Question

Consider a driven damped mechanical oscillator is in resonance. Which of the following statements is true?

• A. Driving frequency is twice the natural frequency of the oscillator
• B. Power transfer from the driving source to system is minimum
• C. Driving frequency is the same as the natural frequency of the oscillator
• D. The force damping the oscillations are at a minimum value
• E. The driving force is in phase with the displacement

Hint:

To remember resonance: Think of it as the "sweet spot" where the system naturally wants to vibrate. When you push a swing at its natural frequency, you get the biggest movement with the least effort because energy transfer is maximized.

Answer 1

The Correct Option is C

Concept: Resonance occurs in a driven damped harmonic oscillator when the frequency of the external driving force matches the natural frequency of the system. At this specific frequency, the amplitude of oscillations reaches its maximum value.

Step 1: {Analyze the condition for resonance.}
Resonance is defined by the condition where the driving frequency ($\omega$) is equal to or very close to the natural frequency ($\omega_0$) of the system. $$\omega = \omega_0$$

Step 2: {Evaluate the energy and power transfer at resonance.}
At resonance, the velocity of the oscillator is in phase with the driving force. This alignment ensures that the power transfer from the external source to the oscillator is at its maximum, not minimum.

Step 3: {Check the phase relationship between force and displacement.}
At resonance, the driving force leads the displacement by exactly $\pi/2$ radians (90°). Therefore, they are not in phase; rather, the force is in phase with the velocity.

Step 4: {Confirm the correct statement.}
Based on the physical definition of resonance in a mechanical system, the statement that the driving frequency is the same as the natural frequency is the fundamental characteristic of the state of resonance.