A plank with a box on it at one end is gradually raised about the other end. As the angle of inclina

Question

A plank with a box on it at one end is gradually raised about the other end. As the angle of inclination with the horizontal reaches $45^{\circ}$, the box starts to slip and slides 5.0 m down the plank in 2.0 s. The coefficients of static and kinetic friction between the box and the plank will be (Take $g = 10 \mathrm{~m/s}^2$)

Professor PC Thomas Classes · Answer

Angle of repose $	heta = 45^{\circ} \implies \mu_s = 	an 45^{\circ} = 1.0$. For sliding, $S = \frac{1}{2}at^2 \implies 5.0 = \frac{1}{2}a(2)^2 \implies a = 2.5 \mathrm{~m/s}^2$. Acceleration $a = g(\sin 	heta - \mu_k \cos 	heta) \implies 2.5 = 10(\frac{1}{\sqrt{2}} - \mu_k \frac{1}{\sqrt{2}})$. $2.5 = \frac{10}{\sqrt{2}}(1 - \mu_k) = 5\sqrt{2}(1 - \mu_k) \approx 7.07(1 - \mu_k) \implies 1 - \mu_k = \frac{2.5}{7.07} = 0.35 \implies \mu_k = 0.65$.