Two Superposed Vibrations of Equal Frequency

Similarly, the difference between the second and third is 30 degrees so it will effect it as per the phase its in at that time. Hence, the amplitude of the resultant displacement is 0.3 mm and its phase relative to the first component is 60 degrees.In summary, the problem involves a particle undergoing three simple harmonic motions in the x direction with amplitudes of 0.25, 0.20, and 0.15 mm respectively. With phase differences of 45 and 30 between the first and second, and second and third motions, the amplitude of the resultant displacement is 0.3 mm and its phase relative to the first component is 60 degrees. The equations used for two vector systems may
  • #1
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The Problem
A particle is simultaneously subjected to three simple harmonic motions, all of the same frequency and in the x direction. If the amplitudes are 0.25, .20, and 0.15 mm, respectively, and the phase difference between the first and second is 45, and between the second and third is 30, find the amplitude of the resultant displacement and its phase relative to the first (0.25 mm amplitude) component.

I drew a diagram and of the vectors but the equations I have are all for two vectors systems. How do I start this problem off?
 
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  • #2
Start by assuming the 0.25mm wave is fully dispacing the particle, so its moving it 0.25mm from equilibrium position. the difference between the first and the second is 45 degrees so it will effect it as per the phase its in at that time.
 
  • #3


As a scientist, the first step in solving this problem would be to use vector addition to find the resultant displacement of the particle. This can be done by breaking down each of the three harmonic motions into their x and y components, and then adding them together. The amplitude of the resultant displacement can then be found using the Pythagorean theorem.

To find the phase of the resultant displacement relative to the first component, we can use trigonometric functions to calculate the angle between the resultant displacement vector and the first component vector.

The equations used for two vector systems can still be applied in this scenario, as we can consider the first two motions as one combined motion and then add the third motion to it. Alternatively, we can also use the principle of superposition to add the individual displacements caused by each of the three motions.

In summary, to solve this problem, we would first use vector addition to find the resultant displacement and then use trigonometric functions to calculate its phase relative to the first component. By breaking down the problem into smaller components and using mathematical principles, we can successfully solve this problem involving three simultaneous simple harmonic motions.
 

1. What is the concept of "Two Superposed Vibrations of Equal Frequency"?

Two superposed vibrations of equal frequency refers to the phenomenon where two waves with the same frequency overlap and interact with each other. This results in a new wave that has a different amplitude and phase compared to the original waves.

2. What is the principle behind superposition in this concept?

The principle of superposition states that when two or more waves meet, the resultant displacement at any point is equal to the sum of the individual displacements caused by each wave. This is applicable to waves of any kind, including sound, light, and mechanical waves.

3. How do the amplitudes of two superposed vibrations affect the resulting wave?

If the amplitudes of the two superposed vibrations are equal, the resulting wave will have an amplitude that is twice that of the individual waves. If the amplitudes are unequal, the resulting wave will have an amplitude that lies between the two individual amplitudes.

4. Can two superposed vibrations of different frequencies form a standing wave?

No, in order to form a standing wave, the two superposed vibrations must have the same frequency. This is because standing waves are formed when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other.

5. What are some real-life applications of the concept of "Two Superposed Vibrations of Equal Frequency"?

This concept has many practical applications, such as in noise cancellation technology, where sound waves with equal frequency but opposite phases are used to cancel out unwanted noise. It is also used in musical instruments, where the superposition of multiple sound waves creates a harmonious sound. Additionally, this concept is important in understanding the behavior of waves in various fields of science, such as optics and acoustics.

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