question_answer

A particle of mass m is attached to a spring (of spring constant\[k\] and has a natural angular frequency\[{{\omega }_{0}}\]. An external force F(t) proportional to \[\cos \omega t(\omega \ne {{\omega }_{0}})\]is applied to the oscillator. The time displacement of the oscillator will be proportional to [AIEEE 2004]

A)  \[\frac{m}{\omega _{0}^{2}-{{\omega }^{2}}}\]        

B)       \[\frac{1}{m(\omega _{0}^{2}-{{\omega }^{2}})}\]

C)  \[\frac{1}{m(\omega _{0}^{2}+{{\omega }^{2}})}\]  

D)       \[\frac{m}{\omega _{0}^{2}+{{\omega }^{2}}}\]

Correct Answer: B

Solution :

[b] Initial angular velocity of particle

and at any instant t, angular velocity

Therefore, for a displacementthe resultant acceleration

...(i)

External force,               ...(ii)

Since,                               (given)

From Eq. (ii),

...(iii)

Now, equation of simple harmonic motion

...(iv)

At

(as)

...(v)

Hence, from Eqs. (iii) and (v). we finally get