C2_ParticleImp - 0 Contents 2 PARTICLE IMPACT 2.1...

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0 Contents 2 PARTICLE IMPACT 1 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Direct central impacts of two spheres . . . . . . . . . . . . . . 3 2.3 Oblique central impacts . . . . . . . . . . . . . . . . . . . . . 7 2.4 Oblique impact of a sphere with a wall . . . . . . . . . . . . . 8 2.5 General laws of impact . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Carnot’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.8 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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PARTICLE IMPACT 1 2 PARTICLE IMPACT 2.1 Introduction A mechanical impact occurs when the linear momentum m v has a suddenly variation or a jump. The velocity v has a discontinuity of first kind, i.e., different values at the beginning and at the end of the impact period. The impact period is very short. The phenomenon of impact may occur when two particles with different velocities collide. Another example is when an elastic sphere collides with a rigid wall and changes suddenly its velocity. The sphere position has a little variation during the collision. A particle of mass m with the velocity v 0 is subjected to an impact at the moment t 0 . The impact period of time is Δ. The velocity after impact is v 1 and corresponds to the moment t 1 = t 0 + Δ. The moments t 0 and t 1 mark the beginning and the end of the impact. The effects of external forces are negligible and the total linear momentum of the particle is conserved. Newton’s second law for the particle gives m d v = F dt. (2.1) Integrating Eq. (2.1) for the collision time, yields m v 1 - m v 0 = t 1 Z t 0 F dt. (2.2) The right-hand side of Eq. (2.2) is the linear impulse of the particle and is denoted by P P = t 1 Z t 0 F dt. (2.3) The impulse applied to the particle during the period of impact is equal to the change of the linear momentum of the particle. m v 1 - m v 0 = P . (2.4) The average with respect to time of the total force acting on the particle from t 0 to t 1 is F av and P = F av ( t 1 - t 0 ) = F av Δ .
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PARTICLE IMPACT 2 In Eq. (2.4), because the velocities v 0 and v 1 have different values during the impact, the linear impulse P must be finite. The impact period Δ is very short. To obtain a finite value for P the force F av must have a greater value. During the impact, the particle displacement can be neglected. The change of the displacement during the impact has the same order as the impact period. This can be proved by integrating the relation d r = v dt for the impact time interval ( t 0 , t 1 ) r 1 - r 0 = t 1 Z t 0 v dt. (2.5) Applying the average theorem for the integral in Eq. (2.5) and assuming for Δ = t 1 - t 0 a very low value then, the value of difference r 1 - r 0 has the same order as Δ . For impact problems, the change in the position of the particle during the negligible impact period can be considered negligible.
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PARTICLE IMPACT 3 2.2 Direct central impacts of two spheres The smooth spheres 1 and 2 move along a straight line with velocities v 1 and v 2 before their impact ( v 1 > v 2 ) , Fig. 2.1(a). The impact force is parallel to the line along which the spheres travel (direct central impact) [Fig. 2.1(b)].
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