22 Spring06 Linear Impulse-Momentum

# 22 Spring06 Linear Impulse-Momentum - M AE 242: Dynamics...

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Slide No. 1 MAE 242: Dynamics Class No. 22 Linear Impulse & Momentum http://www.cemr.wvu.edu/~mae242 Readings: 15.1-3 Examples: 15.1 - 15.7 Homework: 15.35 , 47

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Slide No. 2 i dt m = F v r r i t F 1 t 2 t i F t 2 1 t i t dt F ur t 1 t 2 t i F d d m m dt dt = = = L v a F = L Our starting point is Newton’s 2 nd law: i m = F a r r ( / ) i m d dt = F v r r written as: = = ∆ 2 1 L -L L m = L v = Linear momentum integrate over the time interval t 1 to t 2 2 2 1 1 ( ) t i t dt m = - F v v r r 2 1 2 1 t i t m dt m + = v F v r r 2 1 t i t dt = F Sumof Impulses x x F L = y y F L = z z F L = Linear Impulse & Momentum 2 1 t ix t F dt 2 1 t iy t F dt 2 1 t iz t F dt 2 1 2 1 t x ix x t mv F dt mv + = 2 1 2 1 t y iy y t mv F dt mv + = 2 1 2 1 t z iz z t mv F dt mv + = Sum of linear impulsive forces equals the change in linear momentum. m 2 v 2 t t I dt = F m 1 v = + Momentum is conserved in any direction where all forces are non- impulsive. Impulse is a short time-duration force that act on a body to change its linear momentum.
Slide No. 3 As with FBD, it is important to draw an Impulse-Diagram (ID) that shows all the impulsive forces acting on the body except those forces of negligible magnitudes. Some forces are very large and of short duration, such forces are called impulsive forces. An example is a short-duration impact. We may assume that such impulsive forces are constant over their time duration. We also frequently assume that in the presence of such large impulsive forces, the effect of other nonimpulsive forces such as weight, spring force, friction, etc may be neglected. Example: The effect of weight of a 5 OZ baseball can be neglected compared to the impulse caused by the bat. If the motion of the bat is analyzed, the impulse of the weight of the bat may also be neglected, however the reactions of the player’s hands on the bat should be included since they will be of significant magnitude if the ball is incorrectly hit. Unknown reaction may or may not be impulsive depending on the problem. 2 1 1 2 t t m dt m + = ∑ ∑ ∑ v F v For a system of particles 1 2 m m = ∑ ∑ v v If all external forces are nonimpulsive then:

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Slide No. 4 The crate of mass m is subjected to exponentially decaying impulse F 0 e -bt . If the cart is stationary at time t = 0, determine its velocity as a function of time. What is the value of v for large values of t. mg F N 1 2 0 t mv Fdt mv + = 0 2 0 0 t bt F e dt mv - + = 0 2 (1 ) bt F e mv b - - = 0 2 ) bt F v e mb - = - As t → ∞ 0 2 F v mb =
Slide No. 5 15-13) Assuming that the force acting on a 2-g bullet, as it passes horizontally through the barrel of a rifle, varies with time in the manner shown, determine the maximum net force F 0 , applied to the bullet when it is fired. The muzzle velocity is 500 m/s when t = 0.75 ms. Neglect friction between the bullet and the rifle barrel.

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## 22 Spring06 Linear Impulse-Momentum - M AE 242: Dynamics...

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