A block of mass M rests on a rough horizontal surface as shown. Coefficient of friction between the block and the surface is `mu`. A force F=mg acting at angle `theta` with the vertical side of the block pulls it. In which of the following cases the block can be pulled along the surface?

Options

A. `tan theta ge mu`
B. `tan ((theta)/(2) ge mu)`
C. `cos theta ge mu`
D. `cot ((theta)/(2)) ge mu`

Answer

D

Explanation

(d) `N=Mg-F cos theta`
`=Mg-Mg cos theta=2 Mg sin^(2)"" (theta)/(2)`
Further block can be pulled if,
`F sin theta ge mu N cos theta`
`implies " " 2Mg sin""(theta)/(2).cos""( theta)/(2) ge 2 mu Mg sin^(2)"" (theta)/(2) implies cot"" (theta)/(2) ge mu`