In an acoustics experiment, a piano string with a mass of \(5.00 \mathrm{~g}\) and a length of \(70.0 \mathrm{~cm}\) is held under tension by running the string over a frictionless pulley and hanging a \(250 .-\mathrm{kg}\) weight from it. The whole system is placed in an elevator. a) What is the fundamental frequency of oscillation for the string when the elevator is at rest? b) With what acceleration and in what direction (up or down) should the elevator move for the string to produce the proper frequency of \(440 .\) Hz, corresponding to middle A?

Acoustics experiments often involve studying the properties of sound waves and how they interact with different materials and conditions. One key element in such experiments is the generation of sound through the vibration of a medium, like a string in a musical instrument. Understanding the frequency of a vibrating string is crucial, as it relates to the pitch of the note produced. In the context of our exercise, the fundamental frequency—that is, the lowest frequency at which the string naturally vibrates—is determined under static conditions and also when an external force (acceleration of the elevator) is applied. This illustrates how changes in the physical environment can affect sound production and frequency, offering a practical perspective into the physics of acoustics.

Tension plays a pivotal role in determining the frequency of a string's vibration. Simply put, tension refers to the force that stretches the string, making it taut. The greater the tension, the faster the string vibrates, and thus the higher the pitch of the sound it produces. In an experiment like ours, tension is controlled by suspending a weight from the string, which stretches it by the force of gravity. However, if we alter the system—say, by moving the elevator up or down—we effectively change the tension in the string, either by adding to or subtracting from the force due to gravity, which in turn alters the fundamental frequency of the vibration.

The linear density of a string is a measure of mass per unit length and is a key factor in the wave properties of the string, including its frequency. In our exercise, calculating the linear density requires knowledge of the string's total mass and its length. The linear density, indicated by the Greek letter μ (mu), determines how mass is distributed along the string and affects how quickly a wave can travel through it. A lower linear density means that less mass needs to be moved as the string vibrates, allowing for higher frequencies at the same tension. It's a principle that's utilized in designing musical instruments, as varying the linear density of strings allows for a wide range of pitches.

Newton's second law is fundamental in classical mechanics and states that the acceleration of an object is directly proportional to the net force acting upon it, and inversely proportional to its mass. The formula is expressed as \( F = m \times a \), where \( F \) is the net force, \( m \) is the mass, and \( a \) is the acceleration. This principle is not just confined to solid objects, but also relates to waves in strings. In our example, suspending the weight from the string applies a force that we experience as tension. By moving the elevator, we apply additional force (or reduce it), thereby changing the tension and thus the acceleration of the mass, which, according to Newton's second law, must result in a modified force—and, as a result, a modified frequency of vibration for the string, as observed in the second part of our exercise.