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

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1 Answer
Nov 21, 2016

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.