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Derive the system of equations (1) by applying Newton’s second law,
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The system of equations describing motions of a three mass and four spring system by applying Newton’s second law
Answer to Problem 1P
Solution:
The system of equations describing motions of a three mass and four spring system are
Explanation of Solution
Given information:
Three masses
Newton’s second law is
The springs follow Hooke’s Law: The force exerted by a spring on a mass is directly proportional to the length of its departure from its equilibrium length.
Explanation:
The first mass’s one side is attached to the wall with the spring and the other side is attached to the second body with another spring of spring constant
Then the velocity of the mass is
Then the acceleration of the mass is
By Newton’s law, the forces acting on the mass is
By Hooke’s law, the forces acting on the mass is
Also, a force
Then the equation of motion of the first body is
In the second case, the second body is attached to the first body with the spring and the other side is attached to the third body with another spring of spring constant
Then the velocity of the mass is
Then the acceleration of the mass is
By Newton’s law, the forces acting on the mass is
By Hooke’s law, the forces acting on the mass is
Also, a force
Then the equation of motion of the second body is
Similarly, in the third case, the third mass is attached to the wall with the spring and in the other side, it is attached to the second body with another spring of spring constant
Then the velocity of the mass is
Then the acceleration of the mass is
By Newton’s law, the forces acting on the mass is
By Hooke’s law, the forces acting on the mass is
Also, a force
Then the equation of motion of the third body is
Thus, the system of equations describing the motions of a three mass and four spring system are
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Differential Equations: An Introduction to Modern Methods and Applications
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