Two oscillations of the same direction are added and

Consider two oscillations: x₁ = 2sin(nt) and x₂ = sin(n(t + 0.5)), where t is time in seconds, and x₁ and x₂ - vibration lengths in centimeters.

To find the amplitude and initial phase of the resulting oscillation, we add these functions. To do this, we use the formula for adding functions sin(a + b) = sin(a)cos(b) + cos(a)sin(b):

x = x₁ + x₂ = 2sin(pt) + sin(n(t + 0.5)) =

= 2sin(pt) + sin(pt)cos(0.5p) + cos(pt)sin(0.5p) =

= sin(pt)(2 + cos(0.5p)) + cos(pt)sin(0.5p)

Thus, the equation of the resulting oscillation has the form:

x = Asin(пt + φ), where

A = √((2 + cos(0.5п))² + sin²(0.5p)) ≈ 2.19 - amplitude of the resulting vibration in centimeters;

φ = arctg(sin(0.5p)/(2 + cos(0.5p))) ≈ -0.25 - the initial phase of the resulting oscillation in radians.

The Fluctuations collection is a digital product presented in the digital goods store. This collection includes two vibrations that are added together to form the resulting vibration. Both vibrations have the same direction and period, and are described by mathematical functions.

A beautiful HTML code was used to design the product page, which allows you to visually present mathematical formulas and graphs of fluctuations. The product page provides equations for each of the vibrations, as well as a formula for the resulting vibration. In addition, the page indicates the values ​​of the amplitude and initial phase of the resulting oscillation, which can be used to study this phenomenon in more detail.

The Oscillations collection is an excellent choice for those interested in physics, mathematics and science in general. This digital product can be useful for both educational purposes and scientific research.

The "Oscillations" collection is a digital product that includes two oscillations of the same direction and period: x1=2sinpt and x2 = sinp(t + 0.5) (length in centimeters, time in seconds). To determine the amplitude and initial phase of the resulting oscillation, it is necessary to add these functions.

The addition of functions is carried out according to the addition formula sin(a + b) = sin(a)cos(b) + cos(a)sin(b):

x = x1 + x2 = 2sinпt + sinп(t + 0.5) = 2sinпt + sinпtcos(0.5п) + cosпtsin(0.5п) = sinпt(2 + cos(0.5п)) + cosпt sin(0.5п)

Thus, the equation of the resulting oscillation has the form:

x = Asin(пt + φ),

Where

A = √((2 + cos(0.5p))2 + sin2(0.5p)) ≈ 2.19 - amplitude of the resulting vibration in centimeters;

φ = arctg(sin(0.5p)/(2 + cos(0.5p))) ≈ -0.25 - the initial phase of the resulting oscillation in radians.

Thus, the equation for the resulting oscillation will be:

x = 2.19sin(пt - 0.25)

Such a resulting vibration can be interesting for the study of physics and mathematics, and can be used for educational purposes or for scientific research.

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This product is a description of problem No. 40229, related to finding the amplitude and initial phase of the resulting oscillation, which is obtained by adding two oscillations of the same direction and period: x1=2sinpt and x2 = sinp(t + 0.5).

To solve the problem, the laws of harmonic vibrations and the principle of addition of vibrations are used. The amplitude A and the initial phase of the resulting oscillation are found using the appropriate formulas.

The result of solving the problem is the equation of the resulting oscillation and the values ​​of the amplitude and initial phase.

A detailed solution to the problem can be found in the relevant textbooks and workbooks on physics. If you have additional questions about solving the problem, I am ready to help you solve them.

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