Question

A forced oscillator is acted upon by a force $F=F_{0}sin\omega t.$ The amplitude of oscillation is given by $\dfrac{55}{\sqrt(2\omega^{2}-36\omega+9)}$. The resonant angular frequency is
\[\begin{align}
  & \text{A}\text{.2 units} \\
 & \text{B}\text{.9 units} \\
 & \text{C}\text{.18 units} \\
 & \text{D}\text{.36 units} \\
\end{align}\]

Answer 1

Hint:
We know that oscillations are to and fro motion along the mean position. And are examples of simple harmonic motions or SHM. An oscillation is said to be forced, if it experiences an external force, which is either compressing or expanding the object. Here, $F=F_{0}sin\omega t$ is the external force.

Formula used:
$\dfrac{dA}{d\omega}=0$

Complete answer:
The oscillations are produced but to the restoring force, which opposes the motion, due to the external force.
Resonance occurs, when the natural frequency of the body matches with the frequency of the external force. Resonance frequency is achieved, when the amplitude of the external frequency is maximum .i.e. $\dfrac{dA}{d\omega}=0$
There are various ways in which resonance is obtained, like mechanical, electrical, optical, orbital and atomic resonance. Where the type of system and the external force applied to the system is different in each case.
Given that, the force acting on the oscillator is $F=F_{0}sin\omega t$, also the amplitude of oscillation is given by $A=\dfrac{55}{\sqrt(2\omega^{2}-36\omega+9)}$. Then we know that the resonant frequency is obtained when $A$ is maximum.
Since,$\dfrac{dA}{d\omega}=0$, we can differentiate, $A$ with respect to $\omega$ we get, $\dfrac{d\left(\dfrac{55}{\sqrt(2\omega^{2}-36\omega+9)}\right)}{dw}=\dfrac{4\omega-36}{(2\omega^{2}-36\omega+9)^{\dfrac{3}{2}}}$
As we know, $\dfrac{d}{dx}x^{n}=x^{n+1}$. Here, $n=-\dfrac{1}{2}$
For, the resonant frequency, $\omega_{0}$, the $\dfrac{dA}{d\omega}=0$
Then $\dfrac{4\omega-36}{(2\omega^{2}-36\omega+9)^{\dfrac{3}{2}}}=0$
$\Rightarrow 4\omega_{0}-36=0$
$\Rightarrow \omega_{0}=\dfrac{36}{4}$
$\Rightarrow \omega_{0}=9$
Hence, the resonant frequency is \[\text{9 units}\]

Then the answer is \[\text{B}\text{.9 units}\]

Note:
This sum may seem hard in the first look, but is very easy, if one knows differentiation. At the resonant frequency, when an external force is applied. the body oscillates at a higher amplitude, as compared to when the same force is applied to when, the resonance frequency is not reached.