**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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