The amplitude of velocity of a particle is given by, `V_(m)=V_(0)//(a omega^(2)-b omega+c)` where `V_(0)`, a, b and c are positive :
The condition for a single resonant frequency is

Options

A. `b^(2)lt 4 ac`
B. `b^(2)=4ac`
C. `b^(2)=5ac`
D. `b^(2)=7ac`

Explanation

To find the condition for a single resonant frequency for the given amplitude of velocity of a particle, we start with the expression:

\[ V_m = \frac{V_0}{a \omega^2 - b \omega + c} \]

where \( V_0 \), \( a \), \( b \), and \( c \) are positive constants.

### Step 1: Identify the condition for resonance
For resonance to occur, the amplitude of velocity \( V_m \) must become infinite. This happens when the denominator of the expression becomes zero:

\[ a \omega^2 - b \omega + c = 0 \]

### Step 2: Analyze the quadratic equation
The equation \( a \omega^2 - b \omega + c = 0 \) is a quadratic equation in terms of \( \omega \). For a quadratic equation \( Ax^2 + Bx + C = 0 \), the condition for there to be a single (repeated) root is given by the discriminant being equal to zero.

### Step 3: Calculate the discriminant
The discriminant \( D \) of the quadratic equation \( a \omega^2 - b \omega + c = 0 \) is given by:

\[ D = B^2 - 4AC = (-b)^2 - 4ac = b^2 - 4ac \]

### Step 4: Set the discriminant to zero
For there to be a single resonant frequency, we set the discriminant equal to zero:

\[ b^2 - 4ac = 0 \]

### Step 5: Rearranging the condition
Rearranging this gives us the condition for a single resonant frequency:

\[ b^2 = 4ac \]

### Conclusion
Thus, the condition for a single resonant frequency is:

\[ b^2 = 4ac \]

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