question_answer

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 \[{{30}^{\text{o}}},\] the box starts to slip and slides m down the plank in 4.0s. The coefficients of static and kinetic friction between the box and the plank will be, respectively                                                                        

A)  0.6 and 0.6        

B)  0.6 and 0.5        

C)         0.5 and 0.6        

D)         0.4 and 0.3

Correct Answer: B

Solution :

Given a plank with a box on it one end is gradually raised about the end having angle of  Inclination, the box starts to slip and slides down 4 m the Plank m 4s as shown in figure,     The coefficient of static friction, \[{{\mu }_{s}}=\tan {{30}^{{}^\circ }}=\frac{1}{\sqrt{3}}=0.6\] So, distance covered by a plank,                                 \[s=ut+\frac{1}{2}a{{t}^{2}}\] Here, u= 0 and  \[a=g\,(\sin \theta -\mu \,\cos \theta )\] \[\therefore \]       \[4=\frac{1}{2}g(sin30-{{\mu }_{k}}cos30){{(4)}^{2}}\] \[\Rightarrow \,\,\,\,\,\,\,\,0.5=10\times \frac{1}{2}-{{\mu }_{x}}\times 10\times \frac{\sqrt{3}}{2}\] \[\Rightarrow \,\,\,5\sqrt{3}{{\mu }_{k}}=45\Rightarrow {{\mu }_{k}}=0.51\] Thus, coefficient of kinetic friction between the box and the plank is 0.51