## Question

If the amplitude of velocity of a particle acted on by a force *F* = *F*_{0} cos ω*t*along *x*-axis is given by:

Find the condition of resonance and resonant frequency.

### Solution

*b*^{2} = 4*ac*

At resonance will be max → ∞.

i.e., resonance frequency ω = (b/2a) …(2)

further substituting the value of ω from eqn. (2) in (1), we get

Which is the required condition of resonance

**Alternative solution:-**

At resonance

Now as at resonance ω is single valued, which is possible only if

Which is the required condition of resonance, and so at resonance ω = (*b*/2*a*) [substituting *b*^{2} = 4*ac* in Eqn. (1)].

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