Question

Amplitude of a wave is represented by $A=\dfrac{c}{a+b-c}.$ Then, the resonance will occur when
$\begin{align}
  & \text{A}\text{. }b=-\dfrac{c}{2} \\
 & \text{B}\text{. }b=0\text{ and }c=0 \\
 & \text{C}\text{. }b=-\dfrac{a}{2} \\
 & \text{D}\text{. none of the above} \\
\end{align}$

Answer 1

Hint: When we apply an external force on a vibrating object to oscillate in its natural frequency resonance occurs. For resonance the amplitude of the oscillation should be very large or infinite. Find the value of the quantities which will follow the above condition for resonance.

Complete answer: or Complete step by step answer:
All systems which are oscillating have a natural frequency. Resonance can be defined as the phenomenon when we force an oscillating system to vibrate with natural frequency by applying some external force periodically.

The wave is represented as, $A=\dfrac{c}{a+b-c}$
For the resonance to occur, the amplitude of the wave should be very high. Taking the amplitude as infinity, we can say that the resonance will occur at infinite amplitude. Mathematically we can write,

$\begin{align}
  & A\to \propto \\
 & \dfrac{c}{a+b-c}\to \propto \\
\end{align}$

For the above term to be infinite the denominator of the term should be zero.
So, we can write,
$a+b-c=0$
Again, the term c in the numerator cannot be zero.
If we put $b=-c/2$ , the value of the term will not be infinite. Again, if we put $b=-a/2$ , the value of the term will not be infinite.
When we put $b=0$ and $a=c$
$\dfrac{c}{a+b-c}=\dfrac{a}{a+0-a}=\dfrac{a}{0}=\propto $
For this value, the amplitude will be infinite. So, the resonance will occur only if $b=0$ and $a=c$.

So, the correct answer is “Option B”.

Note:
We are applying external force on the system i.e. we are doing work on the system. According to the work energy theorem, the work done on the system will raise the energy level of the system. We can simply define it as the transform of potential energy to kinetic energy and kinetic energy to potential energy.

But this energy cannot change the natural frequency of the system. This extra energy on the system due to the work done will raise the amplitude of oscillation. This gives the condition for resonance of an oscillation.