A block of mass m is placed on a rough inclined plane. The corfficient of friction between the block and the plane is `mu` and the inclination of the plane is `theta`.Initially `theta=0` and the block will remain stationary on the plane. Now the inclination `theta` is gradually increased . The block presses theinclined plane with a force `mgcostheta`. So welding strength between the block and inclined is `mumgcostheta`, and the pulling forces is `mgsintheta`. As soon as the pulling force is greater than the welding strength, the welding breaks and the blocks starts sliding, the angle `theta` for which the block start sliding is called angle of repose `(lamda)`. During the contact, two contact forces are acting between the block and the inclined plane. The pressing reaction (Normal reaction) and the shear reaction (frictional force). The net contact force will be resultant of both.
Answer the following questions based on above comprehension:
Q. If `mu=(3)/(4)` then what will be frictional force (shear force) acting between the block and inclined plane when `theta=30^@`:

Options

A. `theta gt tan^(-1)mu`
B. `theta lt tan^(-1)mu`
C. `theta gt cot^(-1)mu`
D. `theta lt cot^(-1)mu`

Answer

A

Explanation

Angle `(theta')` fo repose `:`
`m(g+a)sintheta'=F`
`m(g+a)cos theta'=R`
`:. (F)/(R)=tantheta'`
`theta'=tan^(-1)((F)/(R))=alpha`
Hence angle of repose does not change.
To slide `mg sin theta gt mu mg cos theta`
`sin theta gt mu cos theta`
`theta gt tan ^(-1) mu `