6 - Circular Motion and Other Applications of Newton's Laws

This force may depend on position velocity or time ax

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Unformatted text preview: te results can usually be obtained with the help of a computer. Graphs of velocity versus time or position versus time can be displayed to help you visualize the motion. One advantage of the Euler method is that the dynamics is not obscured — the fundamental relationships between acceleration and force, velocity and acceleration, and position and velocity are clearly evident. Indeed, these relationships form the heart of the calculations. There is no need to use advanced mathematics, and the basic physics governs the dynamics. The Euler method is completely reliable for infinitesimally small time increments, but for practical reasons a finite increment size must be chosen. For the finite difference approximation of Equation 6.10 to be valid, the time increment must be small enough that the acceleration can be approximated as being constant during the increment. We can determine an appropriate size for the time in- TABLE 6.3 The Euler Method for Solving Dynamics Problems Step Time 0 1 2 3 t0 t1 t2 t3 n tn t0 t1 t2 Position t t t x0 x1 x2 x3 xn x0 x1 x2 Velocity v0 t v1 t v2 t 171 v0 v1 v2 v3 vn v0 v1 v2 Acceleration a0 t a1 t a2 t a0 a1 a2 a3 an F (x 0 , v0 , t 0)/m F (x 1 , v 1 , t 1)/m F (x 2 , v 2 , t 2)/m F (x 3 , v 3 , t 3)/m See the spreadsheet file “Baseball with Drag” on the Student Web site (address below) for an example of how this technique can be applied to find the initial speed of the baseball described in Example 6.14. We cannot use our regular approach because our kinematics equations assume constant acceleration. Euler’s method provides a way to circumvent this difficulty. A detailed solution to Problem 41 involving iterative integration appears in the Student Solutions Manual and Study Guide and is posted on the Web at http:/ www.saunderscollege.com/physics 172 CHAPTER 6 Circular Motion and Other Applications of Newton’s Laws crement by examining the particular problem being investigated. The criterion for the size of the time increment may need to be changed during the course of the motion. In practice, however, we usually choose a time increment appropriate to the initial conditions and use the same value throughout the calculations. The size of the time increment influences the accuracy of the result, but unfortunately it is not easy to determine the accuracy of an Euler-method solution without a knowledge of the correct analytical solution. One method of determining the accuracy of the numerical solution is to repeat the calculations with a smaller time increment and compare results. If the two calculations agree to a certain number of significant figures, you can assume that the results are correct to that precision. SUMMARY Newton’s second law applied to a particle moving in uniform circular motion states that the net force causing the particle to undergo a centripetal acceleration is Fr mar mv2 r (6.1) You should be able to use this formula in situations where the force providing the centripetal acceler...
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