Question

A cubical block of mass m rests on a rough horizontal surface. $ \mu $ is the coefficient of static friction between the block and the surface. A force mg acting on the cube at an angle $ \theta $ with the vertical side of the cube pulls the block. If the block is to be pulled along the surface, then the value cot ( $ \theta $ /2) is:

• A. less than $ \mu $
• B. greater than $ \mu $
• C. equal to $ \mu $
• D. not dependent on $ \mu $

Answer 1

The Correct Option is B

A force mg acts on a cube of mass m at angle $ \theta $ with the vertical side of cube and pulls the block. The forces acting on the block are: (i) Applied force mg at an angle $ \theta $ with vertical side of cube. (ii) Weight mg of cube vertically downward. (iii) Reaction of surface vertically upward (iv) Friction force $ f $ Resoling the components of mg along horizontal and vertical i.e., $ mg\,\sin \,\theta $ and $ mg\,\cos \,\theta . $ Component $ mg\sin \theta $ moves the block in forward direction for this we have $ mg\sin \theta >f $ ?(i) Also $ mg\cos \theta +R=mg $ or $ R=mg(1-cos\theta ) $ ...(ii) and $ f=\mu R=\mu mg(1-cos\theta ) $ ...(iii) From Eqs. (i) and (iii), we have $ mg\sin \theta >\mu mg(1-\cos \theta ) $ $ \sin \theta >\mu (1-cos\theta ) $ or $ 2\sin \theta /2\cos \theta /2>\mu 2{{\sin }^{2}}\theta /2 $ or $ \frac{\cos \theta /2}{\sin \theta /2}>\mu $ $ \cot \theta /2>\mu $