Decide whether each of the following statements are T-True, or F-False.
A) If you increase the tension in a string, the frequency of its fundamental vibration will get lower
B) If the fundamental vibration of a string has wavelength lambda, then the next higher mode will have wavelength lambda/2
C) If an ideal string can vibrate in a pure standing wave with fundamental frequency f, then (with no changes in length, tension, or material) it can also be made to vibrate in a pure standing wave with a frequency of f/4
D) If an ideal string can vibrate in a pure standing wave with frequency f, then (with no changes in length, tension, or material) it can also be made to vibrate in a pure standing wave with frequency of 3f
E) If you have a standing wave with frequency 5 times the fundamental, there are 4 internal nodes (not counting nodes at the end of the string)
To determine whether each statement is true or false, let's analyze them one by one:
A) If you increase the tension in a string, the frequency of its fundamental vibration will get lower.
To find the answer to this statement, we need to understand the relationship between tension and frequency in a string. According to the equation of wave velocity on a string (v = √(T/μ)), where T is the tension and μ is the linear mass density, tension actually increases the wave velocity. Since the frequency (f) is directly proportional to the wave velocity (v) and inversely proportional to the wavelength (λ) through the equation v = fλ, increasing the tension will increase the frequency, not lower it. Therefore, statement A is False.
B) If the fundamental vibration of a string has wavelength λ, then the next higher mode will have wavelength λ/2.
To determine the correctness of this statement, we need to understand the relationship between the modes of vibration on a string. Each mode of vibration on a string has a specific pattern characterized by the number of antinodes and nodes. For a string fixed at both ends (like a guitar string), the fundamental mode has one antinode at the center, while the next higher mode (the first overtone) has two antinodes and twice the frequency of the fundamental mode.
Now, the wavelength of a wave on a string is twice the distance between two consecutive antinodes. In the fundamental mode, this distance is the entire length of the string, so λ = 2L, where L is the length of the string. In the next higher mode, the distance between two consecutive antinodes is half the length of the string (since it has two antinodes), so λ' = L. Therefore, the wavelength of the next higher mode is equal to the wavelength of the fundamental mode, not half of it. Thus, statement B is False.
C) If an ideal string can vibrate in a pure standing wave with a fundamental frequency f, then (with no changes in length, tension, or material) it can also be made to vibrate in a pure standing wave with a frequency of f/4.
To evaluate this statement, we need to consider the harmonic series. The harmonic series refers to the set of frequencies at which an ideal string can vibrate with a pure standing wave. The fundamental frequency (f) represents the first harmonic. The next harmonic is the second harmonic (2f), then comes the third harmonic (3f), and so on.
Since the frequency of the next harmonic is obtained by doubling the fundamental frequency, we cannot achieve a frequency of f/4, which is one-fourth of the fundamental frequency unless we skip several harmonics. Therefore, statement C is False.
D) If an ideal string can vibrate in a pure standing wave with frequency f, then (with no changes in length, tension, or material) it can also be made to vibrate in a pure standing wave with a frequency of 3f.
This statement is true. A string that can vibrate at the fundamental frequency f can also vibrate at harmonics, such as 2f, 3f, 4f, and so on. These higher harmonics are whole number multiples of the fundamental frequency. Thus, an ideal string can vibrate at frequencies that are integer multiples of the fundamental frequency without any changes in length, tension, or material. Therefore, statement D is True.
E) If you have a standing wave with a frequency 5 times the fundamental, there are 4 internal nodes (not counting nodes at the end of the string).
To assess this statement, we need to understand the relationship between standing waves and nodes. A standing wave on a string has nodes and antinodes. Nodes are points on the string that do not oscillate (remain at rest), while antinodes are points of maximum oscillation.
In a standing wave on a string, the number of nodes is equal to the number of antinodes minus one. Since the fundamental mode has one node in the middle, the next higher mode (first overtone) has two antinodes, yielding one internal node.
For a standing wave with a frequency 5 times the fundamental, we are considering the fifth harmonic. In the fifth harmonic, the string vibrates with five times the frequency of the fundamental and has six antinodes. Therefore, the number of nodes (excluding the nodes at the ends of the string) is equal to the number of antinodes minus one, which is six minus one equals five. Thus, the statement E is False.
Summary:
A) False
B) False
C) False
D) True
E) False
A) False. According to the wave equation, the frequency of a string's fundamental vibration is directly proportional to the tension in the string. Therefore, increasing the tension in a string will result in a higher frequency, not a lower frequency.
B) True. In a string, the wavelength of the next higher mode is always half of the wavelength of the fundamental mode. This is because the next higher mode corresponds to the string vibrating in two segments, resulting in two half-wavelengths.
C) False. In an ideal string, the frequencies of the pure standing waves are integer multiples of the fundamental frequency. This means that the frequency of the next higher mode would be 2f, not f/4.
D) True. Similar to the explanation in statement B, the frequency of the next higher mode in an ideal string is always three times the frequency of the fundamental mode.
E) True. In a standing wave, the number of internal nodes (points of zero amplitude) is equal to the number of half-wavelengths present in the string. If the frequency of the standing wave is 5 times the fundamental frequency, it means there are 5 half-wavelengths in the string. Since the string has two endpoints that are always nodes, there will be 3 internal nodes.