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# lecture+notes+11 - 14:440:222 Dynamics(Lecture Note#11...

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14:440:222 Dynamics (Lecture Note #11) Instructor: Prof. Peng Song Rutgers University Today’s Lecture Linear impulse Linear momentum

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When a stake is struck by a sledgehammer, a large impulse force is delivered to the stake and drives it into the ground. If we know the initial speed of the sledgehammer and the duration of impact, how can we determine the magnitude of the impulsive force delivered to the stake? Linear Impulse: Applications This principle is useful for solving problems that involve force, velocity, and time. It can also be used to analyze the mechanics of impact (taken up in a later section). The result is referred to as the principle of impulse and momentum. It can be applied to problems involving both linear and angular motion. The next method we will consider for solving particle kinetics problems is obtained by integrating the equation of motion with respect to time. Principle of Linear Impulse and Momentum
This equation represents the principle of linear impulse and momentum. It relates the particle’s final velocity ( v 2 ) and initial velocity ( v 1 ) and the forces acting on the particle as a function of time. The principle of linear impulse and momentum is obtained by integrating the equation of motion with respect to time. The equation of motion can be written F = m a = m (d v /dt) Separating variables and integrating between the limits v = v 1 at t = t 1 and v = v 2 at t = t 2 results in m v 2 – m v 1 d v m F dt v 2 v 1 t 2 t 1 = = Principle of Linear Impulse and Momentum (cont’d) Linear impulse: The integral F dt is the linear impulse, denoted I . It is a vector quantity measuring the effect of a force during its time interval of action. I acts in the same direction as F and has units of N·s or lb·s. Linear momentum: The vector m v is called the linear momentum, denoted as L . This vector has the same direction as v . The linear momentum vector has units of (kg·m)/s or (slug·ft)/s. The impulse may be determined by direct integration. Graphically, it can be represented by the area under the force versus time curve. If F is constant, then I = F (t 2 – t 1 ) .

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