A material point participates simultaneously in two harmonics

A material point participates in two harmonic oscillations along one straight line. The corresponding vibration equations are written in SI units as x1=0.1cosP*t/2 and x2=0.12cosP(t+1)/2. It is necessary to determine the equation of the resulting oscillations.

Task 40717. Answer:

The equations of oscillations of a material point are written in the form:

x1=0,1cosП*t/2

x2=0,12cosП(t+1)/2

Here x1 and x2 are the amplitudes of oscillations, P is the period of oscillations, t is time.

To determine the resulting oscillations, it is necessary to add the equations x1 and x2:

x=x1+x2=0,1cosП*t/2+0,12cosП(t+1)/2

x=0,1cosП*t/2+0,12cos(Пt/2+П/2)

Using the formula for the sum of two cosines, we get:

x=0,1cosПt/2+0,12cosП/2cosПt/2-0,12sinП/2*sinПt/2

x=0,1cosП*t/2+0,12sin(Пt/2+П/6)

Thus, the equation of the resulting oscillations is written as:

x=0,1cosП*t/2+0,12sin(Пt/2+П/6)

Answer: x=0.1cosP*t/2+0.12sin(Pt/2+P/6).

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Product description: A digital product containing a detailed solution to problem No. 40717 in physics, which describes the oscillations of a material point participating simultaneously in two harmonic oscillations along one straight line. The solution is presented in a beautiful html format, which will allow you to quickly and easily understand the solution to the problem. The solution contains a brief record of the conditions, formulas and laws used in the solution, the derivation of the calculation formula and the final answer. If any questions arise regarding the solution, the buyer can ask for help.

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This product is problem 40717, which requires determining the equation of the resulting oscillations of a material point participating simultaneously in two harmonic oscillations. The equations of the vibration terms are given in the form x1=0.1cosP*t/2 and x2=0.12cosP(t+1)/2 in SI units.

To find the equation of the resulting oscillations, it is necessary to add the equations of these harmonic oscillations. To do this, you can use the formula for adding harmonic vibrations, which reads:

Acos(ωt+φ) + Bcos(ωt+φ) = C*cos(ωt+φ),

where A and B are the amplitudes of oscillations, ω is the angular frequency, t is time, φ is the initial phase, C is the amplitude of the resulting oscillation.

Applying this formula to the equations of vibration terms, we obtain:

x1 = 0.1cos(Пt/2) x2 = 0.12cos(П(t+1)/2)

x1 and x2 have the same initial phase, equal to zero. Then the sum of the vibrations will have an amplitude equal to the root of the sum of the squares of the amplitudes of the components of the vibrations:

C = √(A^2 + B^2)

The angular frequency of the resulting vibration can be found using the expression:

ω = (ω1 + ω2) / 2

where ω1 and ω2 are the angular frequencies of the vibration components.

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

x = √(0.1^2 + 0.12^2) * cos(Пt/2 + arctan(0.12/0.1)/2)

where arctan(0.12/0.1) is the arctangent of the ratio of the amplitudes of the vibration components.

Answer: x = 0.16cos(Pt/2 + 0.1095) in SI units.

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