Question

Материальная точка совершает колебания в соответствии с уравнением: x(t) = 0,5sin( πt + 1 ) Определить амплитуду, период, частоту, циклическую частоту и начальную фазу колебаний. Хотелось бы увидеть подробное решение

Answer 1

X(t)=xm*sin*(w*t+ф0)
x(t)=0,5*sin(pi*t+1)
После сравнения:
xm=0,5 м    w=pi рад/с  ф0=1
v=w/2*pi=pi/2*pi=0,5 Гц
T=1/v=1/0,5=2 c

Answer 2

Mathematical Analysis of the Oscillation Equation

To determine the amplitude, period, frequency, angular frequency, and initial phase of the oscillations described by the equation x(t) = 0.5sin(πt + 1), we can analyze the equation step by step.

1. Amplitude (A): The amplitude of an oscillation represents the maximum displacement from the equilibrium position. In this equation, the coefficient of the sine function, 0.5, represents the amplitude. Therefore, the amplitude is 0.5.

2. Period (T): The period of an oscillation is the time it takes for one complete cycle. It can be calculated using the formula T = 2π/ω, where ω is the angular frequency. In this equation, the angular frequency is π, so the period can be calculated as T = 2π/π = 2.

3. Frequency (f): The frequency of an oscillation is the number of cycles per unit time. It is the reciprocal of the period and can be calculated using the formula f = 1/T. In this case, the period is 2, so the frequency is f = 1/2 = 0.5.

4. Angular Frequency (ω): The angular frequency is a measure of how quickly the oscillation completes one cycle. It is related to the frequency through the formula ω = 2πf. In this equation, the frequency is 0.5, so the angular frequency can be calculated as ω = 2π(0.5) = π.

5. Initial Phase (φ): The initial phase represents the starting point of the oscillation. In this equation, the constant term 1 represents the initial phase. Therefore, the initial phase is 1.

Summary of the Oscillation Parameters

Based on the given equation x(t) = 0.5sin(πt + 1), we can summarize the parameters as follows:

- Amplitude (A): 0.5 - Period (T): 2 - Frequency (f): 0.5 - Angular Frequency (ω): π - Initial Phase (φ): 1

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