1. Two Coupled oscillators Three ideal, massless springs and two masses are attached to each other as shown in figure 1. The extreme on the left hand side and on the right hand side are attached to fix walls. Neglect the effect due to gravity and consider k₁, k2, k12, M1, M2 > 0. k₁ wwwwwww m₁ x1 K12 wwwwwwwwwwww m₂ x2 wwwwww k₂ Figure 1: System of three ideal, massless springs with two masses. k₁, k2, k12, M1, M2 > 0 (a) Prove that the system in figure 1 always has two well defined normal modes. (b) What is the angular frequency of the oscillation for each normal mode if m₁ = m₂ = m and k₁= k₂= k k12? (c) What are the two normal modes if m₁ = m₂ = m and k₁ = k₂ = k = k12? Is the Center of Mass a normal mode? (d) What are the answers to the last two questions if k₁= k2 = k12 = k, and m₁ = m₂ = m?

1. Two Coupled oscillators Three ideal, massless springs and two masses are attached to each other as shown in figure 1. The extreme on the left hand side and on the right hand side are attached to fix walls. Neglect the effect due to gravity and consider k₁, k2, k12, M1, M2 > 0. k₁ wwwwwww m₁ x1 K12 wwwwwwwwwwww m₂ x2 wwwwww k₂ Figure 1: System of three ideal, massless springs with two masses. k₁, k2, k12, M1, M2 > 0 (a) Prove that the system in figure 1 always has two well defined normal modes. (b) What is the angular frequency of the oscillation for each normal mode if m₁ = m₂ = m and k₁= k₂= k k12? (c) What are the two normal modes if m₁ = m₂ = m and k₁ = k₂ = k = k12? Is the Center of Mass a normal mode? (d) What are the answers to the last two questions if k₁= k2 = k12 = k, and m₁ = m₂ = m?

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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Work and Heat Transfer

It is generally related to thermal engineering which tells about the use, creation, conversion and transfer of heat energy between the systems. It is also considered that to transfer heat from one source to another, transfer of masses is also required. There are various mechanisms by which, transfer of heat takes place i.e. thermal conduction, convection, radiation or by change of phase. All these mechanisms have their own specific features which sometimes occur simultaneously within the same system.

Question

1. Two Coupled oscillators
Three ideal, massless springs and two masses are attached to each other as shown in
figure 1. The extreme on the left hand side and on the right hand side are attached
to fix walls. Neglect the effect due to gravity and consider k₁, k2, k12, M1, M2 > 0.
k₁
m1
=
x1
K12
m₂
=
x2
Figure 1: System of three ideal, massless springs with two masses. k₁, k2, k12, M1, M2 0
(a) Prove that the system in figure 1 always has two well defined normal modes.
(b) What is the angular frequency of the oscillation for each normal mode if m₁ =
m₂ = m and k₁ k₂ = k k12?
(c) What are the two normal modes if m₁
kk12? Is
the Center of Mass a normal mode?
(d) What are the answers to the last two questions if k₁
m₁ = m₂ = m?
k₂
m₂ = m and k₁
=
=
k2
k2
=
=
k12
=
k, and

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