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Unformatted text preview: Impulse and Momentum This is another alternate form of Newton’s Second law X ~ F = m~ a . Integrate with respect to time from t 1 to t 2 to yield Z t 2 t 1 X ~ F dt = Z t 2 t 1 m~a dt . Since there is no inner product, the vector properties are preserved. Define now the “impulse” ~ I as ~ I ≡ Z t 2 t 1 X ~ F dt and the change in “linear momentum” Δ ~ G as Δ ~ G ≡ Z t 2 t 1 m d~v dt dt = Z ~v 2 ~v 1 m d~v = m ( ~v 2- ~v 1 ) = ~ G 2- ~ G 1 . This formulation is most frequently used for impact or explosion type of problems because the events occur too quickly for all the details to be considered. The impulse ~ I can be estimated in many ways experimentally. Impulse and Momentum for a System of Particles Since “Impulse-Momentum” is ideal for collision problems, the more interesting problems would involve more than one particle. Start with the “Principle of Motion of the Center of Mass,” X ~ F = M~a G and integrate with respect to time t to yield Z t 2 t 1 X ~ F dt = M~v G 2- M~v G 1 where ∑ ~ F is the summation of all external forces on the system and ~v G is the velocity of the system’s center of gravity. During a collision, all contact forces are internal forces because they are equal and opposite. In the case of a violent collision, the contact forces are much greater than the external forces (e.g., friction, gravitation, etc), therefore, during the short interval from t 1 to t 2 , the external forces can be neglected, i.e., Z t 2 t 1 X ~ F dt ≈ – 5 / 1 – and that will lead to M~v G 2 = M~v G 1 , or, in terms of individual particles N X i =1 m i ~v i 2 = N X i =1 m i ~v i 1 . The above equation is called “The Conservation of Linear Momentum.” The problem with this equation is that there are many unknowns, it is not sufficient to determine the velocities of the particles after the collision even though all the velocities before the collision are given. Direct Central Impact For two-body collision, an experimental method can be used to help determine the velocities after a direct central impact (head-on collision). Before During After m 1 m 2 m 1 m 2 m 1 m 2 v 1 ’ v 2 ’ v 1 v 2 Note: all the velocities are written as scalars, they must be along a straight line, say the a-axis (line of collision), for the experimental results to be valid. Let e ≡ v 2 a- v 1 a v 1 a- v 2 a = | relative velocity of separation | | relative velocity of approach | . e is called the coefficient of restitution and is regarded as a property of materials and shape of bodies. Actually, handbook values for e are unreliable as it also depends on ( v 2 a- v 1 a ) . Physically, the coefficient of restitution can be explained as e = R t 1 t (Restoration Force) dt R t 1 t (Deformation Force) dt ....
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This note was uploaded on 04/16/2008 for the course CE 325 taught by Professor Docwong during the Spring '08 term at USC.
- Spring '08