Direct Impact and Newton's Law of Restitution

4 Key excerpts on "Direct Impact and Newton's Law of Restitution"

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  eBook - PDF

  Engineering Mechanics

  Dynamics

  • L. G. Kraige, J. N. Bolton(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  This treatment included the direct use of Newton’s second law, the equations of work and energy, and the equations of impulse and momentum. We paid special attention to the kind of problem for which each of the approaches was most appropriate. Several topics of specialized interest in particle kinetics will be briefly treated in Section D: 1. Impact 2. Central-force motion 3. Relative motion These topics involve further extension and application of the fundamental principles of dynamics, and their study will help to broaden your background in mechanics. 3/12 Impact The principles of impulse and momentum have important use in describing the behavior of colliding bodies. Impact refers to the collision between two bodies and is characterized by the generation of relatively large contact forces which act over a very short interval of time. It is important to realize that an impact is a very complex event involving material deformation and recovery and the generation of heat and sound. Small changes in the impact conditions may cause large changes in the impact process and thus in the conditions immediately following the im- pact. Therefore, we must be careful not to rely heavily on the results of impact calculations. Direct Central Impact As an introduction to impact, we consider the collinear motion of two spheres of masses m 1 and m 2 , Fig. 3∕ 17a, traveling with velocities v 1 and v 2 . If v 1 is greater than v 2 , collision occurs with the contact forces directed along the line of centers. This condition is called direct cen- tral impact. Following initial contact, a short period of increasing deforma- tion takes place until the contact area between the spheres ceases to increase. At this instant, both spheres, Fig. 3∕ 17b, are moving with the same velocity v 0 . During the remainder of contact, a period of restoration occurs during which the contact area decreases to zero.

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  eBook - PDF

  Theoretical Mechanics for Sixth Forms

  In Two Volumes

  • C. Plumpton, W. A. Tomkys(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER XI THE IMPACT OF ELASTIC BODIES 11.1. Direct Collision of Elastic Spheres When two spheres moving along the line joining their centres collide, the collision is described as a direct collision. As discussed in Chapter X, § 10.2, the effect of the collision is a temporary distortion of shape of the spheres and a resulting loss of kinetic energy. The magnitude of the loss of energy is related to the velocities of the particles by the Law of Restitution, an experimental law discovered by Newton, and stated here in the form: The relative velocity after impact has a magnitude which is a fixed ratio for the spheres concerned of the relative velocity before impact and it is in the opposite direction to the relative velocity before impact. This fixed ratio, which is less than or equal to 1, is called the coefficient of restitution for the two bodies concerned, and is denoted by e. In the ideal case, unattainable in practice, in which e = 1, the two bodies are said to be perfectly elastic. In the case in which e = 0, when the bodies do not separate after collision, the collision is said to be an inelastic one. The motion of the spheres is subject also to the principle of conserva-tion of momentum. In algebraic terms, if the masses of the spheres are m and m 2 , their velocities before impact Wi and u 2 along the positive direction of the line of centres, and their velocities after impact i and v 2 each in this direction, then mU--m 2 u 2 == wi^i-f m 2 v 2 , (Momentum equation), ( . —e(u 1 —u 2 ) = Vi—v 2 , (Restitution equation). Example 1. Two spheres of masses 5 kg and 2 kg move towards one another in opposite directions along their line of centres with speeds 2 m/s and 5 m/s respectively. Given that the coefficient of restitution between the spheres is |, calculate their veloci-ties after collision. 266 THE IMPACT OF ELASTIC BODIES 267 Figure 11.1 shows the velocities of the spheres before and after the collision.

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  eBook - PDF

  Impact Mechanics

  • W. J. Stronge(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  The contact force and changes in relative velocity at the contact point C will be resolved into components normal and tangential to the common tangent plane. 2.1 Equation of Relative Motion for Direct Impact A direct impact has the centers of mass of both colliding bodies on the line of the common normal passing through the contact point C. The contact force at C acts in the common normal direction. Hence for both bodies, all changes in velocity during impact are in the common normal direction and the analysis is one-dimensional. Consider two colliding bodies B and B 0 that have masses M and M 0 and time-dependent velocities V(t) and V 0 t ð Þ in the direction parallel to n. In a direct collision these bodies are not rotating when they collide so that the velocity is uniform in each body (i.e., the same at every point). 21 During contact there are equal but opposite compressive reaction forces which develop at contact points C and C 0 ; these forces oppose interference or overlap the contact surfaces. In the case of direct impact between collinear bodies any change in the relative velocity between the contact points C and C 0 remains parallel to the common normal direction throughout the contact period. A reaction force develops at the contact point as a conse- quence of compression of the local contact region; this force opposes relative motion during contact. In a direct collision the reaction force acts in the normal direction; i.e., coincident with the velocities of the centers of mass as illustrated in Figure 2.1. If the colliding bodies are hard, the contact force is very large in comparison with any body force; consequently, in rigid body impact theory the body or applied contact forces of finite magnitude are negligibly small in comparison with the reaction at the contact point C. Finite body forces are ignorable because they do no work during the vanishingly small displacements that develop during an instantaneous collision.

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  Meriam's Engineering Mechanics

  Dynamics

  • L. G. Kraige, J. N. Bolton(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  This condition is called direct cen-tral impact . Following initial contact, a short period of increasing deforma-tion takes place until the contact area between the spheres ceases to increase. At this instant, both spheres, Fig. 3 ∕ 17 b , are moving with the same velocity v 0 . During the remainder of contact, a period of restoration occurs during which the contact area decreases to zero. In the final condition shown in part c of the figure, the spheres now have new velocities v 1 ′ and v 2 ′ , where v 1 ′ must be less than v 2 ′ . All velocities are arbitrarily assumed positive to the right, so that with this scalar notation a velocity to the left would carry a negative sign. m 2 m 2 m 1 m 2 m 1 m 1 v 1 v 0 v 2 Before impact ( a ) Maximum deformation during impact ( b ) After impact ( c ) > < v 1 uni02B9 v 2 uni02B9 FIGURE 3/17 100 CHAPTER 3 Kinetics of Particles If the impact is not overly severe and if the spheres are highly elastic, they will regain their original shape following the restoration. With a more severe impact and with less elastic bodies, a permanent deformation may result. Because the contact forces are equal and opposite during impact, the linear momentum of the system remains unchanged, as discussed in Art. 3 ∕ 9. Thus, we apply the law of conservation of linear momentum and write m 1 v 1 + m 2 v 2 = m 1 v 1 ′ + m 2 v 2 ′ (3 ∕ 35) We assume that any forces acting on the spheres during impact, other than the large internal forces of contact, are relatively small and produce negligible im-pulses compared with the impulse associated with each internal impact force. In addition, we assume that no appreciable change in the positions of the mass cen-ters occurs during the short duration of the impact. Coefficient of Restitution For given masses and initial conditions, the momentum equation contains two unknowns, v 1 ′ and v 2 ′ . Clearly, we need an additional relationship to find the final velocities.

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