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- Apr 13, 2013

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I have a question..I am given the following exercise:

Prove that the motion of a mass m on a linear spring with constant $k$, has the form $y (t) = Asin(wt+f)$ , where $t$ is the time and $A, w, f$ are constants. Interpret the physical meaning of the above constants and specify their values if for $t = 0, y(0)=y_{0}$ and $y'(0)=v_{0}$. If,in addition, the mass is subject to external force $F (t) = F_{0}sin (w_{0}t)$, where $F_{0}$ the amplitude and $w_{0}$ the cyclic frequency,calculate the amplitude of the motion and find its dependence from the cyclic frequency $w_{0}$.

I have shown that the motion of the mass has the form $y(t)= Asin(wt+f)$,where $A=\sqrt{\frac{v_{0}^{2}}{w^{2}}+y_{0}^{2}}, \text { where } w=\sqrt{\frac{k}{m}}$ and $f=arctan(\frac{y_{0}w}{v_{0}})$ .But,when the mass is subject to external force $F (t) = F_{0}sin (w_{0}t)$,do we get this differential equation: $y''+w^{2}y=\frac{F_{0}}{m}sin(w_{0}t)$ ,or am I wrong?If it is right,how can I find the amplitude of the motion?