Consider a system of three masses, each with mass m, joined by ideal, massless springs, each with spring's constant k, as shown in figure 1. k m X1 k m x2 k m x3 k Figure 1: 3 identical masses attached with identical ideal, massless springs. The coordi- nates x₁, x2, x3 represent the position in time of each of the masses. Neglect any effect due to gravity. (a) Show the spring forces acting upon each mass when T1, T2, T3 > 0. (b) Find the equations of motion of each mass. That is, a system of second order, coupled differential equations for x1, x2, x3. (c) Find the normal modes of the system and their associated angular frequencies.

Consider a system of three masses, each with mass m, joined by ideal, massless springs, each with spring's constant k, as shown in figure 1. k m X1 k m x2 k m x3 k Figure 1: 3 identical masses attached with identical ideal, massless springs. The coordi- nates x₁, x2, x3 represent the position in time of each of the masses. Neglect any effect due to gravity. (a) Show the spring forces acting upon each mass when T1, T2, T3 > 0. (b) Find the equations of motion of each mass. That is, a system of second order, coupled differential equations for x1, x2, x3. (c) Find the normal modes of the system and their associated angular frequencies.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

Consider a system of three masses, each with mass m, joined by ideal, massless
springs, each with spring's constant k, as shown in figure 1.
k
m
x1
k
m
x2
k
m
X3
k
Figure 1: 3 identical masses attached with identical ideal, massless springs. The coordi-
nates 1, 2, 3 represent the position in time of each of the masses.
Neglect any effect due to gravity.
(a) Show the spring forces acting upon each mass when T1, T2, T3 > 0.
(b) Find the equations of motion of each mass. That is, a system of second order,
coupled differential equations for x1, x2, x3.
(c) Find the normal modes of the system and their associated angular frequencies.

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