In a system under the influence of an external harmonic force, the velocity amplitude of a particle is related to the frequency of the external force through resonance.
The resonance frequency occurs when the frequency of the external force matches the natural frequency of the system. In this case, the velocity amplitude is maximum.
Given that the velocity amplitude is equal to half the maximum value at frequencies w1 and w2, we can set up the following equation:
A(w1) = A_max/2, A(w2) = A_max/2,
where A(w1) and A(w2) are the velocity amplitudes at frequencies w1 and w2, respectively.
For a harmonic oscillator, the velocity amplitude is given by:
A(w) = (F0/m) / sqrt((w0^2 - w^2)^2 + (w * γ)^2),
where F0 is the amplitude of the external force, m is the mass of the particle, w0 is the natural frequency of the system, and γ is the damping factor.
Since the velocity amplitude is equal to half the maximum value at both w1 and w2, we have:
(F0/m) / sqrt((w0^2 - w1^2)^2 + (w1 * γ)^2) = A_max/2, (F0/m) / sqrt((w0^2 - w2^2)^2 + (w2 * γ)^2) = A_max/2.
Dividing the two equations, we can eliminate A_max/2:
sqrt((w0^2 - w1^2)^2 + (w1 * γ)^2) / sqrt((w0^2 - w2^2)^2 + (w2 * γ)^2) = 1.
Simplifying further:
(w0^2 - w1^2)^2 + (w1 * γ)^2 = (w0^2 - w2^2)^2 + (w2 * γ)^2.
Expanding and rearranging the terms:
w0^4 - 2w0^2w1^2 + w1^4 + w1^2γ^2 = w0^4 - 2w0^2w2^2 + w2^4 + w2^2γ^2.
Canceling out the common terms and simplifying:
w1^4 - w2^4 = w1^2w2^2 (w2^2 - w1^2).
Factoring the equation:
(w1^2 + w2^2)(w1^2 - w2^2) = w1^2w2^2 (w2^2 - w1^2),
(w1^2 + w2^2) = w1^2w2^2.
Now, we can solve for the frequency corresponding to the velocity resonance by rearranging the equation:
w1^2 + w2^2 = w1^2w2^2,
w1^2w2^2 - w1^2 - w2^2 = 0.
This equation is quadratic in terms of w1^2. We can solve it using the quadratic formula:
w1^2 = [1 ± sqrt(1 + 4w2^2)] / (2w2^2).
Since w1^2 is positive, we take the positive square root:
w1^2 = [1 + sqrt(1 + 4w2^2)] / (2w2^2).
Finally, the frequency corresponding to the velocity resonance is given by:
w1 = sqrt{[1 + sqrt(1 + 4w2^2)] / (2w2^2)}.
This equation provides the frequency w1 that corresponds to the velocity resonance when w2 is known.