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

Question

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 $\omega$?

Answer 1

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.
You will find that the particular solution has an amplitude increasing without bound (this is pure resonance).