Question

Statement (I): Resonance is a special case of forced vibration in which the natural frequency of the body is the same as the impressed frequency of the external periodic force whereby the amplitude of the forced vibration peaks sharply.

Statement (II): The amplitude of forced vibration of a body increases with an increase in the frequency of the externally impressed periodic force.
1. Both Statement (I) and Statement (II) are individually true and Statement (II) is the correct explanation of Statement (I)
2. Both Statement (I) and Statement (II) are individually true but Statement (II) is not the correct explanation of Statement (I)
3. Statement (I) is true but Statement (II) is false
4. Statement (I) is false but Statement (II) is true

Answer 1

Correct Answer - Option 3 : Statement (I) is true but Statement (II) is false

In case of forced vibration, the amplitude of the steady-state response is given by-

\(A = \frac{{{F_o}/s}}{{\sqrt {\left\{ {1 - {{\left( {\frac{\omega }{{{\omega _n}}}} \right)}^2}} \right\} + {{\left( {\frac{{2\xi \omega }}{{{\omega _n}}}} \right)}^2}} }}\)

Where, s = stiffness of the equivalent spring, m = equivalent mass, F₀ = amplitude of the applied force

ω = frequency of the applied force, ω_n = natural frequency of the system, ξ = damping factor.

Resonance is a special case of forced vibration. When the frequency of an externally applied periodic force on a body is equal to its natural frequency, the body starts vibrating with an increased amplitude as seen from the equation stated above. So, the statement I) is correct.

It can also be noted from the above equation that if the frequency of the applied force increases, the amplitude of the steady-state response will reduce. So, statement II) is false.