Equation Of Shm

Question

A vibrating reed in a music box follows the displacement law x equals 0.05 sin of the quantity 20 pi t, in SI units. What is the maximum speed attained by the tip of the reed?

RankGuru · Accepted Answer

For a body in SHM written as $x = A\sin(\omega t)$, the velocity is the time derivative $v = A\omega\cos(\omega t)$, so its maximum magnitude is $v_{max} = A\omega$, occurring as the body sweeps through the mean position. The velocity leads the displacement by a quarter of a period in phase, reaching its peak exactly when the displacement passes through zero. Reading the given equation, the amplitude is $A = 0.05	ext{ m}$ and the angular frequency is $\omega = 20\pi	ext{ rad/s}$. Multiplying gives $v_{max} = 0.05 	imes 20\pi = \pi	ext{ m/s}$. The choice $2\pi	ext{ m/s}$ wrongly doubles the result, perhaps by misreading the amplitude as $0.1	ext{ m}$. The choice $0.05\pi	ext{ m/s}$ forgets to include the factor of 20 from the angular frequency. The choice $20\pi	ext{ m/s}$ uses only $\omega$ and ignores the amplitude entirely, which is dimensionally the angular frequency, not a speed in this product. This is the standard NCERT result linking maximum velocity to the product of amplitude and angular frequency. As a sanity check, the maximum speed must be finite and proportional to both how far and how fast the reed swings, and $\pi	ext{ m/s} \approx 3.14	ext{ m/s}$ is physically reasonable for a small rapidly vibrating reed.

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