Decide whether each of the following statements are T-True, or F-False. (If the first is T and the rest are F, enter TFFFF)

A) If you decrease the tension in a string, the frequency of its fundamental vibration will get lower
B) 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)
C) If the fundamental vibration of a string has wavelength lambda, then the next higher mode will have wavelength 2*lambda
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 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 frequency of f/2

FTFFT

To evaluate each of these statements, we need to consider the properties and behaviors of vibrating strings. Let's go through each statement and determine whether it is true or false:

A) If you decrease the tension in a string, the frequency of its fundamental vibration will get lower.
To evaluate this statement, we need to understand the relationship between tension and frequency in a vibrating string. According to the equation for the frequency of a vibrating string, f = (1/2L) * sqrt(T/μ), where f is the frequency, L is the length of the string, T is the tension, and μ is the linear mass density of the string. This equation shows that frequency is directly proportional to the square root of tension. As tension decreases, the frequency will also decrease. Therefore, this statement is T - True.

B) 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 evaluate this statement, we need to understand the concept of standing waves and nodes. Nodes are points on a standing wave where the amplitude is always zero. For a standing wave with frequency 5 times the fundamental, we can conclude that it has 5 nodes. However, the statement specifies that we should not count the nodes at the end of the string. Therefore, there are actually 3 internal nodes (not counting the end nodes). This makes the statement False (F).

C) If the fundamental vibration of a string has wavelength lambda, then the next higher mode will have wavelength 2*lambda.
To evaluate this statement, we need to understand how the wavelengths of different modes in a vibrating string are related. In a vibrating string, the wavelengths of different modes are determined by the length of the vibrating string. The fundamental mode has a wavelength equal to twice the length of the string, while the second mode has a wavelength equal to the length of the string, and so on. Therefore, the statement is False (F) as the wavelength of the next higher mode is equal to the length of the string, not 2 times the wavelength of the fundamental mode.

D) If an ideal string can vibrate in a pure standing wave with frequency f, then it can also be made to vibrate in a pure standing wave with frequency of 3f, without any changes in length, tension, or material.
To evaluate this statement, we need to understand the concept of harmonics in vibrating strings. Harmonics are multiples of the fundamental frequency. In a vibrating string, the harmonic frequencies are determined by the length, tension, and material of the string. If a string can vibrate in a pure standing wave with frequency f, it means it can support the fundamental mode. The third harmonic (3f) is three times the frequency of the fundamental mode. However, for an ideal string, the harmonics are fixed and determined by the string's properties. Therefore, without any changes in these properties, the string cannot vibrate in a pure standing wave with a frequency of 3f. Thus, the statement is False (F).

E) If an ideal string can vibrate in a pure standing wave with fundamental frequency f, then it can also be made to vibrate in a pure standing wave with a frequency of f/2, without any changes in length, tension, or material.
To evaluate this statement, we need to understand the concept of the frequency of the fundamental mode in vibrating strings. The frequency of the fundamental mode is determined solely by the properties of the string – the length, tension, and material. If an ideal string can vibrate in a pure standing wave with fundamental frequency f, there is no reason to assume it can also vibrate with a frequency of f/2. The harmonics of an ideal string are integer multiples of the fundamental frequency, so f/2 is not part of the harmonics series. Therefore, the statement is False (F).