The angular frequencies are $ \frac{3\pi}{7} $ and $ \frac{4\pi}{7} $, so periods are $ \frac{2\pi}{3\pi/7} = \frac{14}{3} $, and $ \frac{2\pi}{4\pi/7} = \frac{7}{2} $. The LCM of $ \frac{14}{3} $ and $ \frac{7}{2} $ is found by expressing as rational multiples.

["Understanding Angular Frequencies, Periods, and the LCM in Oscillatory Systems", "In physics and engineering, analyzing periodic motion often hinges on angular frequency and period concepts, especially in systems like oscillators, waves, and signal processing. A key concept arises when angular frequencies are given as fractional multiples of $\pi$, such as $ \frac{3\pi}{7} $ and $ \frac{4\pi}{7} $. Understanding how to derive the associated periods and compute their least common multiple (LCM) reveals deeper insights into the system’s rhythmic behavior.", "### Angular Frequency and Period: The Fundamental Relationship", "Angular frequency, denoted as $ \omega $, measures how rapidly an oscillation cycles per unit time, typically expressed in radians per second. It is directly related to the period $ T $—the time for one complete cycle—via:", "$$\nT = \frac{2\pi}{\omega}\n$$", "Given the angular frequencies $ \omega_1 = \frac{3\pi}{7} $ and $ \omega_2 = \frac{4\pi}{7} $, we calculate the corresponding periods:", "- For $ \omega_1 $:\n $$\n T_1 = \frac{2\pi}{\frac{3\pi}{7}} = \frac{2\pi \cdot 7}{3\pi} = \frac{14}{3}\n $$", "- For $ \omega_2 $:\n $$\n T_2 = \frac{2\pi}{\frac{4\pi}{7}} = \frac{2\pi \cdot 7}{4\pi} = \frac{7}{2}\n $$", "At first glance, $ T_1 = \frac{14}{3} $ and $ T_2 = \frac{7}{2} $ appear as simple fractions, but in periodic systems—particularly when synchronization or phase alignment matters—the least common multiple of these periods determines the system’s fundamental repeating pattern.", "### Calculating the LCM of Rational Periods", "Since $ \frac{14}{3} $ and $ \frac{7}{2} $ are rational numbers, we use a standard method to compute their LCM. The LCM of two rational numbers $ \frac{a}{b} $ and $ \frac{c}{d} $ is given by:", "$$\n\ ext{LCM}\left( \frac{a}{b}, \frac{c}{d} \right) = \frac{\ ext{LCM}(a, c)}{\ ext{GCD}(b, d)}\n$$", "Apply this to $ \frac{14}{3} $ and $ \frac{7}{2} $:", "- $ \ ext{LCM}(14, 7) = 14 $\n- $ \ ext{GCD}(3, 2) = 1 $", "Thus,", "$$\n\ ext{LCM}\left( \frac{14}{3}, \frac{7}{2} \right) = \frac{14}{1} = 14\n$$", "### Interpretation and Physical Meaning", "The LCM of the two periods, $ 14 $ units of time, signifies the smallest duration after which both oscillations fully realign, repeating their combined pattern in unison. This principle applies across disciplines—from alternating current circuits to pendulum systems—where synchronized cycles reveal coherence in periodic behavior.", "Similarly, converting angular frequencies back to frequencies:\n$ f_1 = \frac{\omega_1}{2\pi} = \frac{3\pi/7}{2\pi} = \frac{3}{14} $ Hz,\n$ f_2 = \frac{4\pi/7}{2\pi} = \frac{4}{14} = \frac{2}{7} $ Hz.", "The ratio $ \frac{f_1}{f_2} = \frac{3/14}{2/7} = \frac{3}{4} $ confirms rational frequency synchronization (periodic alignment every 4 synchronization cycles), further reinforcing the role of LCM in harmonic systems.", "### Conclusion", "Identifying angular frequencies like $ \frac{3\pi}{7} $ and $ \frac{4\pi}{7} $ and deriving their periods enables crucial analysis of system periods and resonance. By expressing these as rational multiples and computing the LCM of periods $ \frac{14}{3} $ and $ \frac{7}{2} $, we reliably determine the fundamental cycle length of 14 time units. This computational framework bridges theory and application in oscillatory systems—essential for students, physicists, and engineers alike.", "Keywords: angular frequency, period, LCM, oscillation, physics, engineering, wave motion, harmonic analysis, time domain, phase alignment."]