A forced oscillator is acted by an external periodic force. The amplitude of oscillation is 55/(2p^2 -36p +9)^1/2. 
Find the resonant angular frequency of oscillation.

To find the resonant angular frequency of oscillation, we need to analyze the amplitude formula and identify the conditions that maximize the amplitude.

Given the amplitude equation:
\[
A = \frac{55}{\sqrt{2p^2 - 36p + 9}}
\]
where \( p \) represents the angular frequency of the oscillator.

### Step 1: Maximizing the amplitude
The resonant frequency occurs when the amplitude is maximized. To achieve maximum amplitude, the denominator \( \sqrt{2p^2 - 36p + 9} \) should be minimized, because the amplitude \( A \) is inversely proportional to this term.

Thus, the goal is to minimize the quadratic expression:
\[
f(p) = 2p^2 - 36p + 9
\]

### Step 2: Finding the minimum of the quadratic expression
The minimum of a quadratic function \( ax^2 + bx + c \) occurs at \( p = -\frac{b}{2a} \). For our function \( f(p) = 2p^2 - 36p + 9 \), we have:

- \( a = 2 \)
- \( b = -36 \)
- \( c = 9 \)

The value of \( p \) that minimizes the function is:
\[
p = -\frac{-36}{2(2)} = \frac{36}{4} = 9
\]

### Step 3: Conclusion
The resonant angular frequency of oscillation is \( p = 9 \).

Would you like more details on this process? Here are 5 related questions for deeper understanding:

1. How is resonance defined in terms of mechanical oscillations?
2. Can we analyze the phase shift at resonance in forced oscillators?
3. What is the role of damping in resonance phenomena?
4. How would the amplitude change if the external force was varied?
5. Could you derive this amplitude expression using differential equations?

**Tip:** Resonance occurs when the driving frequency matches the system's natural frequency, leading to maximum energy transfer and amplitude.

A forced oscillator is acted by an external periodic force. The amplitude of oscillation is 55/(2p^2 - 36p + 9)^1/2. Find the resonant angular frequency of oscillation.

Explore the process of finding the resonant angular frequency of a forced oscillator. Learn how to maximize amplitude by minimizing a quadratic function, using key physics and algebraic concepts. This problem involves oscillations, resonance, and quadratic minimization, suitable for advanced high school or undergraduate students.

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