(From 4.2) In a mass-spring system, motion is assumed to occur only in the vertical direction....
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(From 4.2) In a mass-spring system, motion is assumed to occuronly in the vertical direction. That is, the system has one degreeof freedom. When the mass is pulled downward and then released, thesystem will oscillate. If the system is undamped, meaning thatthere are no forces to slow or stop the oscillation, then thesystem will oscillate indefinitely. Applying Newton’s Second Law ofMotion to the mass yield the second-order differential equation ?′′ + ? 2? = 0 where ? is the displacement at time ? and ? is afixed constant called the natural frequency of the system.
a) Verify that the general solution of the above differentialequation is ?(?) = ?1????? + ?2????? where ?1 and ?2 are arbitraryconstants.
b) Show that the set of all functions ?(?) form a vectorspace.
(From 4.2) In a mass-spring system, motion is assumed to occuronly in the vertical direction. That is, the system has one degreeof freedom. When the mass is pulled downward and then released, thesystem will oscillate. If the system is undamped, meaning thatthere are no forces to slow or stop the oscillation, then thesystem will oscillate indefinitely. Applying Newton’s Second Law ofMotion to the mass yield the second-order differential equation ?′′ + ? 2? = 0 where ? is the displacement at time ? and ? is afixed constant called the natural frequency of the system.
a) Verify that the general solution of the above differentialequation is ?(?) = ?1????? + ?2????? where ?1 and ?2 are arbitraryconstants.
b) Show that the set of all functions ?(?) form a vectorspace.
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