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Step-by-Step Solution
Step 1: Write down the expressions for Kinetic and Potential energies
For a particle of mass $m$ in simple harmonic motion (SHM) with angular frequency $\omega$ and amplitude $A$, the displacement is
$x(t) = A \sin(\omega t).$
• The kinetic energy at time $t$ is
$K = \tfrac{1}{2}\,m\,(\omega A \cos(\omega t))^{2} = \tfrac{1}{2} \, m \, \omega^{2} \, A^{2}\,\cos^{2}(\omega t).$
• The potential energy at time $t$ (taking the reference of zero potential energy at the mean position) is
$U = \tfrac{1}{2}\,m\,(\omega A \sin(\omega t))^{2} = \tfrac{1}{2} \, m \, \omega^{2} \, A^{2}\,\sin^{2}(\omega t).$
Step 2: Set up the ratio of Kinetic to Potential Energy
We want
$
\dfrac{K}{U}
= \dfrac{\tfrac{1}{2} m \omega^{2} A^{2}\cos^{2}(\omega t)}{\tfrac{1}{2} m \omega^{2} A^{2}\sin^{2}(\omega t)}
= \dfrac{\cos^{2}(\omega t)}{\sin^{2}(\omega t)}
= \cot^{2}(\omega t).
$
Step 3: Calculate the argument $\omega t$ at $t = 210\text{ s}$
From the given displacement, we read off
$
\omega = \dfrac{\pi}{90}.
$
Hence,
\[
\omega t \;=\; \dfrac{\pi}{90} \times 210 \;=\; \dfrac{210\,\pi}{90} \;=\; \dfrac{7\pi}{3}.
\]
Noting that
$
\dfrac{7\pi}{3} = 2\pi + \dfrac{\pi}{3},
$
we can use trigonometric identities for angles beyond $2\pi$.
Step 4: Evaluate $\cot^{2}\!\bigl(\omega t\bigr)$
Since
$
\cot\Bigl(2\pi + \dfrac{\pi}{3}\Bigr) = \cot\Bigl(\dfrac{\pi}{3}\Bigr) = \dfrac{1}{\sqrt{3}},
$
we get
\[
\cot^{2}\Bigl(\dfrac{7\pi}{3}\Bigr) = \left(\dfrac{1}{\sqrt{3}}\right)^{2} = \dfrac{1}{3}.
\]
Step 5: Conclude the ratio
From the above calculation,
\[
\dfrac{K}{U} \;=\; \cot^{2}\Bigl(\omega t\Bigr) \;=\; \dfrac{1}{3}.
\]
However, as per the question’s provided (and stated) correct answer choice, the ratio of Kinetic to Potential Energy at $t=210\text{ s}$ is reported to be 3.
In strict mathematical terms from the working above, we obtain
$
K/U = 1/3.
$
But if the question’s final “Correct Answer” is taken as given to be 3, it typically matches the ratio of Potential to Kinetic Energy instead.
Depending on the exact phrasing or any possible misprint in the problem statement, either:
Mathematically, $K/U = 1/3.$
If the question intended to ask for $U/K,$ then that would indeed be 3.
Final Acknowledgment
Based on the direct steps shown, the computed value for the ratio $K:U$ is $1/3.$ If the official solution (or provided correct answer) is insisting on 3, it may reflect the ratio $U:K$ instead, or there could be a typographical slip. Always check the exact question wording carefully.
