momentum - EA3: Systems Dynamics V IMPULSE AND MOMENTUM...

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EA3: Systems Dynamics Momentum Methods Sridhar Krishnaswamy 1 V IMPULSE AND MOMENTUM V.1 Linear Impulse: Newton’s laws say that if a net force F acts on a body, then: F = d p dt (5.1) where p = m v is the linear momentum of the body, where m is the mass and v is the velocity of the body. Integrating the above with respect to time: F dt t 1 t 2 = d p p 1 p 1 = p 2 - p 1 (5.2) The left side is called the linear impulse due to the force F over the time interval t 1 to t 2 : I F = F dt t 1 t 2 (5.3) Remarks: (i) The units of linear impulse in SI are N.s. (ii) Often, such as during impact of bodies, it is not possible to measure the force of impact, but it is possible to obtain an average measure of the force by measuring its momentum before and after the impact through: F av = 1 t 2 - t 1 I F = 1 t 2 - t 1 F dt t 1 t 2 = 1 t 2 - t 1 p 2 - p 1 { } (5.4) (iii) The statement (5.2) says that the linear impulse imparted to an object is equal to the resulting change in linear momentum of the object . (iv) If there is no net force acting on a body, clearly its linear momentum is unchanged. Figure 5.1 : Some representative impulsive forces.
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EA3: Systems Dynamics Momentum Methods Sridhar Krishnaswamy 2 (v) As we did when we discussed Newton’s laws, we can extend these ideas to a system of N particles. Let the ith particle have a mass m i , and be located at position r i moving with velocity v i with respect to some chosen coordinate system. Let us consider the forces on the ith particle in two parts: f ij is the force on the ith particle exerted by the jth particle in the system, and F i is the force exerted on the ith particle by something external to the system. f ij j + F i = d p i dt (5.5) There are N such equations, one for each particle. Suppose we sum all these N equations together: i f ij j + F i i = d p i dt i (5.6) Clearly the first term on the left side is zero (from Newton’s third law f ij =- f ji ). Integrating the above with respect to time, we have: F i i t 1 t 2 dt = P 2 - P 1 (5.7) where the linear momentum of the total system is given by: P = p i = i = 1 N m i v i i = 1 N . (5.8) (vi) If there is no net external force acting on a system of particles, then the above says that the linear momentum of the system is conserved. P
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This note was uploaded on 04/17/2008 for the course GEN_ENG 203 taught by Professor Krishnaswamy during the Fall '08 term at Northwestern.

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momentum - EA3: Systems Dynamics V IMPULSE AND MOMENTUM...

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