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(B) The body will not slip if the maximum static frictional force is greater than or equal to the force required to provide the maximum acceleration o

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$\mu=0.075$

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(B) The body will not slip if the maximum static frictional force is greater than or equal to the force required to provide the maximum acceleration of the board.$F_{	ext{friction, max}} \ge m a_{\max}$$\mu m g \ge m \omega^2 A$$\mu \ge \frac{\omega^2 A}{g}$Given,amplitude $A = 0.3 \ m$,period $T = 4 \ s$,and taking $g = 10 \ m/s^2$.The angular frequency $\omega = \frac{2 \pi}{T} = \frac{2 \pi}{4} = \frac{\pi}{2} \ rad/s$.Substituting the values:$\mu = \frac{(\frac{\pi}{2})^2 	imes 0.3}{10} = \frac{\frac{\pi^2}{4} 	imes 0.3}{10} \approx \frac{9.8696 	imes 0.3}{40} \approx \frac{2.96088}{40} \approx 0.07402$.Using $\pi^2 \approx 10$,we get $\mu = \frac{10 	imes 0.3}{4 	imes 10} = \frac{0.3}{4} = 0.075$.

$A$ horizontal board is performing simple harmonic oscillations horizontally with an amplitude of $0.3 \ m$ and a period of $4 \ s$. What is the minimum coefficient of friction between a heavy body placed on the board and the board itself if the body is not to slip?

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