Question

A guitar string vibrates at a frequency of 440 Hz. A point at its center moves in SHM with an amplitude of 3.0 mm and a phase angle of zero. (a) Write an equation for the position of the center of the string as a function of time. (b) What are the maximum values of the magnitudes of the velocity and acceleration of the center of the string? (c) The derivative of the acceleration with respect to time is a quantity called the \(jerk\). Write an equation for the jerk of the center of the string as a function of time, and find the maximum value of the magnitude of the jerk.

Short answer

Position: \(x(t) = 3.0 \cos(880\pi t)\). Max velocity: \(2640\pi \text{ mm/s}\). Max acceleration: \(3.0 \times 774400\pi^2 \text{ mm/s}^2\). Max jerk: \(3.0 \times 680244480\pi^3 \text{ mm/s}^3\).

Step-by-step solution

Write position equation

First, recognize the simple harmonic motion (SHM) equation for the position of the center of the string. The general formula for SHM is given by \( x(t) = A \cos(\omega t + \phi) \), where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase angle. Given the amplitude \(A = 3.0 \text{ mm}\) and a phase angle of zero, the equation simplifies to \( x(t) = 3.0 \cos(\omega t) \).

Calculate angular frequency

The angular frequency \(\omega\) is related to the frequency \(f\) by \(\omega = 2\pi f\). Given the frequency \(f = 440 \text{ Hz}\), calculate \(\omega\) as follows: \(\omega = 2\pi \times 440 = 880\pi \text{ rad/s}\).

Position function equation

Now substitute \(\omega\) into the position equation: \( x(t) = 3.0 \cos(880\pi t) \). This is the position as a function of time for the center of the string.

Max velocity and acceleration

For SHM, the maximum velocity \(v_{\text{max}}\) is given by \(v_{\text{max}} = A\omega\). \(v_{\text{max}} = 3.0 \times 880\pi = 2640\pi \text{ mm/s}\). The maximum acceleration \(a_{\text{max}}\) is given by \(a_{\text{max}} = A\omega^2\). \(a_{\text{max}} = 3.0 \times (880\pi)^2 = 3.0 \times 774400\pi^2 \text{ mm/s}^2\).

Determine jerk function

The jerk is the derivative of acceleration with respect to time. Start by finding the acceleration function: \( a(t) = -A\omega^2 \cos(\omega t)\). Then, the jerk function is \( j(t) = \frac{d}{dt}[-A\omega^2 \cos(\omega t)] = A\omega^3 \sin(\omega t)\).

Maximum jerk calculation

The maximum value of the jerk \(j_{\text{max}}\) occurs when \(\sin(\omega t) = \pm 1\). \(j_{\text{max}} = A\omega^3 = 3.0 \times (880\pi)^3 = 3.0 \times 680244480\pi^3 \text{ mm/s}^3\).

Key concepts

Angular Frequency

Angular frequency, often denoted by the symbol \( \omega \), is an essential concept in the study of Simple Harmonic Motion (SHM). It's related to how quickly an oscillating object cycles through its motion, similar to how regular frequency measures cycles per second. Angular frequency is given by the formula \( \omega = 2\pi f \), where \( f \) is the frequency in hertz (Hz). For example, in the case of a guitar string vibrating at 440 Hz, the angular frequency would be \( \omega = 2\pi \times 440 = 880\pi \) rad/s. This means the string completes \( 880\pi \) radians every second. Angular frequency helps us determine other important properties of SHM, such as velocity and acceleration. Using it, we craft precise equations to describe motion, such as \( x(t) = A \cos(\omega t + \phi) \), aiding in predictions of the oscillating system's behavior over time.

Understanding the concept of angular frequency not only allows us to write accurate position equations but also helps in visualizing the rapidity of oscillation in physical systems. Its unit, radians per second, captures the conversion of periodic motion into rotational parameters, a valued insight for scientists and engineers alike.

Maximum Velocity and Acceleration

In Simple Harmonic Motion, both velocity and acceleration oscillate just like position does, reaching their extremums at specific points in the cycle. The maximum velocity \( v_{\text{max}} \) in SHM is derived from the formula \( v_{\text{max}} = A\omega \), where \( A \) is the amplitude and \( \omega \) is the angular frequency. Applying this to our guitar string example with amplitude of 3.0 mm and \( \omega = 880\pi \) rad/s, the maximum velocity becomes \( 2640\pi \) mm/s. This indicates the highest speed the string’s center reaches while oscillating.

Similarly, the maximum acceleration \( a_{\text{max}} \) is given by \( a_{\text{max}} = A\omega^2 \). For the guitar string, plugging in \( A \) and \( \omega \) gives us \( 3.0 \times 774400\pi^2 \) mm/s². The acceleration is at its fastest when the string passes through its equilibrium position, driven by the restoring force attempting to bring the system back to equilibrium. These extreme values help understand the energy dynamics at play—at maximum velocity the kinetic energy is at its peak, whereas maximum acceleration aligns with peak restoring force.

Jerk in Physics

Jerk, while perhaps a lesser-known term compared to velocity or acceleration, is crucial in understanding dynamic systems in more detail. It represents the rate of change of acceleration over time, defined as the derivative of acceleration. In terms of mathematics for SHM, it is expressed as \( j(t) = \frac{d}{dt}[-A\omega^2 \cos(\omega t)] = A\omega^3 \sin(\omega t) \). Breaking this down for a system with an amplitude of 3.0 mm and an angular frequency \( \omega = 880\pi \) rad/s, the maximum jerk \( j_{\text{max}} \) arises when \( \sin(\omega t) = \pm 1 \).

The result of \( 3.0 \times 680244480\pi^3 \) mm/s³ illustrates the maximum rate of change of the acceleration as the string vibrates. Jerk becomes especially important in scenarios where sudden changes in acceleration can impact the mechanical integrity or user comfort—for instance, the starting and stopping motions in vehicles or amusement parks rides. Accurate jerk calculations help in designing systems optimized for both performance and safety.