I have to figure out if the following are true or false in order to solve an equation. But I am getting at least two wrong so I can't get the next one right.
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)
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To determine if the statements are true or false, we'll need to understand the concepts of tension, frequency, wavelength, and standing waves in a string.
A fundamental vibration refers to the lowest frequency at which a string can vibrate, also known as the first harmonic. To assess the accuracy of statement A, let's now examine its claims.
Statement A: If you increase the tension in a string, the frequency of its fundamental vibration will get lower.
To verify this statement, you can experiment with a real string or visualize it conceptually. When the tension in a string is increased, it becomes tighter. This causes the speed of the waves traveling along the string to increase, which implies a higher frequency. Therefore, statement A is false. Increasing tension in a string results in a higher frequency of the fundamental vibration.
Next, let's analyze statement B.
Statement B: If the fundamental vibration of a string has wavelength lambda, then the next higher mode will have wavelength lambda/2.
Here, "mode" refers to the number of nodes within the string, and "wavelength" represents the distance between two consecutive nodes. Consecutive modes have an additional node, resulting in one more half-wavelength in the string. Therefore, statement B is true. The wavelength of the next higher mode is half that of the fundamental vibration.
Moving on to statement C.
Statement 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.
Given that there are no changes in length, tension, or material, the frequencies of the standing waves are dependent only on the characteristics of the string. Since fundamental frequency corresponds to the first harmonic, which has one antinode, halving the frequency leads to two nodes or two antinodes. In this case, the string would vibrate in the second harmonic, not the fourth harmonic. Therefore, statement C is false.
Continuing with statement D.
Statement 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.
Similar to the previous statement, the frequency of a string's standing wave is determined by its length, tension, and material. Without any changes to these characteristics, the standing wave will maintain its original frequency unless some external factors intervene. Therefore, statement D is false. Without alterations, the string cannot vibrate in a standing wave with a frequency of 3f.
Finally, let's analyze statement E.
Statement 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).
A standing wave with frequency 5 times the fundamental refers to the fifth harmonic. The number of internal nodes (the points on the string that do not move) in a standing wave is equal to one less than the harmonic. Therefore, for the fifth harmonic, there would be four internal nodes. Hence, statement E is true.
To summarize:
A) False: Increasing tension in a string leads to a higher frequency of the fundamental vibration.
B) True: The wavelength of the next higher mode is half that of the fundamental vibration.
C) False: The frequency would correspond to the second harmonic, not the fourth harmonic.
D) False: Without changes in length, tension, or material, the frequency of the standing wave remains constant.
E) True: A standing wave with a frequency 5 times the fundamental has four internal nodes.
Remember, understanding the underlying concepts behind these statements will help you accurately solve the equations and answer the related questions.