Question

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?

Short answer

Answer: The fundamental frequency of the string when the elevator is at rest is approximately 261.1 Hz. To produce a frequency of 440 Hz, the elevator needs to accelerate upwards with an acceleration of approximately 3.1244 m/s².

Step-by-step solution

Calculate tension in the string at rest

When the elevator is at rest, the tension in the string (T) is equal to the weight force acting on the suspended mass (m). To find the tension, we must first convert the mass (250g) to kilograms (0.25kg) and then multiply it by the acceleration due to gravity (g = 9.81 m/s²). T = m * g T = 0.25 kg * 9.81 m/s² T = 2.4525 N

Calculate linear density of the string

To find the linear density (μ) of the string, we must first convert the mass of the string (5.00g) to kilograms (0.005kg) and then divide it by the length of the string (70.0cm or 0.7m). μ = mass_of_string / length_of_string μ = 0.005 kg / 0.7 m μ = 0.007143 kg/m

Calculate the fundamental frequency at rest

Now, we can calculate the fundamental frequency (f) of the string using the formula: f = (1/2L) * sqrt(T/μ) Where L is the length of the string and T is the tension. f = (1 / (2 * 0.7 m)) * sqrt(2.4525 N / 0.007143 kg/m) f ≈ 261.1 Hz The fundamental frequency of oscillation for the string when the elevator is at rest is approximately 261.1 Hz.

Calculate the required tension for 440 Hz

To produce a frequency of 440 Hz, we must modify the tension in the string. Using the formula for fundamental frequency, we can solve for the required tension (T_new). f_new = 440 Hz T_new = (2L * f_new)^2 * μ T_new = (2 * 0.7 m * 440 Hz)^2 * 0.007143 kg/m T_new ≈ 4.0486 N

Calculate the required acceleration of the elevator

Next, we have to consider the force acting on the mass when the elevator is accelerating. The net force (F_net) acting on the mass is the difference between the weight force and the tension (T_new). Then we can use Newton's second law to find the acceleration (a) of the elevator. F_net = m * a = m * g - T_new a = (m * g - T_new) / m a = (0.25 kg * 9.81 m/s² - 4.0486 N) / 0.25 kg a ≈ 3.1244 m/s² Since the acceleration is positive, the elevator should move up for the string to produce the proper frequency of 440 Hz. In conclusion, when the elevator is at rest, the fundamental frequency of the string is approximately 261.1 Hz. To produce a frequency of 440 Hz, the elevator needs to accelerate upwards with an acceleration of approximately 3.1244 m/s².

Key concepts

Acoustics Experiment

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 in the String

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.

Linear Density of the String

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

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.