Math 1320 Lab 6
Name:
Consider a large circular loop of wire with radius 1 m which is fixed to be centered at
the origin and perpindicular to the x-axis.
Suppose that this wire contains a negative charge Q distributed uniformly. If we place a
positive test charge q on the x-axis, the charge will feel a force F (x), which depends on the
distance x in meters to the center of the ring.
F (x) = −F0
x
(1 + x2 )3/2
Here F0 is some positive constant. This function is rather complicated, but if x is small we
can use Taylor series to approximate it with polynomials, which are easier to work with.
The goal of this lab is to use Taylor series to approximate F (x), estimate the error in the
approximation, and use the approximation to compute aspects of how the charge q moves.
1. Write down the first three terms of the Taylor series for F (x) about x = 0. You may
use technology to compute the derivatives of F (x). Hint: Your final answer should look
like Ax + Bx3 + · · · where A and B are constants.
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2. Assume that F0 = 4.2 N. On a single graph, for −.5 ≤ x ≤ .5, plot F (x), the first-order
approximation Ax, and the third-order approximation Ax + Bx3 . Your graph should be
accurate and well labeled. (You may wish to use a computer program).
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3. Based on the first question, F (x) ≈ Ax for some constant A and small x. We can use
Taylor’s inequality to compute the error of this first-order approximation. Assume that
F0 = 4.2 N.
(a) Use Taylor’s inequality to find an upper bound on the error if |x| < .1 m. Write a
short comment about how accurate F (x) ≈ Ax is on this region.
(b) Same question as part (a), but for |x| < .5 m.
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4. Practical Uses of Taylor Expansion. Forces of the form −kx for some positive
constant k cause oscillations, called simple harmonic motion. Most forces do not produce
simple harmonic motion, but a large class of forces are approximately of the form −kx
by using a Taylor expansion. This principle often allows us to figure out how something
will move even when the forces acting on it are very complicated.
(a) Show that for small enough x, the motion caused by F (x) is approximately simple
harmonic.
(b) Under a net force of the form −kx, a particle oscillates with a frequency ω given
by mω 2 = k. Using the approximation from part (a), find the frequency of small
oscillations of the charge q about the center of the ring. Assume that q has a mass
m = 0.1 kg and that F0 = 4.2 N as in problem 2.
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