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 .......... $Hz$ the book will lose contact with the shelf? (Assume that, $g=10 \,ms ^{-2}$ )
KVPY 2010, Medium
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(b)
Book started to loose contact with shelf when maximum acceleration $a$ of shelf is such that,
$a_{\max } \geq g$
$\Rightarrow \quad \omega^2 A \geq g$
$\Rightarrow \quad \omega^2=\frac{g}{A}$.
So, frequency of oscillations is
$f=\frac{\omega}{2 \pi}=\frac{1}{2 \pi} \sqrt{\frac{g}{A}}$
$\Rightarrow \quad f=\frac{1}{2 \pi} \cdot \sqrt{2.5 \times 10^{-2}}=\frac{2 \times 10}{2 \pi}$
$\Rightarrow \quad f \approx 3.18 \,Hz$
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