Question

What is the angular range for `theta`, for which the masses will not move in the following diagram?

Answer 1

First considering the situation when the block A tends to move downward along the inclined plane.In this situation the force of friction as well as weight of block B together will balance the downward force on block A.
Taking acceleration due to gravity `g=9.8ms^-2`

So

`30gxxsintheta-mu_sxx30gxxcostheta=20g`

`=>30sintheta-mu_s30costheta=20`

`=>30sintheta-0.3xx30costheta=20`

`=>10sintheta-3costheta=20/3`

`=>10/sqrt109sintheta-3/sqrt109costheta=20/(3sqrt109)`

Let 10/sqrt109=cosalpha and 3/sqrt109=sinalpha#

This means `tanalpha=0.3=>alpha=tan^-1(0.3)=16.7^@`

The above equation becomes

`=>cosalphasintheta-sinalphacostheta=20/(3sqrt109)`

`=>sin(theta-alpha)=20/(3sqrt109)`

`=>(theta-alpha)=sin^-1(20/(3sqrt109))~~39.68^@`

`theta=39.68+alpha=39.68+16.7=56.38^@`

Again considering the situation when the block A tends to move upward along the inclined plane.In this situation the force of friction as well as downward force on block A together will balance the downward force on block B.

So

`30gxxsintheta+mu_sxx30gxxcostheta=20g`

`=>30sintheta+mu_s30costheta=20`

`=>30sintheta+0.3xx30costheta=20`

Similar manner we get

`=>(theta+alpha)=sin^-1(20/(3sqrt109))~~39.68^@`

Here

`theta=39.68-16.7=22.98^@`

So when `" "22.98^@<=theta<=56.38^@` the combined sytem of blocks will not move.