A guitar string vibrates harmonically with a frequency of 512 Hz (one octave above middle C on the musical scale). If the amplitude of oscillation of the centerpoint of the string is 0.002 m (2 mm), what are the maximum speed and the maximum acceleration at that point?

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Quizlet Admin · Accepted Answer

So, we know the expression for displacement at the point x is connected with it's amplitude A and frequency f as

$$
\begin{align*}
    x &= A \sin \qty( 2 \pi f t ), 	ag{1}
\end{align*}
$$

where t is time at which we observe our point. We can calculate the speed at that point by taking a derivative of equation (1), so we would get

$$
\begin{align*}
    \Dot{x} &= A \cdot \qty(2 \pi f) \cos \qty( 2 \pi f t ). 	ag{2}
\end{align*}
$$

Similarly, we can find the acceleration at that point by taking a derivative of equation (2)

$$
\begin{align*}
    \Ddot{x} &= A \cdot \qty( 2 \pi f )^2 \sin \qty( 2 \pi f t ). 	ag{3}
\end{align*}
$$

Since the problem asks us the value of maximum speed and acceleration (maximum values of equation (2) and (3) are when $\sin \qty(2  \pi ft) =1$), equation (2) and (3) becomes

$$
\begin{align*}
      \Dot{x}_{	ext{max}} &= A \cdot \qty(2 \pi f) 	ag{4}
\end{align*}
$$

and

$$
\begin{align*}
    \Ddot{x}_{	ext{max}}&= A \cdot \qty( 2 \pi f )^2 . 	ag{5}
\end{align*}
$$

So, inserting numbers from the problem we can calculate the maximum speed as

$$
\begin{align*}
     \Dot{x}_{	ext{max}}&= 0.002 \: \mathrm{m} \cdot \qty( 2 \pi \cdot 512 \: \underbrace{\mathrm{Hz}}_{= \mathrm{1/s}} ) \
    \Dot{x}_{	ext{max}} &= \boxed{ 6.43 \: \mathrm{m/s} }
\end{align*}
$$

and maximum acceleration at that point is

$$
\begin{align*}
    \Ddot{x}_{	ext{max}} &= 0.002 \: \mathrm{m} \cdot \qty( 2 \pi \cdot 512 \: \underbrace{\mathrm{Hz}}_{= \mathrm{1/s}} )^2 \
     \Ddot{x}_{	ext{max}}&= \boxed{ 2.07 \cdot 10^4 \: \mathrm{m/s^2} }.
\end{align*}
$$

Related questions with answers

A guitar string vibrates harmonically with a frequency of 512 Hz (one octave above middle C on the musical scale). If the amplitude of oscillation of the centerpoint of the string is 0.002 m (2 mm), what are the maximum speed and the maximum acceleration at that point?

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