JEE Main & Advanced Physics NLM, Friction, Circular Motion Angle of Repose

Angle of Repose Angle of repose is defined as the angle of the inclined plane with horizontal such that a body placed on it is just begins to slide. By definition a  is called the angle of repose. In limiting condition \[F=mg\,\sin \alpha \]                         and \[R=mg\,\cos \alpha \] So \[\frac{F}{R}=\tan \alpha \] \[\therefore \,\,\,\,\frac{F}{R}=\mu =\tan \theta =\tan \alpha \] [As we know \[\frac{F}{R}=\mu =\tan \theta \]] Thus the coefficient of limiting friction is equal to the tangent of angle of repose. As well as \[\alpha =\theta \] i.e. angle of repose = angle of friction. Sample problems based on angle of friction and angle of repose Problem 7. A body of 5 kg weight kept on a rough inclined plane of angle \[{{30}^{o}}\] starts sliding with a constant velocity. Then the coefficient of friction is (assume \[\operatorname{g} = 10 m/{{s}^{2}}\]) [JIPMER 2002] (a) \[1/\sqrt{3}\] (b) \[2/\sqrt{3}\] (c) \[\sqrt{3}\]                (d) \[2\sqrt{3}\] Solution: (a) Here the given angle is called the angle of repose So, \[\mu =\tan {{30}^{o}}=\frac{1}{\sqrt{3}}\]             Problem 8. The upper half of an inclined plane of inclination q is perfectly smooth while the lower half is rough.  A body starting from the rest at top comes back to rest at the bottom if the coefficient of friction for the lower half is given [Pb PMT 2000] (a) \[\operatorname{m}= sin\,\theta \]      (b) \[\operatorname{m}= cot\,\theta \]      (c) \[\operatorname{m}= 2 cos\,\theta \]                   (d) \[\operatorname{m}= 2 tan\,\theta \]             Solution: (d) For upper half by the equation of motion \[{{v}^{2}}={{u}^{2}}+2as\]             \[{{v}^{2}}={{0}^{2}}+2(g\sin \theta )l/2\]\[=gl\,\sin \theta \] [As \[u=0,\,s=l/2,\,a=g\,\sin \theta ]\] For lower half \[0={{u}^{2}}+2g(\sin \theta -\mu \cos \theta )\] 1/2 [As \[v=0,\,s=l/2,a=g\,(\sin \theta -\mu \cos \theta )\]] \[\Rightarrow \,\,\,0=gl\sin \theta +gl(\sin \theta -\mu \cos \theta )\] [As final velocity of upper half will be equal to the initial velocity of lower half] \[\Rightarrow \,\,2\sin \theta =\mu \cos \theta \Rightarrow \mu =2\tan \theta \]