# Undamped forced motion and resonance, find $ω$ given an external force

The position of a certain spring-mass system satisfies the initial value problem

$$6x''+kx=0,x(0) = 5,x'(0)=v$$

The period and amplitude of the resulting motion are observed to be $3π$ and $6,$ respectively.

$1.$Determine the values of $k$ and $v$. Assume $v≥0$.

$2.$ Suppose an external force $F(t)=6\sin(\omega t)$ is introduced to the system. Find the value of $ω$ for which resonance occurs due to the external force.

Hint: It may be helpful to know that if $A$ and $B$ are constants, then $A\sin(\omega t)+B\cos(\omega t)=\sqrt{A^2+B^2}\cos(\omega t−\delta)$ for some $\delta$ that satisfies $\tan\delta=\dfrac{B}{A}$.

I calculated $k$ = $\dfrac{24}{9}$ and $v$ = $2.211$

But how do I find the value of $\omega$?

• what is resonance? – James S. Cook Oct 18 '13 at 4:38

I start from $$mx''+kx=F_0 \sin \omega\,t$$ for an undamped forced oscillation.
If $$\omega=\sqrt {\frac km}$$i.e. if $\omega$ is equal to the natural frequency of the system, then $\,\sin \omega\,t\,$ is a term of the complementary function so you must be clever in applying the method of undetermined coefficients.