of the angular momentum along the rotation axis and I is the rotational inertia for this same axis. If no net external torque acts, then L z must remain constant.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
This is the rotational counterpart to linear momentum being conserved when the external force on a system is zero. • For a rigid body that changes its angular momentum in the absence of a net external torque, Conservation of Angular Momentum gives I f ω f = I i ω i . This equation says that the angular velocity is inversely proportional to the moment of inertia. Thus, if the moment of inertia decreases, the angular velocity must increase to conserve angular momentum. • Systems containing both point particles and rigid bodies can be analyzed using Conservation of Angular Momentum. The angular momentum of all bodies in the system must be taken about a common axis. 11.4 Precession of a Gyroscope • When a gyroscope is set on a pivot near the surface of Earth, it precesses around a vertical axis, since the torque is always horizontal and perpendicular to L → . If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque, and it rotates about a horizontal axis, falling over just as we would expect. • The precessional angular velocity is given by ω P = rMg Iω , where r is the distance from the pivot to the center of mass of the gyroscope, I is the moment of inertia of the gyroscope’s spinning disk, M is its mass, and ω is the angular frequency of the gyroscope disk. CONCEPTUAL QUESTIONS 11.1 Rolling Motion 1. Can a round object released from rest at the top of a frictionless incline undergo rolling motion? 2. A cylindrical can of radius R is rolling across a horizontal surface without slipping. (a) After one complete revolution of the can, what is the distance that its center of mass has moved? (b) Would this distance be greater or smaller if slipping occurred? 3. A wheel is released from the top on an incline. Is the wheel most likely to slip if the incline is steep or gently sloped? 4. Which rolls down an inclined plane faster, a hollow cylinder or a solid sphere? Both have the same mass and radius.
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
296 Chapter 11 Angular Momentum then D L S tot 5 0 (11.19) Equation 11.19 can be written as L S tot 5 constant or L S i 5 L S f (11.20) For an isolated system consisting of a small number of particles, we write this conservation law as L S tot 5 o L S n 5 constant, where the index n denotes the nth par- ticle in the system. If the system consists of a large number of particles, so that it is difficult to evaluate the individual L n , then we can express the magnitude of the angular momentum of the system with Equation 11.16, L = Iv. If an isolated rotating system is deformable so that its mass undergoes redistri- bution in some way, the system’s moment of inertia changes. Combining Equations 11.16 and 11.20, we see that Conservation of Angular Momentum requires that the product of I and v must remain constant. Therefore, a change in I for an isolated system requires a change in v. In this case, we can express the principle of conser- vation of angular momentum as I i v i 5 I f v f 5 constant (11.21) This expression is valid both for rotation about a fixed axis and for rotation about an axis through the center of mass of a moving system as long as that axis remains fixed in direction. We require only that the net external torque be zero. Many examples demonstrate Conservation of Angular Momentum for a deform- able system. You may have observed a figure skater spinning in the finale of a program (Fig. 11.9). The angular speed of the skater is large when his hands and feet are close to the trunk of his body. (Notice the skater’s hair!) Ignoring friction between skater and ice, there are no external torques on the skater. The moment of inertia of his body increases as his hands and feet are moved away from his body, and therefore from the rotation axis, at the finish of the spin. According to the isolated system model for angular momentum, his angular speed must decrease, and he can perform his finishing flourish after coming to rest.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
Conservation of Angular Momentum We can now understand why Earth keeps on spinning. As we saw in the previous example, ΔL = (net τ)Δt . This equation means that, to change angular momentum, a torque must act over some period of time. Because Earth has a large angular momentum, a large torque acting over a long time is needed to change its rate of spin. So what external torques are there? Tidal friction exerts torque that is slowing Earth’s rotation, but tens of millions of years must pass before the change is very significant. Recent research indicates the length of the day was 18 h some 900 million years ago. Only the tides exert significant retarding torques on Earth, and so it will continue to spin, although ever more slowly, for many billions of years. What we have here is, in fact, another conservation law. If the net torque is zero, then angular momentum is constant or conserved. We can see this rigorously by considering net τ = ΔL Δt for the situation in which the net torque is zero. In that case, (10.108) netτ = 0 implying that (10.109) ΔL Δt = 0. If the change in angular momentum ΔL is zero, then the angular momentum is constant; thus, (10.110) L = constant (net τ = 0) or (10.111) L = L′(netτ = 0). Chapter 10 | Rotational Motion and Angular Momentum 369 These expressions are the law of Conservation of Angular Momentum. Conservation laws are as scarce as they are important. An example of Conservation of Angular Momentum is seen in Figure 10.23, in which an ice skater is executing a spin. The net torque on her is very close to zero, because there is relatively little friction between her skates and the ice and because the friction is exerted very close to the pivot point. (Both F and r are small, and so τ is negligibly small.) Consequently, she can spin for quite some time. She can do something else, too. She can increase her rate of spin by pulling her arms and legs in.
- Joseph C. Amato, Enrique J. Galvez(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
But spin is only one example of angular momentum. We will now broaden the definition of I L , and show that all bodies, non-rotating ones, even bodies moving along a straight line, may possess angular momentum. Moreover, we will see that the total angular momentum of an isolated system does not change, giving us a new conservation law to add to those of momentum and energy. Conservation of Angular Momentum is the crux of Kepler’s second law (the equal area law), and we will use it to find the shape of planetary orbits and to predict the trajectories of asteroids, comets, and spacecraft. Like momentum and energy, angu -lar momentum is a powerful tool for studying motion, and it will be employed in Chapters 13 and 14 to study the behavior of rigid bodies in simple lab-scale situations as well as in phenomena that challenge our understanding of the cosmos. 12.2 BROADER DEFINITION OF ANGULAR MOMENTUM Imagine that you are at point A , observing the motion of a small body. Relative to A , the body is located at position I r and moving with momentum I p (see Figure 12.1a ). The angular momentum I L of the body relative to A is formally defined by the cross product I I I L r p ≡ × . (12.1) According to the properties of the cross product, I L is perpendicular to the plane containing I r and I p , and points in the direction determined by the right hand rule. (For a review of cross products, see Section 8.3.) The magnitude of I L is given by L = rp sin θ , where θ is the angle between I r and I p ( θ < 180 ° ). Alternatively, we may express the magnitude as L = rp ⊥ , where p ⊥ = p sin θ is the component of I p perpendicular to I r , or equivalently as L = r ⊥ p (see Figure 12.1b ). EXERCISE 12.1 Show that the angular momentum relative to point A of a body moving with no forces acting on it is a constant, that is, I I I I L r L r ( ( ) ). = ′ A r v r ΄
eBook - PDF- Patrick Hamill(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
In the next section we shall explore this relationship using Lagrange’s equations. 202 8 CONSERVATION LAWS AND SYMMETRIES 8.4 Symmetries and Conserved Physical Quantities Consider a physical system. If it is symmetrical with respect to a rotation, then it is unchanged by the rotation. The system is the same after it was rotated as it was before. A physical system can be described in many ways; for example, you could generate a table giving all of the physical properties of the system (velocity, position, etc.) as functions of time. However, a much better and simpler way to describe a physical system is to give its Lagrangian. If you can write down the Lagrangian, you know all the essential mechanical properties of a physical system. 4 You can use Lagrange’s equations to determine the equations of motion and you can solve them to evaluate how the system will evolve in time. Suppose the Lagrangian of some system does not contain a particular coordinate. Specifically, suppose that the angle φ does not appear in the Lagrangian. Remember that we call such a coordinate “ignorable” and as we showed in Chapter 4, the generalized momentum conjugate to an ignorable coordinate is constant. The generalized momentum conjugate to the angle φ is the angular momentum, so for this system, the angular momentum is constant. That is, if a system is symmetrical with respect to a rotation, then the Lagrangian of the system does not depend in any way on φ and φ will not appear in the Lagrangian. Consequently, the angular momentum associated with rotations through φ will be conserved. Similarly, if a system is symmetrical with respect to a translation, then its linear momentum will be conserved. Let us consider these concepts in more detail. 8.4.1 Conservation of Linear Momentum Imagine a particle in a homogeneous region of space. By spatial homogeneity we mean that the space is the same at all points. A particle in this space will exhibit translational symmetry.
eBook - PDFDynamics of Particles and the Electromagnetic Field
(With CD-ROM)
- Slobodan Danko Bosanac(Author)
- 2005(Publication Date)
- WSPC(Publisher)
Chapter 4 Angular Momenturn 4.1 General Remarks Angular momentum, which is defined in (1.18), is a very important quantity in dynamics of a particle, despite the fact that it is invariant of motion only if the forces are centrally symmetric. By using the probability current (1.13) angular momentum is defined as d3r d3p r ' x $p(r',$, t ) = m d3r r' x j(7, t) J' which is often more convenient for discussing its properties. In particu- lar three typical contributions to the angular momentum are distinguished, and two of them are deduced from the probability current. It should be emphasized though that these contributions are often not easily disentan- gled in real situations. When one talks about various contributions it is meant that there are circumstances when only one is dominant. One typical contribution is when the probability current parametrizes (or approximately parametrizes) as J(r',t) = (i7) P(F,t), where (i?) is the average velocity of the particle and P(r',t) is the probability density for the coordinates, which is defined in (1.13). For this parametrization the average of the probability current is not zero -. (4 = / d 3 r .f(r',t) # 0 and in this case the angular momentum is estimated as 51 52 Dynamics of Particles and the Electromagnetic Field where (F) is the average position of the particle (3 = / d 3 r FP(?,t). This angular momentum is associated with the motion of the probability density P(F, t ) as the whole and will be called the orbital angular momen- tum. Very often one encounters situation when the average probability cur- rent is zero for example when the particle is free and not moving, meaning that its average velocity is zero (see Section 2.2). Angular momentum of the particle is in this case zero. However, the condition (4.1) is not sufficient for the angular momentum to be zero, and one example that shows this is treated in Section 1.2.1.
eBook - PDF- Yoni Kahn, Adam Anderson(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
1.6.1 Effective Potential The fact that our potential has the form U(r) immediately gives us conservation laws which we can put right to use. First, let’s write down the Lagrangian for a particle of mass m mov- ing in the potential U: after writing x, y, and z in spherical coordinates, we find L = 1 2 m˙ r 2 + 1 2 mr 2 ˙ θ 2 + 1 2 mr 2 sin 2 θ ˙ φ 2 − U(r). 20 Classical Mechanics (The polar angle θ shows up in the kinetic energy roughly for the same reason that it shows up in the spherical volume element r 2 sin θ .) Reverting to Newtonian reasoning for a bit, Conservation of Angular Momentum implies the conservation of a whole vector L (whose magnitude is l), and the fact that the direction of this vector is constant means that the particle is confined to a plane. By spherical symmetry, we can choose this plane to be at θ = π/2; the second term (involving ˙ θ ) vanishes since θ is constant, and sin(π/2) = 1 means the third term simplifies as well. We are left with the restricted form: L = 1 2 m˙ r 2 + 1 2 mr 2 ˙ φ 2 − U(r), (1.34) which we will use from now on. Now, since U(r) is indepen- dent of the azimuthal angle φ, so is the Lagrangian, and that gives us conservation of the conjugate momentum to φ, which we identify as the ordinary angular momentum l: l = mr 2 ˙ φ. (1.35) The radial behavior of the orbit is of course described by the Euler–Lagrange equation for the radial coordinate: d dt (m˙ r) = mr ˙ φ 2 − U (r). Substituting for ˙ φ in terms of l using (1.35), we get m¨ r = l 2 mr 3 − U (r). First of all, we have reduced a complex system of partial dif- ferential equations in three dimensions to a single ordinary differential equation, which we may have some hope of under- standing. And secondly, this looks suspiciously like Newton’s second law of motion. We can improve the resemblance by “factoring” the derivative on the right-hand side to find m¨ r = − d dr l 2 2mr 2 + U .
eBook - PDFPrinciples of Continuum Mechanics
Conservation and Balance Laws with Applications
- J. N. Reddy(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
5 CONSERVATION OF MASS AND BALANCE OF MOMENTA AND ENERGY It is the mark of an educated mind to be able to entertain a thought without accepting it. Aristotle An error does not become truth by reason of multiplied propagation, nor does truth become error because nobody sees it. Mahatma Gandhi 5.1 Introduction 5.1.1 Preliminary Comments Most phenomena in nature, whether mechanical, biological, chemical, geologi-cal, or geophysical can be described, based on the goal of the study, in terms of mathematical relations among various quantities of interest. Such relationships are called mathematical models and are based on fundamental scientific laws of physics that are extracted from centuries of research on the behavior of mechan-ical systems subjected to the action of external stimuli. What is most exciting is that the laws of physics also govern biological systems because of mass and energy transports. However, biological systems may require additional laws, yet to be discovered, from biology and chemistry to complete their description. This chapter is devoted to the study of fundamental laws of physics as applied to mechanical systems. The laws of physics are expressed in analytical form with the aid of the concepts and quantities introduced in previous chapters. The laws or principles of physics that we study here are: (1) the principle of conservation of mass, (2) the principle of balance of linear momentum, (3) the principle of bal-ance of angular momentum, and (4) the principle of balance of energy. These laws allow us to write mathematical relationships – algebraic, differential, or integral type – of physical quantities of interest such as displacement, velocity, temper-ature, stress, and strain in mechanical systems. The solution of these equations represents the response of the system, which aids the design and manufacturing of the system.
eBook - PDF- R. Douglas Gregory(Author)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
Part Two MULTI-PARTICLE SYSTEMS AND CONSERVATION PRINCIPLES CHAPTERS IN PART TWO Chapter 9 The energy principle Chapter 10 The linear momentum principle Chapter 11 The angular momentum principle Chapter Nine The energy principle and energy conservation KEY FEATURES The key features of this chapter are the energy principle for a multi-particle system, the poten- tial energies arising from external and internal forces, and energy conservation. This is the first of three chapters in which we study the mechanics of multi-particle systems. This is an important development which greatly increases the range of problems that we can solve. In particular, multi-particle mechanics is needed to solve problems involving the rotation of rigid bodies. The chapter begins by obtaining the energy principle for a multi-particle system. This is the first of the three great principles of multi-particle mechanics ∗ that apply to every mechanical system without restriction. We then show that, under appropriate conditions, the total energy of the system is conserved. We apply this energy conservation principle to a wide variety of systems. When the system has just one degree of freedom, the energy conservation equation is sufficient to determine the whole motion. 9.1 CONFIGURATIONS AND DEGREES OF FREEDOM A multi-particle system S may consist of any number of particles P 1 , P 2 , . . ., P N , with masses m 1 , m 2 . . ., m N respectively. † A possible ‘position’ of the system is called a configuration. More precisely, if the particles P 1 , P 2 , . . ., P N of a system have position vectors r 1 , r 2 , . . ., r N , then any geometrically possible set of values for the posi- tion vectors { r i } is a configuration of the system. If the system is unconstrained, then each particle can take up any position in space (independently of the others) and all choices of the { r i } are possible.
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