Ch 17: Superposition

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 17: Superposition

문제 42

17장, 문제 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?

검증된 단계별 안내

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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주요 개념

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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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