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L9 Linear Impulse and Momentum

# L9 Linear Impulse and Momentum - J Peraire S Widnall 16.07...

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J. Peraire, S. Widnall 16.07 Dynamics Fall 2009 Version 2.0 Lecture L9 - Linear Impulse and Momentum. Collisions In this lecture, we will consider the equations that result from integrating Newton’s second law, F = m a , in time. This will lead to the principle of linear impulse and momentum. This principle is very useful when solving problems in which we are interested in determining the global e ff ect of a force acting on a particle over a time interval. Linear Momentum We consider the curvilinear motion of a particle of mass, m , under the influence of a force F . Assuming that the mass does not change, we have from Newton’s second law, d v d F = m a = m = ( m v ) . dt dt The case where the mass of the particle changes with time (e.g. a rocket) will be considered later on in this course. The linear momentum vector, L , is defined as L = m v . Thus, an alternative form of Newton’s second law is F = L ˙ , (1) which states that the total force acting on a particle is equal to the time rate of change of its linear momentum. Principle of Linear Impulse and Momentum Imagine now that the force considered acts on the particle between time t 1 and time t 2 . Equation (1) can then be integrated in time to obtain t 2 t 1 F dt = t 2 t 1 ˙ L dt = L 2 L 1 = L . (2) Here, L 1 = L ( t 1 ) and L 2 = L ( t 2 ). The term t 2 I = t 1 F dt = L = ( m v ) 2 ( m v ) 1 , is called the linear impulse . Thus, the linear impulse on a particle is equal to the linear momentum change L . In many applications, the focus is on an impulse modeled as a large force acting over a small time. But 1

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in fact, this restriction is unnecessary. All that is required is to be able to perform the integral t 2 F dt . If t 1 the force is a constant F, then L = t 2 F dt = F ( t 2 t 1 ). If the force is given as a function of time, then t 1 L = t t 1 2 F ( t ) dt Note Units of Impulse and Momentum It is obvious that linear impulse and momentum have the same units. In the SI system they are N s or kg · m/s, whereas in the English system they are lb s, or slug ft/s. · · · Example (MK) Average Drag Force The pilot of a 90 , 000-lb airplane which is originally flying horizontally at a speed of 400 mph cuts o ff all engine power and enters a glide path as shown where β = 5 o . After 120 s, the airspeed of the plane is 360 mph. We want to calculate the magnitude of the time-averaged drag force. Aligning the x -axis with the flight path, we can write the x component of equation (2) as follows 120 ( W sin β D ) dt = L x (120) L x (0) . 0 The time-averaged value of the drag force, D ¯ , is 120 1 ¯ D = D dt . 120 0 Therefore, ( W sin β D ¯ )120 = m ( v x (120) v x (0)) . Substituting and applying the appropriate unit conversion factors we obtain, 90 , 000 5280 (90 , 000 sin 5 o D ¯ )120 = 32 . 2 (360 400) 3600 D ¯ = 9 , 210 lb . 2
Impulsive Forces We typically think of impulsive forces as being forces of very large magnitude that act over a very small interval of time, but cause a significant change in the momentum. Examples of impulsive forces are those generated when a ball is hit by a tennis racquet or a baseball bat, or when a steel ball bounces on a steel plate. The

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