Ch 17: Superposition

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 17: Superposition

Problem 42

Chapter 17, Problem 42

INT One end of a 75-cm-long, 2.5 g guitar string is attached to a spring. The other end is pulled, which stretches the spring. The guitar string's second harmonic occurs at 550 Hz when the spring has been stretched by 5.0 cm. What is the value of the spring constant?

Verified step by step guidance

Step 1: Understand the problem. The guitar string vibrates in its second harmonic, meaning the wavelength of the standing wave is equal to the length of the string. The frequency of the second harmonic is given as 550 Hz, and we need to calculate the spring constant of the spring attached to the string.

Step 2: Calculate the wavelength of the second harmonic. For the second harmonic, the wavelength is twice the length of the string. Use the formula: λ = 2L, where L is the length of the string (75 cm or 0.75 m).

Step 3: Use the wave speed formula to find the speed of the wave on the string. The formula is: v = fλ, where f is the frequency (550 Hz) and λ is the wavelength calculated in Step 2.

Step 4: Relate the wave speed to the tension in the string. The wave speed on a string is given by: v = √(T/μ), where T is the tension in the string and μ is the linear mass density of the string. Calculate μ using the formula: μ = m/L, where m is the mass of the string (2.5 g or 0.0025 kg) and L is its length (0.75 m). Rearrange the formula to solve for T.

Step 5: Use Hooke's Law to find the spring constant. The tension in the string is provided by the stretched spring, and Hooke's Law states: T = kx, where k is the spring constant and x is the stretch of the spring (5.0 cm or 0.05 m). Rearrange the formula to solve for k: k = T/x.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Harmonics and Frequency

Harmonics refer to the integer multiples of a fundamental frequency at which a system can oscillate. The second harmonic is the first overtone, occurring at twice the fundamental frequency. In this case, the frequency of 550 Hz indicates that the string vibrates at its second harmonic, which is essential for understanding the relationship between frequency, tension, and the physical properties of the string.

Tension in a String

The tension in a string is a force that affects its vibration and frequency. It is influenced by the mass of the string and the amount it is stretched. In this scenario, the tension is created by the spring's force when it is stretched, which can be calculated using Hooke's Law, where the force exerted by the spring is proportional to its extension.

Spring Constant (k)

The spring constant, denoted as 'k', is a measure of a spring's stiffness. It is defined by Hooke's Law, which states that the force exerted by a spring is directly proportional to its extension or compression. The value of 'k' can be determined by analyzing the force applied to the spring and the amount it is stretched, which is crucial for solving the problem of finding the spring constant in this context.

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Textbook Question

BIO Deep-sea divers often breathe a mixture of helium and oxygen to avoid getting the 'bends' from breathing high-pressure nitrogen. The helium has the side effect of making the divers' voices sound odd. Although your vocal tract can be roughly described as an open-closed tube, the way you hold your mouth and position your lips greatly affects the standing-wave frequencies of the vocal tract. This is what allows different vowels to sound different. The 'ee' sound is made by shaping your vocal tract to have standing-wave frequencies at, normally, 270 Hz and 2300 Hz. What will these frequencies be for a helium-oxygen mixture in which the speed of sound at body temperature is 750m/s ? The speed of sound in air at body temperature is 350m/s .

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Textbook Question

In a laboratory experiment, one end of a horizontal string is tied to a support while the other end passes over a frictionless pulley and is tied to a 1.5 kg sphere. Students determine the frequencies of standing waves on the horizontal segment of the string, then they raise a beaker of water until the hanging 1.5 kg sphere is completely submerged. The frequency of the fifth harmonic with the sphere submerged exactly matches the frequency of the third harmonic before the sphere was submerged. What is the diameter of the sphere?

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Textbook Question

A 2.0-m-long string vibrates at its second-harmonic frequency with a maximum amplitude of 2.0 cm. One end of the string is at x = 0 cm. Find the oscillation amplitude at x = 10, 20, 30, 40, and 50 cm.

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Textbook Question

A string under tension has a fundamental frequency of 220 Hz. What is the fundamental frequency if the tension is doubled?

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Textbook Question

A violinist places her finger so that the vibrating section of a 1.0 g/m string has a length of 30 cm, then she draws her bow across it. A listener nearby in a 20°C room hears a note with a wavelength of 40 cm. What is the tension in the string?

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Textbook Question

Tendons are, essentially, elastic cords stretched between two fixed ends. As such, they can support standing waves. A woman has a 20-cm-long Achilles tendon—connecting the heel to a muscle in the calf—with a cross-section area of 90 mm² . The density of tendon tissue is 1100 kg/m³. For a reasonable tension of 500 N, what will be the fundamental frequency of her Achilles tendon?

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