Question

Два одинаково направленных гармонических колебания одинакового периода Т1=Т2=2с с амплитудами А1=4см и А2=8см имеют разность фаз 45°. Определить амплитуду и начальную фазу результирующего колебания

Answer 1

A=sqrt(A1^2+A2^2+2*A1*A2*cosdф)=sqrt(4^2+8^2+2*4*8*0,71)=11,2 см
ф0=arctg(A2/A1)=arctg(8/4)=63 градуса

Answer 2

Calculation of Resultant Oscillation

To determine the amplitude and initial phase of the resultant oscillation, we can use the concept of vector addition. The two harmonic oscillations can be represented by vectors in the complex plane, where the real part represents the displacement and the imaginary part represents the phase.

Let's denote the first oscillation as A1 with an amplitude of 4 cm and the second oscillation as A2 with an amplitude of 8 cm. The phase difference between them is given as 45°.

We can represent these oscillations as complex numbers:

A1 = 4 * e^(i * φ1) A2 = 8 * e^(i * φ2)

where φ1 and φ2 are the initial phases of the respective oscillations.

To find the resultant oscillation, we can add these complex numbers:

A = A1 + A2

Calculation of Amplitude

The amplitude of the resultant oscillation can be found by taking the magnitude of the complex number A:

|A| = |A1 + A2|

Using the formula for the magnitude of a complex number:

|A| = sqrt((Re(A))^2 + (Im(A))^2)

where Re(A) and Im(A) are the real and imaginary parts of A respectively.

Calculation of Initial Phase

The initial phase of the resultant oscillation can be found by taking the argument of the complex number A:

arg(A) = arg(A1 + A2)

Using the formula for the argument of a complex number:

arg(A) = atan2(Im(A), Re(A))

where atan2(y, x) is the four-quadrant inverse tangent function.

Calculation

Let's calculate the amplitude and initial phase of the resultant oscillation using the given values:

A1 = 4 * e^(i * φ1) A2 = 8 * e^(i * φ2)

Given: A1 = 4 cm A2 = 8 cm Phase difference = 45°

To find the resultant oscillation, we can add these complex numbers:

A = A1 + A2

Now, let's calculate the amplitude and initial phase of the resultant oscillation using the formulas mentioned above.

Solution

The amplitude of the resultant oscillation is given by:

|A| = |A1 + A2|

The initial phase of the resultant oscillation is given by:

arg(A) = arg(A1 + A2)

Using the given values, we can calculate the amplitude and initial phase of the resultant oscillation.