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Question

In Fig. the co-efficient of friction between the walls of block of mass m and the plank of mass M is μ. The same co-efficient of friction is there between the bank and the horizontal floor. The force F is 100N and the masses m and M of 1 kg and 3 kg respectively. Find the value of μ, if the block does not slip along the wall of the plank.
983206_e442f845221340e593f2196428fad5d0.png

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Solution

Equation of motion for m:
Fx=maN=ma
Fy=0
FBD of m and M,
Since m is at rest relation to M at the verge of slipping.
f=mg
Taking "m and M" together as system, f=μN=μ(m+M)g
a=Fμg(M+m)M+m
ffmax
mgμN
mgμma
gμ.Fμg(M+m)M+m
(M+m)gμFμ2g(M+m)
μ2g(M+m)μF+(M+m)g0
2μ25μ+20
For critical , 2μ25μ+2=0
(μ2)(2μ1)=0
μ =2 or 1/2
Practically μ does not become 2. Therefore, μ=0.5
1034963_983206_ans_e2fbd686453643178e1b5099061404fd.png

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