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

Answer: (B)

greater than $\mu$

Answer 2

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 $\frac{\cos \theta /2}{\sin \theta /2}>\mu$ $\cot \theta /2>\mu$