Question

Assertion: Resonance is a special case of forced vibration in which the natural frequency of vibration of the body is the same as the impressed frequency of external periodic force and the amplitude of forced vibration is maximum.

Reason: The amplitude of forced vibrations of a body increases with an increase in the frequency of the externally impressed periodic force.

• A. Both assertion and reason are incorrect
• B. The assertion is correct but the reason is incorrect
• C. Both assertion and reason are correct and the reason is the correct explanation for assertion.
• D. Both assertion and reason are correct but the reason is not the correct explanation for assertion.

Answer 1

The correct option is B

The assertion is correct but the reason is incorrect

Explanation for assertion statement:

1. In forced vibration, the amplitude of the steady-state response is, $\mathrm{A}=\frac{{\mathrm{F}}_{{\displaystyle \circ }}}{\sqrt{{\left\{1-{\left({\displaystyle \frac{\mathrm{\omega }}{{\mathrm{\omega }}_{\mathrm{n}}}}\right)}^2\right\}}^2+{\left({\displaystyle \frac{2\mathrm{\xi \omega }}{{\mathrm{\omega }}_{\mathrm{n}}}}\right)}^2}}$ -------- $\left(1\right)$ Here, $\mathrm{s}$ is the stiffness of the equivalent spring, $\mathrm{m}$ is equivalent mass, ${\mathrm{F}}_{\circ }$ is the amplitude of the applied force, $\mathrm{\omega }$ is the frequency of the applied force, ${\mathrm{\omega }}_{\mathrm{n}}$ is the natural frequency of the system, and $\mathrm{\xi }$ damping factor.
2. So, the resonance is a special case of forced vibration.

Explanation for reason statement:When the frequency of an externally applied periodic force on a body or object is equal to its natural frequency, from the above equation $\left(1\right)$, the object begins vibrating with an increased amplitude. So the assertion is correct.

1. . While, from the equation $\left(1\right)$, if the frequency of the applied force rises, the amplitude of the steady-state response will fall. So, the reason is incorrect.

Hence, option $\mathrm{B}$ is the correct answer.