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Benzoate
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Homework Statement
Find the general solution of the damped SHM equation(5.9) for the special case of critical damping , that is , when K = [tex]\Omega[/tex]. Show that , if the particle is initially relaeased from rest at x= a , then the subsequent motion is given by
x=a*(e^-([tex]\Omega[/tex]*t))*(1+[tex]\Omega[/tex]*t)
Homework Equations
x''+2Kx'+([tex]\Omega[/tex])2*x=0
x=A*cos([tex]\Omega[/tex]*t) + B*sin([tex]\Omega[/tex]*t)
The Attempt at a Solution
x=e[tex]\lambda[/tex]*t
x'= [tex]\lambda[/tex]*e[tex]\lambda[/tex]*t
x''=[tex]\lambda[/tex]2*e[tex]\lambda[/tex]*t
x'= -A*[tex]\Omega[/tex]*sin([tex]\Omega[/tex]*t) + B*[tex]\Omega[/tex]*cos([tex]\Omega[/tex]*t)
x''= -A*[tex]\Omega[/tex]2*cos([tex]\Omega[/tex]*t) - B*[tex]\Omega[/tex]2*sin([tex]\Omega[/tex]*t)
Not sure how to proceed...