Question

The angular frequency of the damped oscillator is given by $\omega = \sqrt {\left( {\dfrac{k}{m} - \dfrac{{{r^2}}}{{4{m^2}}}} \right)}$ where k is the spring constant, m is the mass of the oscillator and r is the damping constant. If the ratio $\dfrac{{{r^2}}}{{mk}}$ is 80%, the change in time period compared to the undamped oscillator is approximately as follows:
A. Increases by8%
B. Decreases by 8%
C. Increases by 1%
D. Decreases by 1%

Answer 1

in this question we are already given the relation between spring constant k, damping constant r, mass of the oscillator m with angular frequency. We will use this relation to find out the change in the time period compared to the undamped oscillator. Further we will study about the basics of oscillators and its types.

Formula used:
$\omega = \sqrt {\left( {\dfrac{k}{m} - \dfrac{{{r^2}}}{{4{m^2}}}} \right)}$

Complete answer:
As, we know that an undamped oscillation can be defined as the electrical oscillation whose amplitude remains constant with time or does not change with time.
Here, we have a relation between spring constant k, damping constant r, mass of the oscillator m with angular frequency as:
\eqalign{ & \omega = \sqrt {\left( {\dfrac{k}{m} - \dfrac{{{r^2}}}{{4{m^2}}}} \right)} \cr & \Rightarrow \omega = \sqrt {\dfrac{k}{m}} \sqrt {1 - \dfrac{{{r^2}}}{{4mk}}} \cr}
\eqalign{ & \Rightarrow \omega \approx {\omega _0}\left( {1 - \dfrac{{{r^2}}}{{8mk}}} \right) \cr & \therefore \omega \approx (1 - 1\% ) \cr}
Therefore, the correct option is D) i.e., the time period decreases by 1% approximately as compared to the undamped oscillator.

Additional information:
From the basics of an oscillator we know that an oscillator can be defined as a circuit which produces a continuous, repeated, alternating waveform without any input. We can say that oscillators basically convert unidirectional current flow from a DC or direct current source into an alternating waveform which can be of required frequency, which is already decided by its circuit components.
Further, there are many different types of oscillators, like crystal Oscillator, SAW oscillators, MEMS oscillators, voltage- controlled oscillators, as per the requirement or applications.
When an oscillator generates a sinusoidal waveform of the desired frequency it is said to be a sinusoidal oscillator. Here, there are two different types of sinusoidal oscillators: damped and undamped oscillators.
Damped oscillations are defined as the electrical oscillations whose amplitude goes on decreasing with time. Oscillator which generates these oscillations i.e., damped oscillations, loses some energy during each oscillation. Damped oscillator loses its energy because it has no means to compensate for the losses and as a result the amplitude of the generated wave decreases gradually.
We should know that the frequency of oscillations remains unchanged since it depends upon the constant of the electrical system.
Now, an undamped oscillation can be defined as the electrical oscillation whose amplitude remains constant with time or does not change with time. Although the electrical system in which these oscillations are being generated also has losses, it generates the right amount of energy which is being supplied to overcome the losses. So, the amplitude of the generated wave in this case remains constant.

Note:
Here, we should remember that in damped oscillations, the amplitude of the generated wave gradually decreases with time, whereas in case of an undamped oscillation the amplitude of the generated wave does not change with time. Further, an oscillator is mainly required to produce undamped oscillations for utilizing various electronics equipment.