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(B) The book will lose contact with the shelf when the downward acceleration of the shelf exceeds the acceleration due to gravity $g$.The condition fo

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$3.18$

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(B) The book will lose contact with the shelf when the downward acceleration of the shelf exceeds the acceleration due to gravity $g$.The condition for losing contact is $a_{\max} \geq g$.Since the shelf undergoes simple harmonic motion,the maximum acceleration is given by $a_{\max} = \omega^2 A$,where $\omega$ is the angular frequency and $A$ is the amplitude.Setting $a_{\max} = g$,we get $\omega^2 A = g$,which implies $\omega = \sqrt{\frac{g}{A}}$.Given $A = 2.5 \,cm = 0.025 \,m$ and $g = 10 \,m/s^2$,we have $\omega = \sqrt{\frac{10}{0.025}} = \sqrt{400} = 20 \,rad/s$.The frequency $f$ is given by $f = \frac{\omega}{2\pi} = \frac{20}{2\pi} = \frac{10}{\pi} \approx 3.18 \,Hz$.

$A$ book is resting on a shelf that is undergoing vertical simple harmonic oscillations with an amplitude of $2.5 \,cm$. What is the minimum frequency of oscillation of the shelf for which the book will lose contact with the shelf (in $,Hz$)? (Assume that $g = 10 \,m/s^2$)

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