Question

Figure shows a small block A of mass m kept at the left end of a plank B of mass M=2m and length λ. The system can slide on a horizontal road. The system is started towards right with the initial velocity v. The friction coefficients between the road and the plank is 1/2 and that between the plank and the block is 1/4

(b) displacement of block and plank with respect to ground till that moment?

(b) displacement of block and plank with respect to ground till that moment?

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Solution

Ref. image I

Let a1 and a2 be acceleration of mass m and 2m also a1>a2 so mass m moves on 2m.Let after time t mass m is separated from 2m.

using equation of motion. during this time, mass m covers vt+12a1t2 & 5m=vt+12a2t2 for mass m to separate from 2m, we have vt+12a1t2=vt+12a2t2+1

Ref. image II

As 12μ1=14μ2

μ=μ2

from free body diagram we have, ma1+μ2R=0

ma1=−μ2mg

=μ2g

a1=−5μ

again ,

(2m)a2+μ(2m+m)g−(μ2)mg=0

⇒2(2m)a2+2μ(2m+m)g−μmg=0

⇒2(2m)a2=μmg−2μmg−2μmg

a2=−μmg−2μ(2m)g2(2m)

Substituting values of a1 & a2 in eq. (1)

t=4(2m)l(2m+m)μg

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