Question

Складываются два гармонических колебания, совпадающие по направлению и выражаемые уравнениями х1 = sin πt; х2 = соs πt.Определить амплитуду и начальную фазу результирующего коле- бания, написать его уравнение и дать векторную диаграмму сложе- ния амплитуд.

Answer 1

Воспользуемся формулой sin(al)+cos(al)=1.414*cos(0.25*pi-al), теперь для нашего случая sin(pi*t)+cos(pi*t)=1.414*cos(0.25*pi-pi*t). Амплитуда суммарного колебания равна 1,414, начальная фаза при t=0 равна 0,25*pi .

Answer 2

Resulting Amplitude and Initial Phase of the Combined Oscillation:

The resulting amplitude and initial phase of the combined oscillation can be determined by adding the individual oscillations together. Given the equations for the two harmonic oscillations: - \( x_1 = \sin(\pi t) \) - \( x_2 = \cos(\pi t) \)

The resulting amplitude and initial phase can be calculated using the following formulas: - Amplitude: \( A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\delta)} \), where \( A_1 \) and \( A_2 \) are the amplitudes of the individual oscillations, and \( \delta \) is the phase difference between them. - Initial phase: \( \phi = \arctan\left(\frac{A_1\sin(\phi_1) + A_2\sin(\phi_2)}{A_1\cos(\phi_1) + A_2\cos(\phi_2)}\right) \), where \( \phi_1 \) and \( \phi_2 \) are the initial phases of the individual oscillations.

Let's calculate the resulting amplitude and initial phase using the given equations.

Resulting Amplitude Calculation:

Using the given equations, the amplitudes of the individual oscillations are: - \( A_1 = 1 \) (for \( x_1 = \sin(\pi t) \)) - \( A_2 = 1 \) (for \( x_2 = \cos(\pi t) \))

Substituting these values into the amplitude formula: - \( A = \sqrt{1^2 + 1^2 + 2(1)(1)\cos(\delta)} \) - \( A = \sqrt{2 + 2\cos(\delta)} \)

Initial Phase Calculation:

The initial phases of the individual oscillations are: - \( \phi_1 = 0 \) (for \( x_1 = \sin(\pi t) \)) - \( \phi_2 = \frac{\pi}{2} \) (for \( x_2 = \cos(\pi t) \))

Substituting these values into the initial phase formula: - \( \phi = \arctan\left(\frac{1\sin(0) + 1\sin\left(\frac{\pi}{2}\right)}{1\cos(0) + 1\cos\left(\frac{\pi}{2}\right)}\right) \) - \( \phi = \arctan\left(\frac{0 + 1}{1 + 0}\right) \) - \( \phi = \arctan(1) \) - \( \phi = \frac{\pi}{4} \)

Resulting Amplitude and Initial Phase:

- Resulting Amplitude (A): \( A = \sqrt{2 + 2\cos(\delta)} \) - Initial Phase (\( \phi \)): \( \phi = \frac{\pi}{4} \)

Equation of the Resulting Oscillation:

The equation of the resulting oscillation can be expressed as: - \( x = A\sin(\pi t + \phi) \) - Substituting the values of \( A \) and \( \phi \): - \( x = \sqrt{2 + 2\cos(\delta)}\sin(\pi t + \frac{\pi}{4}) \)

Vector Diagram of Amplitude Addition:

The vector diagram for the addition of amplitudes can be represented graphically to illustrate the resulting amplitude and phase.

This information provides a comprehensive understanding of the combined oscillation and its characteristics based on the given harmonic oscillations.