two collinear harmonic oscillation x1=8sin(100pie(t)) and x2=12sin(96pie(t)) are superposed.calculate the value of time when the amplitude of the resultant oscillation will be *maximum and minimum?
Look mainly at the low "beat" frequency.
4sin96pit + 8(sin 100pit +sin99pit)
4 sin96pit +16 sin100pitcos4pit
saying 199/2 is about 100
that is about
4 sin 100 pi t [1+ 4cos 2 pi t]
the low beat frequency is max when
2 pi t = 0, 2pi, etc
min when 2 pi t = pi/2, pi/2 + 2 pi etc
Solution of ignou physics
To calculate the maximum and minimum amplitude of the resultant oscillation, we need to find the phase difference between the two oscillations.
Given:
x1 = 8sin(100πt)
x2 = 12sin(96πt)
The general equation for the superposition of two harmonic oscillations of the same frequency is:
x = A1sin(ωt + φ1) + A2sin(ωt + φ2)
Comparing this equation with our given values:
x1 = 8sin(100πt) = A1sin(ωt + φ1)
x2 = 12sin(96πt) = A2sin(ωt + φ2)
From the given equations, we can extract the frequency and phase angles for both oscillations:
For x1: A1 = 8, ω = 100π, φ1 = 0
For x2: A2 = 12, ω = 96π, φ2 = 0
Now, let's calculate the resultant amplitude of the superposed oscillation:
The equation for the superposition is:
x = A1sin(ωt + φ1) + A2sin(ωt + φ2)
Substituting the given values:
x = 8sin(100πt) + 12sin(96πt)
To find the maximum and minimum values, we need to find the extrema points by differentiating the equation with respect to time (t) and setting it to zero:
dx/dt = 8(100πcos(100πt)) + 12(96πcos(96πt)) = 0
To solve this equation, we need to find the values of t that satisfy it.
For the maximum amplitude, we need to find the point where the first derivative changes sign from positive to negative. This will give us the maximum value. To find the exact value of t, we need to solve the equation:
8(100πcos(100πt)) + 12(96πcos(96πt)) = 0
Similarly, for the minimum amplitude, we need to find the point where the first derivative changes sign from negative to positive.
By solving the above equations, you can find the values of t when the amplitude of the resultant oscillation is maximum and minimum.